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Bos +maarten@snap.com +Snap Inc. +Santa Monica, California, USA +ABSTRACT +Culture shapes people’s behavior, both online and offline. Surpris- +ingly, there is sparse research on how cultural context affects net- +work formation and content consumption on social media. We an- +alyzed the friendship networks and dyadic relations between con- +tent producers and consumers across 73 countries through a cul- +tural lens in a closed-network setting. Closed networks allow for +intimate bonds and self-expression, providing a natural setting to +study cultural differences in behavior. We studied three theoreti- +cal frameworks of culture - individualism, relational mobility, and +tightness. We found that friendship networks formed across dif- +ferent cultures differ in egocentricity, meaning the connectedness +between a user’s friends. Individualism, mobility, and looseness +also significantly negatively impact how tie strength affects con- +tent consumption. Our findings show how culture affects social +media behavior, and we outline how researchers can incorporate +this in their work. Our work has implications for content recom- +mendations and can improve content engagement. +CCS CONCEPTS +• Human-centered computing → Social networks; Social me- +dia; Social network analysis. +KEYWORDS +Social media platforms, Cross-cultural analysis, Social ties, User +Behavior Modeling, relationship modeling, tie strength +ACM Reference Format: +Agrima Seth, Jiyin Cao, Xiaolin Shi, Ron Dotsch, Yozen Liu, and Maarten +W. Bos. 2023. Cultural Differences in Friendship Network Behaviors: A +Snapchat Case Study. In Proceedings of the 2023 CHI Conference on Human +Factors in Computing Systems (CHI ’23), April 23–28, 2023, Hamburg, Ger- +many. ACM, New York, NY, USA, 14 pages. https://doi.org/10.1145/3544548.3581074 +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full cita- +tion on the first page. Copyrights for components of this work owned by others than +the author(s) must be honored. Abstracting with credit is permitted. To copy other- +wise, or republish, to post on servers or to redistribute to lists, requires prior specific +permission and/or a fee. Request permissions from permissions@acm.org. +CHI ’23, April 23–28, 2023, Hamburg, Germany +© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM. +ACM ISBN 978-1-4503-9421-5/23/04...$15.00 +https://doi.org/10.1145/3544548.3581074 +1 +INTRODUCTION +In the past two decades, social media platforms have transformed +how individuals build and maintain their relationships. These plat- +forms are increasingly becoming the preferred method for initiat- +ing intimate relationships [53], seeking advice [32], and commu- +nity building [33]. With social media platforms becoming an inte- +gral part of social life for many of us (there are 4.26 billion social +media users as of 2021 [42]), understanding the drivers of user be- +haviors is imperative. +Directly engaging with others (e.g., sending messages) and con- +suming their content (e.g., viewing, replying, and reacting to Sto- +ries and posts) are often studied to understand behavioral patterns +on social media platforms. User behavior on online social media +platforms can be said to be broadly driven by a complex combina- +tion of (a) user identity (personality, demographics), (b) the norms +(descriptive and prescriptive) that the users in a network collec- +tively subscribe to, (c) the relationship between users (friends, ac- +quaintances, strangers), (d) usage intent; for example, professional +(LinkedIn) vs. curated self-presentation (Instagram), and (e) plat- +form affordances. While any particular platform usually provides +the same affordances to all users on that platform, users bring their +different backgrounds, experiences, expectations, beliefs, and val- +ues to the platform. As a result, different behaviors on the same +platform are culturally influenced [2, 7, 11, 36]. +Most studies on social media user behavior are based on data +that is west-centric [39, 54], and thus, their results have an implied +context of western cultural norms. These findings fail to account +for the heterogeneity in user behavior that arises from different +cultural contexts [4, 32]. Hence, to further understand how cul- +tural values affect behavior on these platforms, our work focuses +on how users from different cultural backgrounds interact differ- +ently on a platform. Specifically, we use theoretical frameworks of +cultural values to study the differences in the formation of friend- +ship networks and the moderation of differential behavior of con- +tent consumption within these friendship networks. This paper +uses three theoretical frameworks of cultural values: Hofstede’s +concept of Individualism [20], Thomson and colleagues’ concept +of Relational Mobility [46], and Gelfand and colleagues’ concept +of Tightness [15]. The data we used for our analyses is from the + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Seth, et al. +camera and messaging platform Snapchat. Snapchat is used in al- +most 150 countries and has 347 million daily active users world- +wide [41]. Snapchat is a closed network, meaning that a lot of the +content shared by individuals on Snapchat is only available to a +limited set of trusted users. Past work on eliciting the motivations +for Snapchat usage has shown that Snapchat is used to commu- +nicate with close relationships and is viewed as a platform with +a relatively lower emphasis on self-presentation and impression +management compared to platforms like Instagram [3, 6, 35, 50]. +Because closed networks have less formal pressures and allow for +intimate bonds and self-expression, they provide us with a cleaner +setting to study differences in human behavior. +Specifically, we focus on 1) how culture influences network cre- +ation and 2) how culture influences content consumption behav- +iors embedded in the network. In particular, for the second ques- +tion, we are interested in how culture moderates the effect of tie +strength (i.e., the closeness between individuals) on content con- +sumption. Past work has shown that tie strength strongly predicts +a variety of user behaviors on platform, including what informa- +tion will be exchanged [31, 51, 54], the likelihood to change one’s +actions [8], the attention given to content [52], and the preferred +behavior to signal engagement [5]. We will explore how tie strength +moderates tie strength’s effect on content consumption. To study +content consumption behavior, we use the metric of dwell time, +i.e., the time a user spends consuming content that another user +creates. +In sum, we ask the following research questions: +(1) How do friendship networks differ in countries with differ- +ent cultural values? +(2) How do cultural values change the effect of tie strength on +dwell time? +To answer our first research question, we studied the network +properties of friendship networks across 73 countries, which have +been surveyed by either Hofstede [20], Thomson et al. [46], or +Gelfand et al. [15], and have different cultural values that lie on a +continuum of the three cultural values of individualism, mobility, +and tightness. We analyzed how friendship network size and ego- +centricity — the extent to which a person’s friends are connected +with each other — vary across cultures in the closed network set- +ting of Snapchat. We find that users from more individualistic, mo- +bile, and loose cultures have a more extensive friendship network +and are less egocentric. Next, we analyzed within these networks +how tie-strength between users impacts engagement with content +(dwell time) and the role of cultural values as a moderator. We +found that individualism, mobility, and looseness negatively mod- +erate the effect of tie strength on content consumption. +Where previous work on culture and social media platforms has +primarily been limited to a small sample size [2, 39], this paper +contributes by studying cultural differences in user behavior on a +large scale, analyzing hundreds of thousands of users across many +countries. Further, where other quantitative works are usually lim- +ited to open or broadcast networks, this study explores relatively +under-studied closed-network settings [23]. +From an HCI and design perspective, our work can advance +our understanding of behavior patterns across cultures. We dis- +cuss the implications of understanding users’ engagement with +content to design better experiences for the user. When applied to +platform design, our work would help user-retention of platforms +without compromising the user experience, in turn creating better +outcomes for both users and platforms. Our work furthers the re- +search that helps answer the question: What does it mean to under- +stand and support users from diverse cultures on online platforms? +[14]. Most of the designs and practices of online platforms have a +‘one-size-fits-all’ approach and do not actively account for different +user preferences across geographies. Our results provide evidence +of differential behavioral patterns in online friendship networks +across cultures and suggest how algorithm design can be cultur- +ally inclusive. +1.1 +Privacy and Ethics +The data for this study was taken from Snapchat, and the study was +conducted within Snapchat in accordance with Snapchat’s policies +and procedures with respect to Snapchat data. This analysis only +uses the metadata of the user behavior. It does not analyze the ac- +tual content of the communication between the users. +2 +RELATED WORK +2.1 +Ties and user behavior +Interpersonal relationships make social media platforms social. Like +in offline social networks, an individual’s online network consists +of individuals, with each of whom one shares a different type of re- +lationship. Each dyadic relationship is different based on the close- +ness and the purpose they serve to the individual. Social network +analysis literature uses the term tie strength to differentiate be- +tween relations of different closeness. This term was coined by +Granovetter[17], who analyzed the role of different ties in differ- +ent situations. The two types of ties characterized were strong and +weak. The four dimensions determining a tie’s strength were: the +amount of time spent on a tie, the intimacy, the intensity, and re- +ciprocal services [17]. Although researchers have used different +operationalizations to conceptualize tie strength depending on the +purpose of the study, many works on social media platforms op- +erationalize tie strength as proportional to the total number of ex- +changes in the dyad. This operationalization of tie strength has +been used to study various phenomena, like promoting mental +well-being [25], increased diffusion of information, and access to +novel information [17, 49]. While these works analyze the role of +tie strength in reaping social benefits, studies have also focused on +justifying Granovetter’s hypothesis that the two ties elicit differ- +ent interaction patterns, for which they analyze how information +from different ties is received [19, 23, 24, 52]. These studies find +evidence that individuals spend more time on the content received +from stronger ties. +2.2 +Cultural values +One primary aspect of culture is that it’s the normative value sys- +tem that dictates acceptable practices and helps differentiate one +group from another. Culture is both a result of the accepted past +actions and the determinant of acceptable future actions. One of +the ways to reason about attitudes and actions is to understand +the culture people are in. Prior studies have shown that an indi- +vidual’s behavior in the online space is influenced by their culture + +Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study +CHI ’23, April 23–28, 2023, Hamburg, Germany +in the same way as offline behaviors. With cultural values shaping +actions, we must first understand how culture can be measured +and then how culture affects behavior. While prior work usually +focuses on groups and their specialized culture, we introduce liter- +ature from cultural psychology in our work. Culture is often opera- +tionalized through dimensions where a dimension is defined as “an +aspect of a culture that can be measured relative to other cultures.” +In this paper, we bring in concepts from three dominant cultural +psychology theories, namely, individualism-collectivism [20], re- +lational mobility[46], and tightness-looseness[15], to explore how +culture impacts network creation and content consumption behav- +iors within a network. Below, we briefly introduce each of the cul- +tural dimensions. +2.2.1 +Hofstede’s Individualism-collectivism. Hofstede [20] analyzed +data from over 50 countries and identified six critical dimensions of +national culture. Individualism-collectivism is one dimension that +has drawn the most research attention. Typically, individualism +leads to loose ties among the individuals of a society. Individual- +ists focus on "I" as opposed to "we." Because groups are less im- +portant to them, individualists also tend to show no difference in +their behaviors and attitudes toward ingroups versus outgroups. In +contrast, collectivism leads to a collective identity, and the welfare +of an individual is implicitly assumed to be linked to the interests +of the larger group. Hence, collectivists focus on "we." Because of +their particular focus on "we," collectivists are known to have differ- +ent norms and behaviors towards ingroups versus outgroups and +place greater emphasis on harmony. +Because of the "I" nature, individualists need to constantly reach +out to build networks and also tend to see relationships as fluid. +In contrast, collectivists see relationships as given, and thus, they +are less active in building networks. As a result, we predict that +individualism will be positively correlatedwith friendship network +size. +Individualists are less likely to treat other people based on re- +lationship strength and group membership, whereas collectivists +tend to have a strong tendency to favor ingroup members and peo- +ple they are close to. This should also be manifested in how tie +strength drives content engagement behavior in different cultures. +As a result, we predict that individualism will negatively moderate +the positive effect of tie strength on content engagement, such that +the effect of tie strength on content engagement will be weaker for +individualists than for collectivists. +Hence, we hypothesize that: +H1a: The friendship network for individualisticcultures is larger than +the friendship network for collectivistic cultures +H1b: Individualism negatively moderates the effect of tie strength on +content engagement. +2.2.2 +Relational Mobility. Thomson et al. [46] conducted a survey +across 39 countries using a set of 12 questions to construct their +dimension of culture. Relational mobility indicates the degree of +freedom and opportunities the members of a culture have to form +and terminate relationships. The two opposing poles on this in- +dex are high and low relational mobility. For example, relational +mobility is high in North America and low in Japan. Because re- +lationships in high-mobility cultures are less stable and easier to +change than those in low-mobility cultures, they are more fragile. +It also requires more effort to maintain committed relationships. +Prior work has shown that cultures with higher relational mobil- +ity tend to share more about themselves (self-disclosure), are more +active in giving support, and tend to have more trust in the mem- +bers of the society [46, 55]. Because cultures high in mobility have +more opportunities to form relationships, it allows individuals to +have a larger network. In a similar vein, because in high mobil- +ity cultures, individuals see relationships as more fragile and fluid, +they are less likely to adjust their interpersonal behaviors based on +tie strength. As such, we predict that relational mobility will neg- +atively moderate the effect of tie strength on content engagement, +such that the effect of tie strength on content engagement will be +weaker in high-mobility cultures than in low-mobility cultures. +Hence, we hypothesize that: +H2a: The friendship network for high mobility cultures is larger than +the friendship network of low mobility cultures +H2b: Relational mobilitynegatively moderates the effect of tie strength +on content engagement. +2.2.3 +Tightness. Gelfand et al. [15] conducted a survey across 33 +countries using 12 behaviors across 15 situations to construct their +dimension. Tightness-looseness is about the extent to which a so- +ciety tolerates norm-deviant behaviors. The two opposing poles +on this index are tight and loose. For example, Looseness is high +in North America and low in Japan. Tight cultures have stronger +norms and are less tolerant of behavior that deviates from the norm. +In contrast, loose cultures have relatively weaker norms and are +more tolerant of behavior that deviates from the norm. As such, +we predict that tightness should be negatively correlated with net- +work size because a tight culture makes it hard for people to bring +new members to a social network. Cultural tightness is often con- +sidered a selection criterion to test whether a new member can fit +in. In contrast, the level of scrutiny will be much lower in a loose +culture, making it easier for an individual to expand their network. +Similarly, we predict that tie strength’s effect on content engage- +ment will be weaker in loose cultures than in tight cultures. In a +loose culture, tie strength is less likely to be seen as a criterion +that individuals rely upon to decide how they approach a person. +In contrast, in a tight culture, tie strength is a monitoring mech- +anism that powerfully regulates people. As a result, people draw +more influence from tie strength, including content engagement +behavior. +Hence, we hypothesize that: +H3a: The friendship networks for tighter cultures are smaller than +friendship networks of looser cultures +H3b: Tightness positively moderates the effect of tie strength on con- +tent engagement. +Although the three cultural dimensions originated from differ- +ent theories, they are often conceptually related. Prior work has +shown that individualism, relational mobility, and looseness are +often moderately correlated (Thomson et al., 2018, Appendix Ta- +ble S8, p. 51 [46]). For example, the U.S. is a culture that is in- +dividualistic, high mobility, and loose at the same time, whereas +Japan is a culture that is collectivistic, low mobility, and tight. How- +ever, while Germany ranks higher in individualism and mobility, it + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Seth, et al. +ranks lower in looseness, whereas Brazil, though less individualis- +tic, is more mobile and loose. Thus, while the three theories are +conceptually related and can serve as a robustness check for one +another, they each touch upon a unique cultural aspect. When re- +searchers study the effect of one of the cultural values on individu- +als, they also tend to include the other two as a way of robustness +check [44, 46]. As a result, although the three dimensions are from +different theories, we see them as a whole package. +In sum, culture provides an important context about the shared +common knowledge to its members on how to behave in a given +context and how others will interpret their behavior. Comparative +work on interpersonal relationships across cultures has shown that +the same relationships elicit different behaviors in different cul- +tures, implying that the same relationships across cultures are not +similarly perceived [13, 16, 18, 30, 37, 47]. Our work aims to ana- +lyze if user behavior on the same online platform provides empiri- +cal evidence that the impact of tie strength on their behavior varies +across cultures. +3 +DATA +We conduct our study on the Snapchat platform. Snapchat is an +online messaging platform where content shared between users +is ephemeral. Like most platforms, Snapchat allows users to ex- +change content in the form of text, images, and videos. The inter- +actions between users can be one-to-one, one-to-group, or one-to- +all friends (a broadcast interaction). Interactions are identified by +different names and are introduced below: +• Snaps: A direct or personal interaction of image or video +content type between users, which may be one-to-one or +one-to-group. Depending on the receiver’s chosen settings, +Snaps disappear immediately after viewing or 24 hours later. +In our analysis, we only consider Snaps that are exchanged +between dyads (just two users), which are termed ‘direct +Snaps.’ We do not analyze Snaps sent to groups. +• Chats: A text message between users. Akin to Snaps, de- +pending on the receiver’s chosen settings, chats disappear +immediately after viewing or 24 hours later. In our analy- +sis, we only consider the chats that are exchanged between +dyads (just two users), which are termed ’direct chats.’ We +do not analyze group chats. +• Stories: A broadcast interaction (with all of one’s friends) +having an image orvideo as the content type. Users on Snapchat +(posters) can create Stories for their friends (viewers) to con- +sume. Stories constitute a pull communication wherein friends +decide to either engage with a Story in part or whole or ig- +nore it. Unlike Snaps and chats that disappear after watch- +ing, Stories are available for 24 hrs after posting and can be +viewed multiple times. +We analyze users on Snapchat who share a friend connection. +Friendships on Snapchat are bidirectional and are unlike the ‘fol- +low model’ that platforms like Instagram and Twitter allow (i.e., +both individuals need to add each other as friends in Snapchat). +For each of the 73 countries (Refer appendix D), we randomly sam- +pled 10,000 unique users (egos), their associated Story viewing ac- +tivity for one month, and their complete one-hop friend network. +Though users may have friends across geographies, we filtered the +data only to include those friend pairs where both friends resided +in the same country. Aggregated over all 73 countries, cross-country +friendships accounted for 21.8% of the data. The filtering resulted +in a total dataset of approx 600,000 users per country. Each user +can view Stories from multiple friends, with each of whom they +share a different level of closeness. This results in a data set of +unique dyadic relations between a Story viewer and a Story poster. +For each dyadic interaction, we calculate aggregated statistics of +the total time spent by a viewer on each of the poster’s Stories, the +total number of Stories shared by a poster, and the total number of +Snaps and chats exchanged between the two in the dyadic commu- +nication. To avoid noise from users who rarely engage with each +other, we only keep those dyadic pairs where at least one direct +chat or Snap has been exchanged by both the Story poster and the +viewer during the one month we analyzed. To control for effects +unrelated to the cultural values but caused by the economic devel- +opment and platform reach in a country, we include each country’s +GDP [22], which is a measure of a country’s economic standing, +GINI [45], which is a measure of economic inequality within a na- +tion, and Snap’s market penetration 1, which measures the user- +base of Snapchat for a country. Section 4 details the process used +to answer each research question. The three cross-cultural theo- +ries that inform our study did not survey all the same countries. +Thus, while the three theories do not have a perfect overlap with +each other (Refer appendix D), using all three allows us to cover +73 unique countries. +4 +METHOD +We use the observational data from Section 3 and create statistical +models to understand the role of culture on users’ network forma- +tion and content engagement (dwell time). Building on and align- +ing with prior cross-cultural work, we consider a country a repre- +sentative unit of one culture [15, 20, 46] and analyze the users at +the group level of a country. +4.1 +RQ1: How do friendship networks differ in +countries with different cultural values? +We first measured each country’s average friendship network size +to determine whether people from different cultures have differ- +ent friendship networks. For this, we calculated the total number +of friends per user in each country and averaged it over the total +number of users in the country. +Next, for each country under study, we reconstruct the ego net- +work (egonet) for that country’s randomly sampled 10,000 users. +An ego network consists of the user (the ego), the user’s friends +(the alters), and the friendship relations between the alters. The +egonets formed were independent, i.e., the users’ egonets did not +overlap. We filter out networks that consist of only two nodes +(users who are only connected to the default Snapbot and do not +have other friends on the platform) or star graphs (a pattern where +a user is connected to other users, but none of those other users are +connected, which is a pattern mainly shown by bots [38, 48]). Since +all friendships on Snapchat are bidirectional, we convert the graph +to a simple graph by removing the multiple edges (edges that are +incident on the same pair of nodes). For each of the egonets, we +1Internal Snap INC. marketing data + +Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study +CHI ’23, April 23–28, 2023, Hamburg, Germany +calculate measures of egocentricity - the density, transitivity, and +the betweenness centrality of the ego using the igraph package in +R [12]. +Ego betweenness measures the percentage of shortest paths be- +tween two alters. In a social network setting, it allows us to mea- +sure the importance of the ego node. The higher the betweenness +centrality, the more the ego node is the binding factor between its +friends. Since centrality is sensitive to network size, we normalized +it by the maximum possible betweenness of the ego node. This ap- +proach is in line with prior work on measuring betweenness in +egonets Na et al.,[28]. +Betweenness centrality of node i = +� +푖≠푗≠푘 +푔푗푘 (푖) +푔푗푘 +Where 푔푗푘 is the number of shortest paths that connect node j and +node k, 푔푗푘 (푖) is the number of these shortest paths that include +node i. +Network density is the ratio of the edges in the user’s network +to the edges of the same user’s hypothetical network where every +node is connected to every other node. Likewise, transitivity is the +number of triads relative to the number of possible triads. In our +setting, density and transitivity measure the tendency of the users +to cluster or connect. The higher the density and transitivity, the +more the tendency of the group to cluster. +Density for an undirected graph = +� +푗≠푘 푧푗푘 +푛∗(푛−1) +2 +Where n is the number of nodes in a network, and 푧푗푘 is equal to +1 if the alters j and k are connected. +Transitivity for an undirected graph = +3 ∗ +number of triangles in the network +number of connected triples of nodes in the network +A high density and transitivity are indicative of people connect- +ing with friends of friends; a low betweenness, on the other hand, +implies a reduced tendency of nodes to cluster together. Prior work +by Na et al. [28] on self-reported Facebook networks in East Asia +and the USA found that users from the USA were more egocen- +tric than users from East Asia (had higher Ego Betweenness and +lower Density and Transitivity). We use the same methodology — +to analyze data across more countries — to explore whether these +findings generalize across platforms and for data that is not self- +reported but an individual’s actual network data from a social me- +dia platform. To maintain consistency with Na et al.,[28], we log- +transform density and transitivity and then inverse the transforma- +tion by multiplying minus one; we transform betweenness using +푙표푔(1 + 푀푎푥(푥) − 푥) and then inverse the transformation by mul- +tiplying minus one. +4.2 +RQ2: How do cultural values change the +effect of tie strength on dwell time? +Online social media platforms continually aim to remove obsta- +cles for content creation and consumption; this has allowed for +a myriad of content to be available for consumption by users on +all platforms. With the multitude of content available, attention +from one’s social network has become a valuable and competitive +resource. Here, we analyze how users allocate their attention to so- +cial connections with varying degrees of closeness and how this al- +location is moderated by culture. We study attention in the context +of Stories posted by friends in one’s network. We examine whether +tie strength predicts one’s dwell time on a Story and whether cul- +ture moderates the relationship. +4.2.1 +Measuring interest. Attention to a poster’s Story is a proxy +for the interest in the information shared by the user. Attention +towards a friend who posts Stories (p) is measured by the total +time they spend on viewing their Story; longer attention (dwell +time) for a Story indicates a stronger interest towards that friend. +To measure total time spent on content consumption (TC), we refer +to the formulation proposed in prior works on measuring content +dwell time [23]. +푇퐶(푣,푝) = +� +푠∈푆푝→푣 +훿(푠) +where푆푝→푣 denotes the set of Stories postedby p and consumed by +v, s denotes (without loss of generality) one such Story sample, and +훿(푠) indicates the time spent by v in viewing the Story. This mea- +sures the relative difference in the viewer’s interest across different +posters. However, as pointed out in prior literature, a viewer’s total +view time on a poster’sStory can be skewed by the frequency of the +posting activity of the Story creator, i.e., given the equal likelihood +to consume Stories from different poster’s푇퐶(푣, 푝1) > 푇퐶(푣, 푝2) if +|푆푝1→푣| > |푆푝2→푣|. Hence, we model dwell time towards a sender +s as the average time spent by a viewer on the sender’s Stories. +퐷푇 (푣, 푝) = +� +푠∈푆푝→푣 훿(푠) +|푆푝→푣| +Dwell time is measured in seconds. While Stories vary in dura- +tion and can, in turn, influence dwell times, our initial analysis +of viewing time distribution showed that most viewing activities +were short and independent of content duration. This finding is in +line with prior works on dwell time in closed network settings [23] +- thus, we do not control for this variable. +4.2.2 +Measuring social tie strength between two users. Tie strength +between two users is a complex concept, subject to user percep- +tions and emotions; hence a direct quantitative measure of tie strength +between users is challenging. However, measuring the activity of +direct conversations between two users on social media platforms +has proven to be an effective proxy in estimating tie strength: the +higher the number of dyadic message exchanges, the closer the +two users are. Some users send burst messages while others send +fewer but longer messages; thus, we model tie strength (TS) as the +total number of direct Snaps and chats exchanged between a pair +of users. +푇푆(푣, 푝) = |퐷퐶푝→푣| + |퐷퐶푣→푝| + |퐷푆푝→푣| + |퐷푆푣→푝| +where 퐷퐶푝→푣 denotes the set of direct chats sent by the Story +poster to the Story viewer, 퐷퐶푣→푝 denotes the set of direct chats +sent by the Story viewer to the Story poster, 퐷푆푝→푣 denotes the set +of direct Snaps sent by the Story poster to the viewer, and 퐷푆푣→푝 +denotes the set of direct Snaps sent by Story viewer to the poster. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Seth, et al. +Preliminary analysis of tie strength in each country showed varia- +tion; hence, we standardize tie strengths within each country and +use the standardized version for analysis. +4.2.3 +Measuring culture of each user. We use the results from Hof- +stede’s Individualism [20], Thomson et al.’s Relational Mobility [46], +and Gelfand et al.’s Tightness [15] dimensions, discussed in Sec- +tion 2 as the measure of cultural values (CV) for the country that +an individual belongs to. These measures have been widely used +in the literature. Hofstede’s work has attracted over 45,000 cita- +tions, Thomson et al.’s (more recent) work has already been cited +178 times, and Gelfand et al.’s work has more than 2000 citations. +Since each value system is on a different scale (Appendix D)— In- +dividualism ranges from 6 to 91, Relational Mobility ranges from +3.886 to 4.607, and Tightness ranges from 1.6 to 12.3 — we indepen- +dently standardize each value system across countries and use the +standardized version for analysis. +4.2.4 +Mixed effects model to analyze dwell time as a function of tie +strength and cultural values . We used a linear mixed-effects model +to address the research question of how cultural values moderate +the impact of tie strength on the time spent consuming content +(dwell time) in closed network settings. Since the sets of countries +surveyed by Hofstede [20], Thomson et al. [46], and Gelfand et al. +[15] do not have perfect overlap, we created three multilevel mod- +els to understand how cultural values moderate the effect of tie +strength on Story dwell time. The models included terms for tie +strength (dyad level), cultural value (country level), and their in- +teraction as fixed effects, with random intercepts for country and +viewer, and the number of friends, the GDP, GINI, and Snap’s mar- +ket penetration (MP) for a country as control variables. We stan- +dardized each value system across countries and used the standard- +ized version for analysis. Since we have multiple observations per +country and a viewer views multiple posters, we include the ran- +dom effects due to the country and the viewer. +퐷푇 (푣, 푝) = 푇푆(푣, 푝) 푋 퐶푉 (푣) + |푣푓 | + 퐺퐷푃 + 퐺퐼푁퐼 + 푀푃+ +(1|푐표푢푛푡푟푦) + (1|푉푖푒푤푒푟) +where |푣푓 | refers to the number of friends a viewer has, 푇푆(푣, 푝) is +the tie strength between a pair of viewers and a poster, and 퐶푉 (푣) +is the cultural value of the viewer, which is the same as the cultural +value of the poster. +Since each dyad contains the dwell time of multiple Stories, we +model random effects for the dyad. However, users in a dyad can +have two roles: sometimes a user is a viewer, and sometimes a +poster. A user who is a viewer (v) for a poster p can be a poster (푝′) +for some other node (푣′). This directionality complicates modeling. +To simplify, we randomly regard one person as the viewer and the +other as a poster, disregarding the Stories of that dyad where the +viewer posted and the poster viewed. To ensure that the results 5 +are robust against role assignment, we bootstrapped the analysis; +on each run, for each dyad, viewer and poster roles were randomly +assigned before fitting the model. The bootstrapped results are in +Appendix B. +5 +RESULTS +5.1 +RQ1: How do friendship networks differ in +countries with different cultural values? +We report zero-order Pearson correlations between cultural values +and friendship network size in Table 1. We find that countries that +rank higher in individualism, mobility, and looseness tend to have +a bigger friendship network than collectivistic, less mobile, and +tighter countries. This means that people in the higher ranking +countries are connected to more friends on Snapchat, supporting +H1a, H2a, and H3a. To check for robustness, we ran the same analy- +ses with GDP, GINI, and Snapchat’s market penetration as control +variables. The addition of control variables reduced the sample size +of countries, but the results corroborate those reported here A. +Next, the structural analysis of the ego networks of users from +different cultures (Table 3) shows that the ego centrality of user +networks on Snapchat varies with cultural values. Akin to Na et +al.,[28], we find that the individual structural measures, namely +density, transitivity, and betweenness, are highly correlated (Ta- +ble 2), and thus we average the standardized values and report the +results for this averaged index of ego-centrality. The results show +that mobility and individualism are negatively correlated with ego- +centricity, and tightness is positively correlated with egocentrality. +This means that in countries that rank higher on mobility and indi- +vidualism, people’s friends on Snapchat are more likely to be con- +nected to each other, and in countries that rank higher on tightness, +people’s friends on Snapchat are less likely to be connected to each +other. +Table 1: Pearson correlation between cultural values and +friendship network size (∗푝 < 0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001) +Cultural Value +Correlation +Number of countries +Individualism +0.68** +65 +Relational Mobility +0.31* +37 +Tightness +-0.37* +30 +Table 2: Pearson correlation between network structural +measures for data across different cultural values after con- +trolling for GDP, GINI, and market penetration (∗푝 < 0.05, ∗∗ +푝 < 0.01, ∗ ∗ ∗푝 < 0.001) +Cultural +Value +Betweenness +and +Transitivity +Betweenness +and Density +Density +and +Transitivity +Individualism +0.74*** +0.504*** +0.92 *** +Mobility +0.76 *** +0.49** +0.85*** +Tightness +0.82*** +0.45* +0.83 *** + +Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study +CHI ’23, April 23–28, 2023, Hamburg, Germany +Table 3: Pearson correlation between cultural values and +egocentrality(∗푝 < 0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001) +Cultural Value +averaged index of ego-centrality +Individualism +-0.07 *** +Relational Mobility +-0.04*** +Tightness +0.06*** +5.2 +RQ2: How do cultural values change the +effect of tie strength on dwell time? +Given that the friendship network structures are different across +cultures, using multilevel modeling, we analyzed how cultural val- +ues moderate the effect of tie strength on the viewer’s dwell time +(Tables 4, 5, 6). We see that an increase in the strength of ties in- +creases the dwell time, a result in line with prior works [23, 52]. +Having more friends reduces a viewer’s dwell time on content, +which is likely because an increase in the number of friends leads +to more potential Story content to consume. Though the cultural +values do not have a significant main effect, they significantly mod- +erate the effect of tie strength on dwell time across all three cultural +values. We find that tie strength negatively moderates the effect of +tie strength for more individualistic, mobile, and looser cultures. +Thus confirming H1b, H2b, and H3b. The bootstrap results from +100 runs corroborate the findings reported here in Appendix B. +Our work focuses on understanding (and not predicting) within- +dyad level dwell time from theories of country-level cultural val- +ues, which may not fully account for a lot of individual-level vari- +ation. However, a significant moderation effect allows us to argue +for a substantiative effect of cultural values on individual-level be- +havior [27]. Using only the intersection of countries present across +all three measures of culture, we check for robustness of these re- +sults (Appendix C), and the results corroborate the results reported +in Tables 4, 5, 6. Because the effects we found are on the smaller +side, there is still a lot of unexplained variance, and we can not fully +account for all individual-level and item (Story) level variation. +Table 4: Coefficients from Multilevel Modeling for the ef- +fect of Individualism as a moderator on Dwell Time (∗푝 < +0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001), Sample size: country = 47, +dyads = 460000, RMSE = 4.9, AIC = 2793115, BIC = 279226, R2 +conditional = 0.04, R2 marginal = 0.01 +Fixed Effects +Estimate +Standard Error +Intercept +3.741*** +0.078 +Strength of Ties +0.092*** +0.007 +Individualism +0.035 +0.074 +Strength of Ties : Individualism +-0.014*** +0.007 +Control variables +Number of Friends +-0.338*** +0.008 +GDP +-0.036 +0.068 +GINI +-0.040 +0.060 +Market Penetration +0.065* +0.059 +Table 5: Coefficients From Multilevel Modeling for the effect +of Mobility as a moderator on Dwell Time (∗푝 < 0.05, ∗ ∗ 푝 < +0.01, ∗∗∗푝 < 0.001) Sample size: country = 26, dyads = 128800, +RMSE= 3.12, AIC = 1438399, BIC = 1438504, R2 conditional = +0.27, R2 marginal = 0.01 +Fixed Effects +Estimate +Standard Error +Intercept +3.835 *** +0.097 +Strength of Ties +0.116*** +0.008 +High Mobility +0.092 +0.071 +Strength of Ties : High Mobility +-0.012* +0.006 +Control variables +Number of Friends +-0.35*** +0.011 +GDP +-0.051 +0.108 +GINI +-0.02 +0.102 +Market Penetration +0.108 +0.091 +Table 6: Coefficients From Multilevel Modeling for the effect +of Tightness as a moderator on Dwell Time (∗푝 < 0.05, ∗∗푝 < +0.01, ∗∗∗푝 < 0.001), Sample size: country = 25, dyads = 100000, +RMSE=2.19, AIC = 731754.3, BIC = 731850.8, R2 conditional += 0.80, R2 marginal = 0.01 +Fixed Effects +Estimate +Standard Error +Intercept +3.725*** +0.1 +Strength of Ties +0.129*** +0.010 +Tightness +-0.060 +-0.082 +Strength of Ties : Tightness +0.058*** +0.010 +Control variables +Number of Friends +-0.283 *** +0.011 +GDP +-0.154* +0.077 +GINI +-0.171 +0.069 +Market Penetration +0.179* +0.077 +6 +DISCUSSION +Most social media platforms were introduced in the Global North +before they started gaining a user base in other countries. As a +result, studies on understanding users on social media platforms +primarily draw from west-centric populations, which leads to un- +intended biases. Using data from 10,000 users per country from +nearly 73 countries, our work studied how individuals across cul- +tures differ in their behavior on the same platform. We control +for confounders like the platform’s market penetration, countries’ +GDP, and GINI score, which may have influenced the platform’s +user base size and composition. Our main findings are: +Structure of friendship network. The analysis of the egocentrality of +the friendship networks showed that individualistic, more mobile, +and looser cultures are negatively correlated with egocentrality. +This result is unlike the prior survey-based network analysis by Na +et al. [28], which found that individualism is positively correlated +with ego centrality. Na et al. [28] recruited individuals through a +call for survey participants on the Facebook platform, which re- +sulted in a substantially varied number of respondents from each + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Seth, et al. +country and thus could be sensitive to selection and conformity +bias. In our study, we randomly sampled users and analyzed the +metadata of the user behavior, which provides a relatively cleaner +signal for a user’s choices. Apart from a more balanced number of +users from different countries, we also analyzed data from a sub- +stantially higher number of countries. Apart from data collection +and sample size differences, another potential source for the dif- +ferences in findings could arise from who is befriended on these +platforms. +Adams and Plaut posited that friendship’s meaning varies sub- +stantially across cultures [1]. Markus and Kitayama [26] argued +that familial ties form an important part of a user’s social network +in collectivist cultures compared to individualistic cultures. With +the demographics on Snapchat skewing towards a younger popu- +lation [9, 10] and motivations differing from Facebook [3, 34, 50], it +is plausible that (a) the ’younger users’ do not ’friend’ familial ties +due to the difference in how they make sense of ’friendship’ and +whom they ’friend,’ and (b) the ’elder’ familial members are ab- +sent from the platform. Since family ties form an important part of +collectivist cultures, not including them on their Snapchat friend- +ship network could be the reason for differences in our findings +when compared to Na et al. [28]. While our results differ from Na +et al., [28], they agree with the findings from Igarashi et al.[21] +that user’s from collectivist cultures had more egocentric networks. +Given that very few studies have explored how culture affects net- +work structures, future work in this domain will help establish +a stronger understanding of how culture influences the network +structures formed on social media platforms. +Our findings bear important implications for future work that +aims to study user interaction patterns on a platform. Firstly, stud- +ies should elicit and validate the network structure formed for their +population of interest because the network structures vary across +subpopulations on the same platform and across platforms, and +relying on metrics from prior work with a mismatched popula- +tion might lead to incorrect inferences. Next, the differences in +friendship networks bear importance for context-aware friendship +recommendation engines, which we discuss under design implica- +tions. +Cultural Values and user behavior. Culture is a complex societal- +level phenomenon that guides individual behavior. Various studies +have tried to study culture through a system of ’cultural values.’ In +this project, we chose three dominant theories in cultural psychol- +ogy, ranging from Hofstede’s dimensions published in 2001 [20] to +more recent theories on Tightness and Mobility published in 2011 +and 2018 [15, 46], respectively. Consistent with our hypothesis, we +found that each cultural value (i.e., individualism, looseness, mobil- +ity) significantly moderates the effect of tie strength on dwell time, +highlighting the significance of considering culture in understand- +ing behavior patterns on social media. In addition, we found that +individualism, looseness, and mobility moderates the relationship +between tie strength and dwell time in the same direction. Theo- +retically, it is logical because in societies where people have more +freedom to make friends and move between different circles (i.e., +high relational mobility), a looser norm (i.e., looseness) is likely to +develop, and a comparatively more self-focused mindset (i.e., indi- +vidualism) is likely to rise. Indeed, prior work has also predicted +that these three variables would have a similar impact on individ- +ual cognition and behavior [46]. Thus, we extend the prior work in +cultural psychology by adopting a cultural lens in understanding +user behaviors on social media. +6.1 +Design Implications +The diversity of content on platforms has made good recommen- +dation systems a necessity. While these recommendation systems +are becoming increasingly personalized, they fail to distinguish the +varied meanings that different types of social ties have for users +from different cultures. For example, if we consider the dyadic pair +of user A, their strongest tie, and user B, their strongest tie, such +that user A and B belong to different cultures, the influence of the +respective strongest tie may be different. Our study, through evi- +dence, argues for treating users and their friendship relations from +different cultures differently when designing recommendation sys- +tems. Analyzing users at a cultural level may reduce the complexity +of recommendation systems and make the recommendation sys- +tem more culturally sensitive. By doing so, they may be able to +better rank the content the user is more likely to engage with at +a reduced cost. For example, our result suggests that when design- +ing recommendation systems, tie strength should be given greater +weight for users in less mobile, tighter, and collectivistic countries +because our results show that tie strength is more strongly corre- +lated to content dwell time in these countries. +Friendship recommendation engines that are unaware of ’how’ +and ’why’ network structures differ across cultures run the risk of +treating friending activities across different cultures as the same, +resulting in a suboptimal platform experience. For instance, the +motivations of individuals from tight cultures could differ from +those from loose cultures, i.e., in contrast to individuals from loose +cultures, individuals in tight cultures might feel forced to friend +not only those whom they want to but also those whom they have +to - say befriending familial ties. A recommendation engine that +captures behavior from loose cultures might not be able to recom- +mend users with whom one shares common friends. Similarly, a +recommendation engine that focuses on tight cultures would ex- +plore less and over-recommend users with whom one shares com- +mon friends. Hence, using the behavioral understanding from only +either of the cultures risks the failure of the algorithms ( and, in +turn, platform experience) in the other cultures. Thus, while our +work takes a step in highlighting ’how’ the network structures +differ, future work that provides insights into ’why’ the network +structures differ can further enrich the understanding of design- +ing friendship recommendation algorithms. +7 +LIMITATIONS +Our study is subject to a few important limitations. First, our work +uses data from Snapchat, which encompasses a significant but lim- +ited amount of people’s online communications. We could only +use available data for our study, and some of Snapchat’s user data +is only available for a limited time. Additionally, the actual con- +tent of Snapchat communications is not available for analysis. The +Snapchat user group skews young [10], and studies have found +that younger people have shifted away from traditional values [29, +43]. Second, recommendation algorithms play an important role + +Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study +CHI ’23, April 23–28, 2023, Hamburg, Germany +in network formation on the platform. We did not have access to +the friend recommendation algorithm for this study, and we could, +therefore, not control for any potential confounding effects. Get- +ting an insight into the algorithm and its impact on users across +geographies could further enrich future work. Further, the focus of +this study was to understand the friendship network and behavior +on the online social network, which may differ from an individ- +ual’s offline friendship networks and their interactions on these +networks. Next, not every country has been equally surveyed in +prior research on cultural values. There is a non-perfect overlap be- +tween countries that have been studied for mobility and countries +that have been studied for tightness. Once data from more coun- +tries becomes available, our analyses could be extended to include +those countries. Future work can further build on ours by analyz- +ing how content type interacts with cultural values and impacts +dwell time. By using a large random sample of users across coun- +tries, country-level measures of economic growth, and inequity, +we tried to limit selection bias and account for variations across +countries. GDP and GINI measures help us control for country- +level socioeconomic status. However, it is plausible that a given +stratum of society is overrepresented on the platform, and country- +level socioeconomic measures might not fully control for the plat- +form user’s socioeconomic status. The lack of finer-grained mea- +sures could be a limitation of the study. +Human behavior is complex and subject to factors that have +individual-level variation. Hence, it is difficult to fully predict hu- +man behavior in the social sciences. The focus of our work was to +test the theory of the effect of culture, as measured at the country +level, on individual behavior. Like prior works, we can not fully +account for all individual-level and item (Story) level variation. As +brought out in the Introduction 1, individual behavior is affected +by a host of other variables, and content engagement is no differ- +ent. For example, the Story’s content might be an important fac- +tor; however, we could not study this due to Snap Inc.’s policies +on not retaining information about the content. Future studies can +help make the model more complete by operationalizing the type +of content and other variables that might affect the dwell time on +content. While the cultural theories used in this study span a large +geographic region, the identities of the researchers who created +these measures could be a source of bias for these measures. As +argued by Shweder [40] (p. 409), these studies can largely benefit +from a more emic expansion approach, which would help remove +biases from future empirical studies. +8 +CONCLUSION +We examined the friendship network and the dwell time behavior +of users across 73 cultures on the online platform Snapchat. We +studied one month’s data from 10K users from each culture. First, +we found that the friendship networks curated by individuals from +different cultures vary in size and egocentricity. We found evidence +that individuals from individualistic, high mobility, and loose cul- +tures tend to form larger friendship networks. We analyzed how +cultural values moderate the relation between tie strength and users’ +content engagement behavior. We found that individualism, high +mobility, and looseness negatively moderate this effect. This pro- +vides evidence for psychological theories which posit that relation- +ships are not perceived similarly across different cultures, and thus +their effect on user behavior is not uniform across cultures. Our +work could advance the understanding of engagement with con- +tent on online platforms and how using this insight can improve +recommendation systems. 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Current opinion in psychology 32 (2020), 129–132. +A +CULTURAL VALUE AND FRIEND +NETWORK SIZE WITH CONTROL +VARIABLES +Table 7: Pearson correlation between cultural values and +friendship network size with GDP, GINI, and Market Pen- +etration as control variables (∗푝 < 0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < +0.001) +Cultural Value +Correlation +Number of countries +Individualism +0.6** +47 +Relational Mobility +0.27 +26 +Tightness +-0.51* +24 + +Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study +CHI ’23, April 23–28, 2023, Hamburg, Germany +B +BOOTSTAPPED RESULTS FOR MIXED +EFFECTS MODEL (ACROSS 100 RUNS) +Table 8: Bootstrapped Coefficients From Multilevel Model- +ing for the effect of Individualism as a moderator on Dwell +Time +Fixed Effects +Estimate +퐶퐼 (95%) +Intercept +3.740 +[3.718,3.762] +Strength of Ties +0.103 +[0.100,0.106] +Individualism +0.033 +[0.028,0.038] +Strength of Ties : Individualism +-0.008 +[-0.011,-0.005] +Control variables +Number of Friends +-0.329 +[-0.341,-0.317] +GDP +-0.031 +[-0.038,-0.024] +GINI +-0.040 +[-0.042,-0.039] +Market Penetration +0.56 +[0.038,0.074] +Table 9: Bootstrapped Coefficients From Multilevel Model- +ing for the effect of Mobility as a moderator on Dwell Time +Fixed Effects +Estimate +퐶퐼 (95%) +Intercept +3.820 +[3.801,3.839] +Strength of Ties +0.114 +[0.111,0.117] +High Mobility +0.092 +[0.089,0.095] +Strength of Ties : High Mobility +-0.010 +[-0.014,-0.007] +Control variables +Number of Friends +-0.347 +[-0.356,-0.338] +GDP +-0.058 +[-0.062,0.054] +GINI +-0.020 +[-0.020,-0.015] +Market Penetration +0.109 +[0.104,0.114] +Table 10: Bootstrapped Coefficients From Multilevel Model- +ing for the effect of Tightness as a moderator on Dwell Time +Fixed Effects +Estimate +퐶퐼 (95%) +Intercept +3.740 +[3.737,3.743] +Strength of Ties +0.116 +[0.111,0.121] +Tightness +-0.061 +[-0.061, -0.060] +Strength of Ties : Tightness +0.008 +[0.006, 0.012] +Control variables +Number of Friends +-0.291 +[-0.295,-0.285] +GDP +-0.156 +[-0.161,-0.150] +GINI +-0.17 +[-0.171,-0.162] +Market Penetration +0.170 +[0.169,0.170] +C +MIXED EFFECTS MODEL FOR THE +INTERSECTION OF COUNTRIES PRESENT +ACROSS ALL THREE MEASURES (FOR 1 +RUN) +Table 11: Coefficients from Multilevel Modeling for the ef- +fect of Individualism as a moderator on Dwell Time (∗푝 < +0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001), Sample size: country = 18, +dyads = 82800, RMSE = 2.947, AIC = 45373.1, BIC = 453824.6, +R2 conditional =0.27, R2 marginal = 0.01 +Fixed Effects +Estimate +Standard Error +Intercept +3.699*** +0.126 +Strength of Ties +0.147*** +0.017 +Individualism +0.085 +0.116 +Strength of Ties : Individualism +-0.042*** +0.011 +Control variables +Number of Friends +-0.322*** +0.018 +GDP +-0.168* +0.074 +GINI +-0.171 +0.063 +Market Penetration +0.238 +0.128 +Table 12: Coefficients From Multilevel Modeling for the ef- +fect of Mobility as a moderator on Dwell Time (∗푝 < 0.05, ∗ ∗ +푝 < 0.01, ∗ ∗ ∗푝 < 0.001) Sample size: country = 18, dyads = +82800 RMSE = 3.07, AIC = 476936.1, BIC = 477029.3, R2 con- +ditional = 0.09, R2 marginal = 0.01 +Fixed Effects +Estimate +Standard Error +Intercept +3.969*** +0.123 +Strength of Ties +0.126*** +0.014 +High Mobility +0.008 +0.083 +Strength of Ties : High Mobility +-0.037*** +0.009 +Control variables +Number of Friends +-0.30*** +0.017 +GDP +-0.152 +0.077 +GINI +-0.141 +0.060 +Market Penetration +0.099*** +0.010 + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Seth, et al. +Table 13: Coefficients From Multilevel Modeling for the ef- +fect of Tightness as a moderator on Dwell Time (∗푝 < 0.05, ∗∗ +푝 < 0.01, ∗ ∗ ∗푝 < 0.001), Sample size: country = 18, dyads = +82800, RMSE = 2.36, AIC = 482736.7, BIC = 482830, R2 condi- +tional = 0.48, R2 marginal = 0.01 +Fixed Effects +Estimate +Standard Error +Intercept +3.767*** +0.135 +Strength of Ties +0.164*** +0.014 +Tightness +-0.021 +0.084 +Strength of Ties : Tightness +0.094*** +0.012 +Control variables +Number of Friends +-0.383*** +0.024 +GDP +-0.183 +0.081 +GINI +-0.180 +0.070 +Market Penetration +0.191 +0.105 + +Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study +CHI ’23, April 23–28, 2023, Hamburg, Germany +D +LIST OF COUNTRIES ANALYZED +Country +Individualism +Mobility +Tightness +Argentina +46 +× +× +Australia +90 +4.308 +4.4 +Austria +55 +× +6.8 +Belgium +75 +× +5.6 +Brazil +38 +4.419 +3.5 +Canada +× +4.404 +× +Chile +23 +4.3 +× +China +excluded from analysis since Snapchat is banned +Colombia +13 +4.483 +× +Costa Rica +15 +× +× +Czech Republic +58 +× +× +Denmark +74 +× +× +cEcuador +8 +× +× +Egypt +38 +3.971 +× +El Salvador +19 +× +× +Estonia +× +4.233 +2.6 +Ethiopia +27 +× +× +Finland +63 +× +× +France +71 +4.451 +6.3 +Germany +67 +4.194 +7 +Ghana +20 +× +× +Greece +35 +× +3.9 +Guatemala +6 +× +× +Hong Kong +25 +4.043 +6.3 +Hungary +55 +3.893 +2.9 +Iceland +× +× +6.4 +India +48 +× +11 +Indonesia +14 +× +× +Iran +excluded from analysis since Snapchat is banned +Iraq +38 +× +× +Ireland +70 +× +× +Israel +54 +4.336 +3.1 +Italy +76 +× +6.8 +Jamaica +39 +× +× +Japan +46 +3.934 +8.6 +Jordan +× +3.96 +× +Kenya +27 +× +× +Kuwait +38 +× +× +Lebanon +38 +4.079 +× +Libya +38 +4.015 +× +Malaysia +26 +3.886 +11.8 +Mauritius +× +4.385 +× +Mexico +30 +4.607 +7.2 +Morocco +× +4.062 +×s +Netherlands +80 +4.448 +3.3 +New Zealand +79 +4.287 +3.9 +Nigeria +20 +× +× +Norway +69 +× +9.5 +Pakistan +14 +× +12.3 +Panama +11 +× +× +Peru +16 +× +× +Philippines +32 +4.158 +× +Poland +60 +4.415 +6.0 + +CHI ’23, April 23–28, 2023, Hamburg, Germany +Seth, et al. +Portugal +27 +4.236 +7.8 +Puerto Rico +× +4.603 +× +Saudi Arabia +38 +× +× +Sierra Leone +20 +× +× +Singapore +20 +4.133 +10.4 +South Africa +65 +× +× +South Korea +18 +4.089 +10.0 +Spain +51 +4.415 +5.4 +Sweden +71 +4.364 +× +Switzerland +68 +× +× +Taiwan +17 +4.118 +× +Tanzania +27 +× +× +Thailand +20 +× +× +Trinidad and Tobago +× +4.421 +× +Tunisia +× +3.954 +× +Turkey +37 +4.122 +9.2 +Ukraine +excluded from analysis due to geo-political instability +United Arab Emirates +38 +× +× +United Kingdom +89 +4.315 +6.9 +United States +91 +4.382 +5.1 +Uruguay +36 +× +× +Venezuela +12 +4.508 +3.7 +Zambia +27 +× +× +Table 14: List of Countries and the cultural values that they were surveyed for; × signifies country not surveyed for that cultural +value + diff --git a/-NFST4oBgHgl3EQfcDge/content/tmp_files/load_file.txt b/-NFST4oBgHgl3EQfcDge/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d5e6b441dd0d8dd2e050d2c0b4af94cb0371909 --- /dev/null +++ b/-NFST4oBgHgl3EQfcDge/content/tmp_files/load_file.txt @@ -0,0 +1,1009 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf,len=1008 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='13801v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='SI] 29 Jan 2023 Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study Agrima Seth agrima@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='edu School of Information, University of Michigan, Ann Arbor, Michigan, USA Jiyin Cao jiyincao@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='com Stony Brook University Stony Brook, New York, USA Xiaolin Shi Xiaolin@snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='com Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Santa Monica, California, USA Ron Dotsch rdotsch@snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='com Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Santa Monica, California, USA Yozen Liu yliu2@snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='com Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Santa Monica, California, USA Maarten W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Bos maarten@snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='com Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Santa Monica, California, USA ABSTRACT Culture shapes people’s behavior, both online and offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Surpris- ingly, there is sparse research on how cultural context affects net- work formation and content consumption on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We an- alyzed the friendship networks and dyadic relations between con- tent producers and consumers across 73 countries through a cul- tural lens in a closed-network setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Closed networks allow for intimate bonds and self-expression, providing a natural setting to study cultural differences in behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We studied three theoreti- cal frameworks of culture - individualism, relational mobility, and tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We found that friendship networks formed across dif- ferent cultures differ in egocentricity, meaning the connectedness between a user’s friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Individualism, mobility, and looseness also significantly negatively impact how tie strength affects con- tent consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our findings show how culture affects social media behavior, and we outline how researchers can incorporate this in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our work has implications for content recom- mendations and can improve content engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' CCS CONCEPTS Human-centered computing → Social networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Social me- dia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Social network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' KEYWORDS Social media platforms, Cross-cultural analysis, Social ties, User Behavior Modeling, relationship modeling, tie strength ACM Reference Format: Agrima Seth, Jiyin Cao, Xiaolin Shi, Ron Dotsch, Yozen Liu, and Maarten W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Bos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In Proceedings of the 2023 CHI Conference on Human Factors in Computing Systems (CHI ’23), April 23–28, 2023, Hamburg, Ger- many.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1145/3544548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3581074 1 INTRODUCTION In the past two decades, social media platforms have transformed how individuals build and maintain their relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' These plat- forms are increasingly becoming the preferred method for initiat- ing intimate relationships [53], seeking advice [32], and commu- nity building [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' With social media platforms becoming an inte- gral part of social life for many of us (there are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='26 billion social media users as of 2021 [42]), understanding the drivers of user be- haviors is imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Directly engaging with others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', sending messages) and con- suming their content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', viewing, replying, and reacting to Sto- ries and posts) are often studied to understand behavioral patterns on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' User behavior on online social media platforms can be said to be broadly driven by a complex combina- tion of (a) user identity (personality, demographics), (b) the norms (descriptive and prescriptive) that the users in a network collec- tively subscribe to, (c) the relationship between users (friends, ac- quaintances, strangers), (d) usage intent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' for example, professional (LinkedIn) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' curated self-presentation (Instagram), and (e) plat- form affordances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' While any particular platform usually provides the same affordances to all users on that platform, users bring their different backgrounds, experiences, expectations, beliefs, and val- ues to the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As a result, different behaviors on the same platform are culturally influenced [2, 7, 11, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Most studies on social media user behavior are based on data that is west-centric [39, 54], and thus, their results have an implied context of western cultural norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' These findings fail to account for the heterogeneity in user behavior that arises from different cultural contexts [4, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, to further understand how cul- tural values affect behavior on these platforms, our work focuses on how users from different cultural backgrounds interact differ- ently on a platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Specifically, we use theoretical frameworks of cultural values to study the differences in the formation of friend- ship networks and the moderation of differential behavior of con- tent consumption within these friendship networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This paper uses three theoretical frameworks of cultural values: Hofstede’s concept of Individualism [20], Thomson and colleagues’ concept of Relational Mobility [46], and Gelfand and colleagues’ concept of Tightness [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The data we used for our analyses is from the CHI ’23, April 23–28, 2023, Hamburg, Germany Seth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' camera and messaging platform Snapchat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Snapchat is used in al- most 150 countries and has 347 million daily active users world- wide [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Snapchat is a closed network, meaning that a lot of the content shared by individuals on Snapchat is only available to a limited set of trusted users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Past work on eliciting the motivations for Snapchat usage has shown that Snapchat is used to commu- nicate with close relationships and is viewed as a platform with a relatively lower emphasis on self-presentation and impression management compared to platforms like Instagram [3, 6, 35, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Because closed networks have less formal pressures and allow for intimate bonds and self-expression, they provide us with a cleaner setting to study differences in human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Specifically, we focus on 1) how culture influences network cre- ation and 2) how culture influences content consumption behav- iors embedded in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In particular, for the second ques- tion, we are interested in how culture moderates the effect of tie strength (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', the closeness between individuals) on content con- sumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Past work has shown that tie strength strongly predicts a variety of user behaviors on platform, including what informa- tion will be exchanged [31, 51, 54], the likelihood to change one’s actions [8], the attention given to content [52], and the preferred behavior to signal engagement [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We will explore how tie strength moderates tie strength’s effect on content consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To study content consumption behavior, we use the metric of dwell time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', the time a user spends consuming content that another user creates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In sum, we ask the following research questions: (1) How do friendship networks differ in countries with differ- ent cultural values?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' (2) How do cultural values change the effect of tie strength on dwell time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To answer our first research question, we studied the network properties of friendship networks across 73 countries, which have been surveyed by either Hofstede [20], Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [46], or Gelfand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [15], and have different cultural values that lie on a continuum of the three cultural values of individualism, mobility, and tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We analyzed how friendship network size and ego- centricity — the extent to which a person’s friends are connected with each other — vary across cultures in the closed network set- ting of Snapchat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We find that users from more individualistic, mo- bile, and loose cultures have a more extensive friendship network and are less egocentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Next, we analyzed within these networks how tie-strength between users impacts engagement with content (dwell time) and the role of cultural values as a moderator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We found that individualism, mobility, and looseness negatively mod- erate the effect of tie strength on content consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Where previous work on culture and social media platforms has primarily been limited to a small sample size [2, 39], this paper contributes by studying cultural differences in user behavior on a large scale, analyzing hundreds of thousands of users across many countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Further, where other quantitative works are usually lim- ited to open or broadcast networks, this study explores relatively under-studied closed-network settings [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' From an HCI and design perspective, our work can advance our understanding of behavior patterns across cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We dis- cuss the implications of understanding users’ engagement with content to design better experiences for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' When applied to platform design, our work would help user-retention of platforms without compromising the user experience, in turn creating better outcomes for both users and platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our work furthers the re- search that helps answer the question: What does it mean to under- stand and support users from diverse cultures on online platforms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Most of the designs and practices of online platforms have a ‘one-size-fits-all’ approach and do not actively account for different user preferences across geographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our results provide evidence of differential behavioral patterns in online friendship networks across cultures and suggest how algorithm design can be cultur- ally inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Privacy and Ethics The data for this study was taken from Snapchat, and the study was conducted within Snapchat in accordance with Snapchat’s policies and procedures with respect to Snapchat data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This analysis only uses the metadata of the user behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' It does not analyze the ac- tual content of the communication between the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Ties and user behavior Interpersonal relationships make social media platforms social.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Like in offline social networks, an individual’s online network consists of individuals, with each of whom one shares a different type of re- lationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Each dyadic relationship is different based on the close- ness and the purpose they serve to the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Social network analysis literature uses the term tie strength to differentiate be- tween relations of different closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This term was coined by Granovetter[17], who analyzed the role of different ties in differ- ent situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The two types of ties characterized were strong and weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The four dimensions determining a tie’s strength were: the amount of time spent on a tie, the intimacy, the intensity, and re- ciprocal services [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Although researchers have used different operationalizations to conceptualize tie strength depending on the purpose of the study, many works on social media platforms op- erationalize tie strength as proportional to the total number of ex- changes in the dyad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This operationalization of tie strength has been used to study various phenomena, like promoting mental well-being [25], increased diffusion of information, and access to novel information [17, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' While these works analyze the role of tie strength in reaping social benefits, studies have also focused on justifying Granovetter’s hypothesis that the two ties elicit differ- ent interaction patterns, for which they analyze how information from different ties is received [19, 23, 24, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' These studies find evidence that individuals spend more time on the content received from stronger ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2 Cultural values One primary aspect of culture is that it’s the normative value sys- tem that dictates acceptable practices and helps differentiate one group from another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Culture is both a result of the accepted past actions and the determinant of acceptable future actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' One of the ways to reason about attitudes and actions is to understand the culture people are in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Prior studies have shown that an indi- vidual’s behavior in the online space is influenced by their culture Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study CHI ’23, April 23–28, 2023, Hamburg, Germany in the same way as offline behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' With cultural values shaping actions, we must first understand how culture can be measured and then how culture affects behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' While prior work usually focuses on groups and their specialized culture, we introduce liter- ature from cultural psychology in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Culture is often opera- tionalized through dimensions where a dimension is defined as “an aspect of a culture that can be measured relative to other cultures.” In this paper, we bring in concepts from three dominant cultural psychology theories, namely, individualism-collectivism [20], re- lational mobility[46], and tightness-looseness[15], to explore how culture impacts network creation and content consumption behav- iors within a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Below, we briefly introduce each of the cul- tural dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Hofstede’s Individualism-collectivism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hofstede [20] analyzed data from over 50 countries and identified six critical dimensions of national culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Individualism-collectivism is one dimension that has drawn the most research attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Typically, individualism leads to loose ties among the individuals of a society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Individual- ists focus on "I" as opposed to "we.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='" Because groups are less im- portant to them, individualists also tend to show no difference in their behaviors and attitudes toward ingroups versus outgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In contrast, collectivism leads to a collective identity, and the welfare of an individual is implicitly assumed to be linked to the interests of the larger group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, collectivists focus on "we.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='" Because of their particular focus on "we," collectivists are known to have differ- ent norms and behaviors towards ingroups versus outgroups and place greater emphasis on harmony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Because of the "I" nature, individualists need to constantly reach out to build networks and also tend to see relationships as fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In contrast, collectivists see relationships as given, and thus, they are less active in building networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As a result, we predict that individualism will be positively correlatedwith friendship network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Individualists are less likely to treat other people based on re- lationship strength and group membership, whereas collectivists tend to have a strong tendency to favor ingroup members and peo- ple they are close to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This should also be manifested in how tie strength drives content engagement behavior in different cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As a result, we predict that individualism will negatively moderate the positive effect of tie strength on content engagement, such that the effect of tie strength on content engagement will be weaker for individualists than for collectivists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, we hypothesize that: H1a: The friendship network for individualisticcultures is larger than the friendship network for collectivistic cultures H1b: Individualism negatively moderates the effect of tie strength on content engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2 Relational Mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [46] conducted a survey across 39 countries using a set of 12 questions to construct their dimension of culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Relational mobility indicates the degree of freedom and opportunities the members of a culture have to form and terminate relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The two opposing poles on this in- dex are high and low relational mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For example, relational mobility is high in North America and low in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Because re- lationships in high-mobility cultures are less stable and easier to change than those in low-mobility cultures, they are more fragile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' It also requires more effort to maintain committed relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Prior work has shown that cultures with higher relational mobil- ity tend to share more about themselves (self-disclosure), are more active in giving support, and tend to have more trust in the mem- bers of the society [46, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Because cultures high in mobility have more opportunities to form relationships, it allows individuals to have a larger network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In a similar vein, because in high mobil- ity cultures, individuals see relationships as more fragile and fluid, they are less likely to adjust their interpersonal behaviors based on tie strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As such, we predict that relational mobility will neg- atively moderate the effect of tie strength on content engagement, such that the effect of tie strength on content engagement will be weaker in high-mobility cultures than in low-mobility cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, we hypothesize that: H2a: The friendship network for high mobility cultures is larger than the friendship network of low mobility cultures H2b: Relational mobilitynegatively moderates the effect of tie strength on content engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 Tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Gelfand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [15] conducted a survey across 33 countries using 12 behaviors across 15 situations to construct their dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Tightness-looseness is about the extent to which a so- ciety tolerates norm-deviant behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The two opposing poles on this index are tight and loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For example, Looseness is high in North America and low in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Tight cultures have stronger norms and are less tolerant of behavior that deviates from the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In contrast, loose cultures have relatively weaker norms and are more tolerant of behavior that deviates from the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As such, we predict that tightness should be negatively correlated with net- work size because a tight culture makes it hard for people to bring new members to a social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Cultural tightness is often con- sidered a selection criterion to test whether a new member can fit in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In contrast, the level of scrutiny will be much lower in a loose culture, making it easier for an individual to expand their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Similarly, we predict that tie strength’s effect on content engage- ment will be weaker in loose cultures than in tight cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In a loose culture, tie strength is less likely to be seen as a criterion that individuals rely upon to decide how they approach a person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In contrast, in a tight culture, tie strength is a monitoring mech- anism that powerfully regulates people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As a result, people draw more influence from tie strength, including content engagement behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, we hypothesize that: H3a: The friendship networks for tighter cultures are smaller than friendship networks of looser cultures H3b: Tightness positively moderates the effect of tie strength on con- tent engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Although the three cultural dimensions originated from differ- ent theories, they are often conceptually related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Prior work has shown that individualism, relational mobility, and looseness are often moderately correlated (Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', 2018, Appendix Ta- ble S8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 51 [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For example, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' is a culture that is in- dividualistic, high mobility, and loose at the same time, whereas Japan is a culture that is collectivistic, low mobility, and tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' How- ever, while Germany ranks higher in individualism and mobility, it CHI ’23, April 23–28, 2023, Hamburg, Germany Seth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' ranks lower in looseness, whereas Brazil, though less individualis- tic, is more mobile and loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Thus, while the three theories are conceptually related and can serve as a robustness check for one another, they each touch upon a unique cultural aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' When re- searchers study the effect of one of the cultural values on individu- als, they also tend to include the other two as a way of robustness check [44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As a result, although the three dimensions are from different theories, we see them as a whole package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In sum, culture provides an important context about the shared common knowledge to its members on how to behave in a given context and how others will interpret their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Comparative work on interpersonal relationships across cultures has shown that the same relationships elicit different behaviors in different cul- tures, implying that the same relationships across cultures are not similarly perceived [13, 16, 18, 30, 37, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our work aims to ana- lyze if user behavior on the same online platform provides empiri- cal evidence that the impact of tie strength on their behavior varies across cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 3 DATA We conduct our study on the Snapchat platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Snapchat is an online messaging platform where content shared between users is ephemeral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Like most platforms, Snapchat allows users to ex- change content in the form of text, images, and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The inter- actions between users can be one-to-one, one-to-group, or one-to- all friends (a broadcast interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Interactions are identified by different names and are introduced below: Snaps: A direct or personal interaction of image or video content type between users, which may be one-to-one or one-to-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Depending on the receiver’s chosen settings, Snaps disappear immediately after viewing or 24 hours later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In our analysis, we only consider Snaps that are exchanged between dyads (just two users), which are termed ‘direct Snaps.’ We do not analyze Snaps sent to groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Chats: A text message between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Akin to Snaps, de- pending on the receiver’s chosen settings, chats disappear immediately after viewing or 24 hours later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In our analy- sis, we only consider the chats that are exchanged between dyads (just two users), which are termed ’direct chats.’ We do not analyze group chats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Stories: A broadcast interaction (with all of one’s friends) having an image orvideo as the content type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Users on Snapchat (posters) can create Stories for their friends (viewers) to con- sume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Stories constitute a pull communication wherein friends decide to either engage with a Story in part or whole or ig- nore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Unlike Snaps and chats that disappear after watch- ing, Stories are available for 24 hrs after posting and can be viewed multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We analyze users on Snapchat who share a friend connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Friendships on Snapchat are bidirectional and are unlike the ‘fol- low model’ that platforms like Instagram and Twitter allow (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', both individuals need to add each other as friends in Snapchat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For each of the 73 countries (Refer appendix D), we randomly sam- pled 10,000 unique users (egos), their associated Story viewing ac- tivity for one month, and their complete one-hop friend network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Though users may have friends across geographies, we filtered the data only to include those friend pairs where both friends resided in the same country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Aggregated over all 73 countries, cross-country friendships accounted for 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='8% of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The filtering resulted in a total dataset of approx 600,000 users per country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Each user can view Stories from multiple friends, with each of whom they share a different level of closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This results in a data set of unique dyadic relations between a Story viewer and a Story poster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For each dyadic interaction, we calculate aggregated statistics of the total time spent by a viewer on each of the poster’s Stories, the total number of Stories shared by a poster, and the total number of Snaps and chats exchanged between the two in the dyadic commu- nication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To avoid noise from users who rarely engage with each other, we only keep those dyadic pairs where at least one direct chat or Snap has been exchanged by both the Story poster and the viewer during the one month we analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To control for effects unrelated to the cultural values but caused by the economic devel- opment and platform reach in a country, we include each country’s GDP [22], which is a measure of a country’s economic standing, GINI [45], which is a measure of economic inequality within a na- tion, and Snap’s market penetration 1, which measures the user- base of Snapchat for a country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Section 4 details the process used to answer each research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The three cross-cultural theo- ries that inform our study did not survey all the same countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Thus, while the three theories do not have a perfect overlap with each other (Refer appendix D), using all three allows us to cover 73 unique countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 4 METHOD We use the observational data from Section 3 and create statistical models to understand the role of culture on users’ network forma- tion and content engagement (dwell time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Building on and align- ing with prior cross-cultural work, we consider a country a repre- sentative unit of one culture [15, 20, 46] and analyze the users at the group level of a country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 RQ1: How do friendship networks differ in countries with different cultural values?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We first measured each country’s average friendship network size to determine whether people from different cultures have differ- ent friendship networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For this, we calculated the total number of friends per user in each country and averaged it over the total number of users in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Next, for each country under study, we reconstruct the ego net- work (egonet) for that country’s randomly sampled 10,000 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' An ego network consists of the user (the ego), the user’s friends (the alters), and the friendship relations between the alters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The egonets formed were independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', the users’ egonets did not overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We filter out networks that consist of only two nodes (users who are only connected to the default Snapbot and do not have other friends on the platform) or star graphs (a pattern where a user is connected to other users, but none of those other users are connected, which is a pattern mainly shown by bots [38, 48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Since all friendships on Snapchat are bidirectional, we convert the graph to a simple graph by removing the multiple edges (edges that are incident on the same pair of nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For each of the egonets, we 1Internal Snap INC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' marketing data Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study CHI ’23, April 23–28, 2023, Hamburg, Germany calculate measures of egocentricity - the density, transitivity, and the betweenness centrality of the ego using the igraph package in R [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Ego betweenness measures the percentage of shortest paths be- tween two alters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In a social network setting, it allows us to mea- sure the importance of the ego node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The higher the betweenness centrality, the more the ego node is the binding factor between its friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Since centrality is sensitive to network size, we normalized it by the maximum possible betweenness of the ego node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This ap- proach is in line with prior work on measuring betweenness in egonets Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=',[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Betweenness centrality of node i = � 푖≠푗≠푘 푔푗푘 (푖) 푔푗푘 Where 푔푗푘 is the number of shortest paths that connect node j and node k, 푔푗푘 (푖) is the number of these shortest paths that include node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Network density is the ratio of the edges in the user’s network to the edges of the same user’s hypothetical network where every node is connected to every other node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Likewise, transitivity is the number of triads relative to the number of possible triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In our setting, density and transitivity measure the tendency of the users to cluster or connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The higher the density and transitivity, the more the tendency of the group to cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Density for an undirected graph = � 푗≠푘 푧푗푘 푛∗(푛−1) 2 Where n is the number of nodes in a network, and 푧푗푘 is equal to 1 if the alters j and k are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Transitivity for an undirected graph = 3 ∗ number of triangles in the network number of connected triples of nodes in the network A high density and transitivity are indicative of people connect- ing with friends of friends;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' a low betweenness, on the other hand, implies a reduced tendency of nodes to cluster together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Prior work by Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [28] on self-reported Facebook networks in East Asia and the USA found that users from the USA were more egocen- tric than users from East Asia (had higher Ego Betweenness and lower Density and Transitivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We use the same methodology — to analyze data across more countries — to explore whether these findings generalize across platforms and for data that is not self- reported but an individual’s actual network data from a social me- dia platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To maintain consistency with Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=',[28], we log- transform density and transitivity and then inverse the transforma- tion by multiplying minus one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' we transform betweenness using 푙표푔(1 + 푀푎푥(푥) − 푥) and then inverse the transformation by mul- tiplying minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2 RQ2: How do cultural values change the effect of tie strength on dwell time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Online social media platforms continually aim to remove obsta- cles for content creation and consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' this has allowed for a myriad of content to be available for consumption by users on all platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' With the multitude of content available, attention from one’s social network has become a valuable and competitive resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Here, we analyze how users allocate their attention to so- cial connections with varying degrees of closeness and how this al- location is moderated by culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We study attention in the context of Stories posted by friends in one’s network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We examine whether tie strength predicts one’s dwell time on a Story and whether cul- ture moderates the relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Measuring interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Attention to a poster’s Story is a proxy for the interest in the information shared by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Attention towards a friend who posts Stories (p) is measured by the total time they spend on viewing their Story;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' longer attention (dwell time) for a Story indicates a stronger interest towards that friend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To measure total time spent on content consumption (TC), we refer to the formulation proposed in prior works on measuring content dwell time [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 푇퐶(푣,푝) = � 푠∈푆푝→푣 훿(푠) where푆푝→푣 denotes the set of Stories postedby p and consumed by v, s denotes (without loss of generality) one such Story sample, and 훿(푠) indicates the time spent by v in viewing the Story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This mea- sures the relative difference in the viewer’s interest across different posters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' However, as pointed out in prior literature, a viewer’s total view time on a poster’sStory can be skewed by the frequency of the posting activity of the Story creator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', given the equal likelihood to consume Stories from different poster’s푇퐶(푣, 푝1) > 푇퐶(푣, 푝2) if |푆푝1→푣| > |푆푝2→푣|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, we model dwell time towards a sender s as the average time spent by a viewer on the sender’s Stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 퐷푇 (푣, 푝) = � 푠∈푆푝→푣 훿(푠) |푆푝→푣| Dwell time is measured in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' While Stories vary in dura- tion and can, in turn, influence dwell times, our initial analysis of viewing time distribution showed that most viewing activities were short and independent of content duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This finding is in line with prior works on dwell time in closed network settings [23] thus, we do not control for this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2 Measuring social tie strength between two users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Tie strength between two users is a complex concept, subject to user percep- tions and emotions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' hence a direct quantitative measure of tie strength between users is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' However, measuring the activity of direct conversations between two users on social media platforms has proven to be an effective proxy in estimating tie strength: the higher the number of dyadic message exchanges, the closer the two users are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Some users send burst messages while others send fewer but longer messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' thus, we model tie strength (TS) as the total number of direct Snaps and chats exchanged between a pair of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 푇푆(푣, 푝) = |퐷퐶푝→푣| + |퐷퐶푣→푝| + |퐷푆푝→푣| + |퐷푆푣→푝| where 퐷퐶푝→푣 denotes the set of direct chats sent by the Story poster to the Story viewer, 퐷퐶푣→푝 denotes the set of direct chats sent by the Story viewer to the Story poster, 퐷푆푝→푣 denotes the set of direct Snaps sent by the Story poster to the viewer, and 퐷푆푣→푝 denotes the set of direct Snaps sent by Story viewer to the poster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany Seth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Preliminary analysis of tie strength in each country showed varia- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' hence, we standardize tie strengths within each country and use the standardized version for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 Measuring culture of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We use the results from Hof- stede’s Individualism [20], Thomson et al.’s Relational Mobility [46], and Gelfand et al.’s Tightness [15] dimensions, discussed in Sec- tion 2 as the measure of cultural values (CV) for the country that an individual belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' These measures have been widely used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hofstede’s work has attracted over 45,000 cita- tions, Thomson et al.’s (more recent) work has already been cited 178 times, and Gelfand et al.’s work has more than 2000 citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Since each value system is on a different scale (Appendix D)— In- dividualism ranges from 6 to 91, Relational Mobility ranges from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='886 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='607, and Tightness ranges from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='6 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 — we indepen- dently standardize each value system across countries and use the standardized version for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='4 Mixed effects model to analyze dwell time as a function of tie strength and cultural values .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We used a linear mixed-effects model to address the research question of how cultural values moderate the impact of tie strength on the time spent consuming content (dwell time) in closed network settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Since the sets of countries surveyed by Hofstede [20], Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [46], and Gelfand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [15] do not have perfect overlap, we created three multilevel mod- els to understand how cultural values moderate the effect of tie strength on Story dwell time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The models included terms for tie strength (dyad level), cultural value (country level), and their in- teraction as fixed effects, with random intercepts for country and viewer, and the number of friends, the GDP, GINI, and Snap’s mar- ket penetration (MP) for a country as control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We stan- dardized each value system across countries and used the standard- ized version for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Since we have multiple observations per country and a viewer views multiple posters, we include the ran- dom effects due to the country and the viewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 퐷푇 (푣, 푝) = 푇푆(푣, 푝) 푋 퐶푉 (푣) + |푣푓 | + 퐺퐷푃 + 퐺퐼푁퐼 + 푀푃+ (1|푐표푢푛푡푟푦) + (1|푉푖푒푤푒푟) where |푣푓 | refers to the number of friends a viewer has, 푇푆(푣, 푝) is the tie strength between a pair of viewers and a poster, and 퐶푉 (푣) is the cultural value of the viewer, which is the same as the cultural value of the poster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Since each dyad contains the dwell time of multiple Stories, we model random effects for the dyad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' However, users in a dyad can have two roles: sometimes a user is a viewer, and sometimes a poster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' A user who is a viewer (v) for a poster p can be a poster (푝′) for some other node (푣′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This directionality complicates modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To simplify, we randomly regard one person as the viewer and the other as a poster, disregarding the Stories of that dyad where the viewer posted and the poster viewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To ensure that the results 5 are robust against role assignment, we bootstrapped the analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' on each run, for each dyad, viewer and poster roles were randomly assigned before fitting the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The bootstrapped results are in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 5 RESULTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 RQ1: How do friendship networks differ in countries with different cultural values?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We report zero-order Pearson correlations between cultural values and friendship network size in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We find that countries that rank higher in individualism, mobility, and looseness tend to have a bigger friendship network than collectivistic, less mobile, and tighter countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This means that people in the higher ranking countries are connected to more friends on Snapchat, supporting H1a, H2a, and H3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' To check for robustness, we ran the same analy- ses with GDP, GINI, and Snapchat’s market penetration as control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The addition of control variables reduced the sample size of countries, but the results corroborate those reported here A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Next, the structural analysis of the ego networks of users from different cultures (Table 3) shows that the ego centrality of user networks on Snapchat varies with cultural values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Akin to Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=',[28], we find that the individual structural measures, namely density, transitivity, and betweenness, are highly correlated (Ta- ble 2), and thus we average the standardized values and report the results for this averaged index of ego-centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The results show that mobility and individualism are negatively correlated with ego- centricity, and tightness is positively correlated with egocentrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This means that in countries that rank higher on mobility and indi- vidualism, people’s friends on Snapchat are more likely to be con- nected to each other, and in countries that rank higher on tightness, people’s friends on Snapchat are less likely to be connected to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Table 1: Pearson correlation between cultural values and friendship network size (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗ ∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001) Cultural Value Correlation Number of countries Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='68** 65 Relational Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='31* 37 Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='37* 30 Table 2: Pearson correlation between network structural measures for data across different cultural values after con- trolling for GDP, GINI, and market penetration (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001) Cultural Value Betweenness and Transitivity Betweenness and Density Density and Transitivity Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='74*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='504*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='92 *** Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='76 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='49** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='85*** Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='82*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='45* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='83 *** Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study CHI ’23, April 23–28, 2023, Hamburg, Germany Table 3: Pearson correlation between cultural values and egocentrality(∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗ ∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001) Cultural Value averaged index of ego-centrality Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='07 *** Relational Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='04*** Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='06*** 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2 RQ2: How do cultural values change the effect of tie strength on dwell time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Given that the friendship network structures are different across cultures, using multilevel modeling, we analyzed how cultural val- ues moderate the effect of tie strength on the viewer’s dwell time (Tables 4, 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We see that an increase in the strength of ties in- creases the dwell time, a result in line with prior works [23, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Having more friends reduces a viewer’s dwell time on content, which is likely because an increase in the number of friends leads to more potential Story content to consume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Though the cultural values do not have a significant main effect, they significantly mod- erate the effect of tie strength on dwell time across all three cultural values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We find that tie strength negatively moderates the effect of tie strength for more individualistic, mobile, and looser cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Thus confirming H1b, H2b, and H3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The bootstrap results from 100 runs corroborate the findings reported here in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our work focuses on understanding (and not predicting) within- dyad level dwell time from theories of country-level cultural val- ues, which may not fully account for a lot of individual-level vari- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' However, a significant moderation effect allows us to argue for a substantiative effect of cultural values on individual-level be- havior [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Using only the intersection of countries present across all three measures of culture, we check for robustness of these re- sults (Appendix C), and the results corroborate the results reported in Tables 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Because the effects we found are on the smaller side, there is still a lot of unexplained variance, and we can not fully account for all individual-level and item (Story) level variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Table 4: Coefficients from Multilevel Modeling for the ef- fect of Individualism as a moderator on Dwell Time (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗ ∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001), Sample size: country = 47, dyads = 460000, RMSE = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='9, AIC = 2793115, BIC = 279226, R2 conditional = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='04, R2 marginal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01 Fixed Effects Estimate Standard Error Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='741*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='078 Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='092*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='007 Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='074 Strength of Ties : Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='014*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='007 Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='338*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='008 GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='068 GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='060 Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='065* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='059 Table 5: Coefficients From Multilevel Modeling for the effect of Mobility as a moderator on Dwell Time (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗ ∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗∗∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001) Sample size: country = 26, dyads = 128800, RMSE= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='12, AIC = 1438399, BIC = 1438504, R2 conditional = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='27, R2 marginal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01 Fixed Effects Estimate Standard Error Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='835 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='097 Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='116*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='008 High Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='071 Strength of Ties : High Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='012* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='006 Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='35*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='011 GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='108 GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='102 Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='091 Table 6: Coefficients From Multilevel Modeling for the effect of Tightness as a moderator on Dwell Time (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗∗∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001), Sample size: country = 25, dyads = 100000, RMSE=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='19, AIC = 731754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3, BIC = 731850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='8, R2 conditional = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='80, R2 marginal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01 Fixed Effects Estimate Standard Error Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='725*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='129*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='010 Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='082 Strength of Ties : Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='058*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='010 Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='283 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='011 GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='154* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='077 GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='069 Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='179* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='077 6 DISCUSSION Most social media platforms were introduced in the Global North before they started gaining a user base in other countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As a result, studies on understanding users on social media platforms primarily draw from west-centric populations, which leads to un- intended biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Using data from 10,000 users per country from nearly 73 countries, our work studied how individuals across cul- tures differ in their behavior on the same platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We control for confounders like the platform’s market penetration, countries’ GDP, and GINI score, which may have influenced the platform’s user base size and composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our main findings are: Structure of friendship network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The analysis of the egocentrality of the friendship networks showed that individualistic, more mobile, and looser cultures are negatively correlated with egocentrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This result is unlike the prior survey-based network analysis by Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [28], which found that individualism is positively correlated with ego centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [28] recruited individuals through a call for survey participants on the Facebook platform, which re- sulted in a substantially varied number of respondents from each CHI ’23, April 23–28, 2023, Hamburg, Germany Seth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' country and thus could be sensitive to selection and conformity bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In our study, we randomly sampled users and analyzed the metadata of the user behavior, which provides a relatively cleaner signal for a user’s choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Apart from a more balanced number of users from different countries, we also analyzed data from a sub- stantially higher number of countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Apart from data collection and sample size differences, another potential source for the dif- ferences in findings could arise from who is befriended on these platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Adams and Plaut posited that friendship’s meaning varies sub- stantially across cultures [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Markus and Kitayama [26] argued that familial ties form an important part of a user’s social network in collectivist cultures compared to individualistic cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' With the demographics on Snapchat skewing towards a younger popu- lation [9, 10] and motivations differing from Facebook [3, 34, 50], it is plausible that (a) the ’younger users’ do not ’friend’ familial ties due to the difference in how they make sense of ’friendship’ and whom they ’friend,’ and (b) the ’elder’ familial members are ab- sent from the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Since family ties form an important part of collectivist cultures, not including them on their Snapchat friend- ship network could be the reason for differences in our findings when compared to Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' While our results differ from Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', [28], they agree with the findings from Igarashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [21] that user’s from collectivist cultures had more egocentric networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Given that very few studies have explored how culture affects net- work structures, future work in this domain will help establish a stronger understanding of how culture influences the network structures formed on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our findings bear important implications for future work that aims to study user interaction patterns on a platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Firstly, stud- ies should elicit and validate the network structure formed for their population of interest because the network structures vary across subpopulations on the same platform and across platforms, and relying on metrics from prior work with a mismatched popula- tion might lead to incorrect inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Next, the differences in friendship networks bear importance for context-aware friendship recommendation engines, which we discuss under design implica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Cultural Values and user behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Culture is a complex societal- level phenomenon that guides individual behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Various studies have tried to study culture through a system of ’cultural values.’ In this project, we chose three dominant theories in cultural psychol- ogy, ranging from Hofstede’s dimensions published in 2001 [20] to more recent theories on Tightness and Mobility published in 2011 and 2018 [15, 46], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Consistent with our hypothesis, we found that each cultural value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', individualism, looseness, mobil- ity) significantly moderates the effect of tie strength on dwell time, highlighting the significance of considering culture in understand- ing behavior patterns on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In addition, we found that individualism, looseness, and mobility moderates the relationship between tie strength and dwell time in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Theo- retically, it is logical because in societies where people have more freedom to make friends and move between different circles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', high relational mobility), a looser norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', looseness) is likely to develop, and a comparatively more self-focused mindset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', indi- vidualism) is likely to rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Indeed, prior work has also predicted that these three variables would have a similar impact on individ- ual cognition and behavior [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Thus, we extend the prior work in cultural psychology by adopting a cultural lens in understanding user behaviors on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Design Implications The diversity of content on platforms has made good recommen- dation systems a necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' While these recommendation systems are becoming increasingly personalized, they fail to distinguish the varied meanings that different types of social ties have for users from different cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For example, if we consider the dyadic pair of user A, their strongest tie, and user B, their strongest tie, such that user A and B belong to different cultures, the influence of the respective strongest tie may be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our study, through evi- dence, argues for treating users and their friendship relations from different cultures differently when designing recommendation sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Analyzing users at a cultural level may reduce the complexity of recommendation systems and make the recommendation sys- tem more culturally sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' By doing so, they may be able to better rank the content the user is more likely to engage with at a reduced cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For example, our result suggests that when design- ing recommendation systems, tie strength should be given greater weight for users in less mobile, tighter, and collectivistic countries because our results show that tie strength is more strongly corre- lated to content dwell time in these countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Friendship recommendation engines that are unaware of ’how’ and ’why’ network structures differ across cultures run the risk of treating friending activities across different cultures as the same, resulting in a suboptimal platform experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For instance, the motivations of individuals from tight cultures could differ from those from loose cultures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=', in contrast to individuals from loose cultures, individuals in tight cultures might feel forced to friend not only those whom they want to but also those whom they have to - say befriending familial ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' A recommendation engine that captures behavior from loose cultures might not be able to recom- mend users with whom one shares common friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Similarly, a recommendation engine that focuses on tight cultures would ex- plore less and over-recommend users with whom one shares com- mon friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, using the behavioral understanding from only either of the cultures risks the failure of the algorithms ( and, in turn, platform experience) in the other cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Thus, while our work takes a step in highlighting ’how’ the network structures differ, future work that provides insights into ’why’ the network structures differ can further enrich the understanding of design- ing friendship recommendation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 7 LIMITATIONS Our study is subject to a few important limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' First, our work uses data from Snapchat, which encompasses a significant but lim- ited amount of people’s online communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We could only use available data for our study, and some of Snapchat’s user data is only available for a limited time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Additionally, the actual con- tent of Snapchat communications is not available for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The Snapchat user group skews young [10], and studies have found that younger people have shifted away from traditional values [29, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Second, recommendation algorithms play an important role Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study CHI ’23, April 23–28, 2023, Hamburg, Germany in network formation on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We did not have access to the friend recommendation algorithm for this study, and we could, therefore, not control for any potential confounding effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Get- ting an insight into the algorithm and its impact on users across geographies could further enrich future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Further, the focus of this study was to understand the friendship network and behavior on the online social network, which may differ from an individ- ual’s offline friendship networks and their interactions on these networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Next, not every country has been equally surveyed in prior research on cultural values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' There is a non-perfect overlap be- tween countries that have been studied for mobility and countries that have been studied for tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Once data from more coun- tries becomes available, our analyses could be extended to include those countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Future work can further build on ours by analyz- ing how content type interacts with cultural values and impacts dwell time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' By using a large random sample of users across coun- tries, country-level measures of economic growth, and inequity, we tried to limit selection bias and account for variations across countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' GDP and GINI measures help us control for country- level socioeconomic status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' However, it is plausible that a given stratum of society is overrepresented on the platform, and country- level socioeconomic measures might not fully control for the plat- form user’s socioeconomic status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The lack of finer-grained mea- sures could be a limitation of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Human behavior is complex and subject to factors that have individual-level variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Hence, it is difficult to fully predict hu- man behavior in the social sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The focus of our work was to test the theory of the effect of culture, as measured at the country level, on individual behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Like prior works, we can not fully account for all individual-level and item (Story) level variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As brought out in the Introduction 1, individual behavior is affected by a host of other variables, and content engagement is no differ- ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' For example, the Story’s content might be an important fac- tor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' however, we could not study this due to Snap Inc.’s policies on not retaining information about the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Future studies can help make the model more complete by operationalizing the type of content and other variables that might affect the dwell time on content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' While the cultural theories used in this study span a large geographic region, the identities of the researchers who created these measures could be a source of bias for these measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' As argued by Shweder [40] (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 409), these studies can largely benefit from a more emic expansion approach, which would help remove biases from future empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 8 CONCLUSION We examined the friendship network and the dwell time behavior of users across 73 cultures on the online platform Snapchat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We studied one month’s data from 10K users from each culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' First, we found that the friendship networks curated by individuals from different cultures vary in size and egocentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We found evidence that individuals from individualistic, high mobility, and loose cul- tures tend to form larger friendship networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We analyzed how cultural values moderate the relation between tie strength and users’ content engagement behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' We found that individualism, high mobility, and looseness negatively moderate this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' This pro- vides evidence for psychological theories which posit that relation- ships are not perceived similarly across different cultures, and thus their effect on user behavior is not uniform across cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Our work could advance the understanding of engagement with con- tent on online platforms and how using this insight can improve recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='com/statistics/545967/snapchat-app-dau/, Last accessed on 2022-09-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [42] Statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Number of social media users world- wide from 2018 to 2022, with forecasts from 2023 to 2027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='statista.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Bots and online hate during the COVID-19 pandemic: case studies in the United States and the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Jour- nal of computational social science 3, 2 (2020), 445–468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [49] Brian Uzzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Embeddedness in the making of financial capital: How social relations and networks benefit firms seeking financing.' metadata={'source': 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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Attention on weak ties in social and communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' In Complex spreading phenomena in social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Springer, 213–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [53] Naomi Whiteside, Torgeir Aleti, Jason Pallant, John Zeleznikow, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Help- ful or harmful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Exploring the impact of social media usage on intimate relation- ships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Australasian Journal of Information Systems 22 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' [54] REBECCA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' YU, RYAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' MCCAMMON, NICOLE B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' ELLISON, and KEN- NETH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' LANGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' The relationships that matter: social network site use and social wellbeing among older adults in the United States of America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Ageing and Society 36, 9 (2016), 1826–1852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1017/S0144686X15000677 [55] Masaki Yuki and Joanna Schug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Psychological consequences of relational mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Current opinion in psychology 32 (2020), 129–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' A CULTURAL VALUE AND FRIEND NETWORK SIZE WITH CONTROL VARIABLES Table 7: Pearson correlation between cultural values and friendship network size with GDP, GINI, and Market Pen- etration as control variables (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗ ∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001) Cultural Value Correlation Number of countries Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='6** 47 Relational Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='27 26 Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='51* 24 Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study CHI ’23, April 23–28, 2023, Hamburg, Germany B BOOTSTAPPED RESULTS FOR MIXED EFFECTS MODEL (ACROSS 100 RUNS) Table 8: Bootstrapped Coefficients From Multilevel Model- ing for the effect of Individualism as a moderator on Dwell Time Fixed Effects Estimate 퐶퐼 (95%) Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='740 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='718,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='762] Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='103 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='100,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='106] Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='033 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='028,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='038] Strength of Ties : Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='008 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='011,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='005] Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='329 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='341,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='317] GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='031 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='038,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='024] GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='040 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='042,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='039] Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='56 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='038,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='074] Table 9: Bootstrapped Coefficients From Multilevel Model- ing for the effect of Mobility as a moderator on Dwell Time Fixed Effects Estimate 퐶퐼 (95%) Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='820 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='801,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='839] Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='114 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='111,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='117] High Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='092 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='089,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='095] Strength of Ties : High Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='010 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='014,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='007] Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='347 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='356,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='338] GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='058 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='062,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='054] GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='020 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='020,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='015] Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='109 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='104,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='114] Table 10: Bootstrapped Coefficients From Multilevel Model- ing for the effect of Tightness as a moderator on Dwell Time Fixed Effects Estimate 퐶퐼 (95%) Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='740 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='737,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='743] Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='116 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='111,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='121] Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='061 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='061, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='060] Strength of Ties : Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='008 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='006, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='012] Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='291 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='295,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='285] GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='156 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='161,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='150] GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='17 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='171,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='162] Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='170 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='169,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='170] C MIXED EFFECTS MODEL FOR THE INTERSECTION OF COUNTRIES PRESENT ACROSS ALL THREE MEASURES (FOR 1 RUN) Table 11: Coefficients from Multilevel Modeling for the ef- fect of Individualism as a moderator on Dwell Time (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗ ∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001), Sample size: country = 18, dyads = 82800, RMSE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='947, AIC = 45373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1, BIC = 453824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='6, R2 conditional =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='27, R2 marginal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01 Fixed Effects Estimate Standard Error Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='699*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='126 Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='147*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='017 Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='116 Strength of Ties : Individualism 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='042*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='011 Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='322*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='018 GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='168* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='074 GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='063 Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='128 Table 12: Coefficients From Multilevel Modeling for the ef- fect of Mobility as a moderator on Dwell Time (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗ ∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001) Sample size: country = 18, dyads = 82800 RMSE = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='07, AIC = 476936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1, BIC = 477029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3, R2 con- ditional = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='09, R2 marginal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01 Fixed Effects Estimate Standard Error Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='969*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='123 Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='126*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='014 High Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='083 Strength of Ties : High Mobility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='037*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='009 Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='30*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='017 GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='077 GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='060 Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='099*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='010 CHI ’23, April 23–28, 2023, Hamburg, Germany Seth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Table 13: Coefficients From Multilevel Modeling for the ef- fect of Tightness as a moderator on Dwell Time (∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='05, ∗∗ 푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01, ∗ ∗ ∗푝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='001), Sample size: country = 18, dyads = 82800, RMSE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='36, AIC = 482736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='7, BIC = 482830, R2 condi- tional = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='48, R2 marginal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='01 Fixed Effects Estimate Standard Error Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='767*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='135 Strength of Ties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='164*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='014 Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='084 Strength of Ties : Tightness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='094*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='012 Control variables Number of Friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='383*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='024 GDP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='081 GINI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='070 Market Penetration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='105 Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study CHI ’23, April 23–28, 2023, Hamburg, Germany D LIST OF COUNTRIES ANALYZED Country Individualism Mobility Tightness Argentina 46 × × Australia 90 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='308 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='4 Austria 55 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='8 Belgium 75 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='6 Brazil 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='419 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='5 Canada × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='404 × Chile 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 × China excluded from analysis since Snapchat is banned Colombia 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='483 × Costa Rica 15 × × Czech Republic 58 × × Denmark 74 × × cEcuador 8 × × Egypt 38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='971 × El Salvador 19 × × Estonia × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='233 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='6 Ethiopia 27 × × Finland 63 × × France 71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='451 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 Germany 67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='194 7 Ghana 20 × × Greece 35 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='9 Guatemala 6 × × Hong Kong 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='043 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 Hungary 55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='893 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='9 Iceland × × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='4 India 48 × 11 Indonesia 14 × × Iran excluded from analysis since Snapchat is banned Iraq 38 × × Ireland 70 × × Israel 54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='336 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Italy 76 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='8 Jamaica 39 × × Japan 46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='934 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='6 Jordan × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='96 × Kenya 27 × × Kuwait 38 × × Lebanon 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='079 × Libya 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='015 × Malaysia 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='886 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='8 Mauritius × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='385 × Mexico 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='607 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2 Morocco × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='062 ×s Netherlands 80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='448 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 New Zealand 79 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='287 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='9 Nigeria 20 × × Norway 69 × 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='5 Pakistan 14 × 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='3 Panama 11 × × Peru 16 × × Philippines 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='158 × Poland 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='415 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='0 CHI ’23, April 23–28, 2023, Hamburg, Germany Seth, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' Portugal 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='236 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='8 Puerto Rico × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='603 × Saudi Arabia 38 × × Sierra Leone 20 × × Singapore 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='133 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='4 South Africa 65 × × South Korea 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='089 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='0 Spain 51 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='415 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='4 Sweden 71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='364 × Switzerland 68 × × Taiwan 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='118 × Tanzania 27 × × Thailand 20 × × Trinidad and Tobago × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='421 × Tunisia × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='954 × Turkey 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='122 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='2 Ukraine excluded from analysis due to geo-political instability United Arab Emirates 38 × × United Kingdom 89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='315 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='9 United States 91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='382 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='1 Uruguay 36 × × Venezuela 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='508 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content='7 Zambia 27 × × Table 14: List of Countries and the cultural values that they were surveyed for;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} +page_content=' × signifies country not surveyed for that cultural value' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFST4oBgHgl3EQfcDge/content/2301.13801v1.pdf'} diff --git a/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf b/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c4e45c171a61a507ca484e31701ecad88430bf59 --- /dev/null +++ b/-dAyT4oBgHgl3EQfRPaF/content/2301.00062v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2030679606ffe9d886cd2b2eb6b9324d5baf96af03b42d9be25d232d0165ccdb +size 796957 diff --git a/-dAyT4oBgHgl3EQfRPaF/vector_store/index.pkl b/-dAyT4oBgHgl3EQfRPaF/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a30513da315cba92ebbeb533c5f2aa5a80f8e643 --- /dev/null +++ b/-dAyT4oBgHgl3EQfRPaF/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:06740851568214dd70fa9fe375d76c1abdffdd776de84af861e291cb2502a0a1 +size 83618 diff --git a/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/2301.01479v1.pdf.txt b/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/2301.01479v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b0c23ad0268bcf3e5f11af1b855168a570e84eb --- /dev/null +++ b/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/2301.01479v1.pdf.txt @@ -0,0 +1,1197 @@ +arXiv:2301.01479v1 [math.OC] 4 Jan 2023 +Generalizations of R0 and SSM properties; Extended Horizontal Linear +Complementarity Problem +Punit Kumar Yadav +Department of Mathematics +Malaviya National Instiute of Technology, Jaipur, 302017, India +E-mail address: punitjrf@gmail.com +K. Palpandi +Department of Mathematics +Malaviya National Instiute of Technology, Jaipur, 302017, India +E-mail address: kpalpandi.maths@mnit.ac.in +Abstract +In this paper, we first introduce R0-W and SSM-W property for the set of matrices which +is a generalization of R0 and the strictly semimonotone matrix. We then prove some existence +results for the extended horizontal linear complementarity problem when the involved matrices +have these properties. With an additional condition on the set of matrices, we prove that the +SSM-W property is equivalent to the unique solution for the corresponding extended horizontal +linear complementarity problems. Finally, we give a necessary and sufficient condition for the +connectedness of the solution set of the extended horizontal linear complementarity problems. +1 +Introduction +The standard linear complementarity problem (for short LCP), LCP(C, q), is to find vectors x, y +such that +x ∈ Rn, y = Cx + q ∈ Rn and x ∧ y = 0, +(1) +where C ∈ Rn×n, q ∈ Rn and ′∧′ is a min map. The LCP has numerous applications in numerous +domains, such as optimization, economics, and game theory. +Cottle and Pang’s monograph [1] +is the primary reference for standard LCP. Various generalisations of the linear complementarity +problem have been developed and discussed in the literature during the past three decades (see, +[7, 10, 11, 13, 14, 16]). The extended horizontal linear complementarity problem is one of the most +important extensions of LCP, which various authors have studied; see [4, 6, 7] and references therein. +For a given ordered set of matrices C := {C0, C1, ..., Ck} ⊆ Rn×n, vector q ∈ Rn and ordered set of +positive vectors d := {d1, d2, ..., dk} ⊆ Rn, the extended horizontal linear complementarity problem +(for short EHLCP), denoted by EHCLP(C, d, q), is to find a vector x0, x1, ..., xk ∈ Rn such that +C0x0 =q + +k +� +i=1 +Cixi, +x0 ∧ x1 = 0 and (dj−xj) ∧ xj+1 = 0, 1 ≤ j ≤ k − 1. +(2) +If k = 1, then EHLCP becomes the horizontal linear complementarity problem (for short HLCP), +that is, +C0x0 − C1x1 = q and x0 ∧ x1 = 0. +Further, HLCP reduces to the standard LCP by taking C0 = I. Due to its widespread applications in +numerous domains, the horizontal linear complementarity problem has received substantial research +attention from many academics; see [13, 14, 16, 18] and reference therein. +Various writers have presented new classes of matrices for analysing the structure of LCP solution +sets in recent years; see for example, [1, 2, 4]. The classes of R0, P0, P, and strictly semimonotone +1 + +(SSM) matrices play a crucial role in the existence and uniqueness of the solution to LCP. For +instance, P matrix (if [x ∈ Rn, x ∗ Ax ≤ 0 =⇒ x = 0]) gives a necessary and sufficient condition +for the uniqueness of the solution for the LCP (see, Theorem 3.3.7 in [1]). To get a similar type of +existence and uniqueness results for the generalized LCPs, the notion of P matrix was extended for +the set of matrices as the column W-property by Gowda et al. [4]. They proved that column W- +property gives the solvability and the uniqueness for the extended horizontal linear complementarity +problem (EHLCP). Also, they have generalized the concept of the P0-matrix as the column W0- +property. +Another class of matrix, the so-called SSM matrix, has importance in LCP theory. This class of +matrices provides a unique solution to LCP on Rn ++ and also gives the existence of the solution for the +LCP (see, [1]). For a Z matrix (if all the off-diagonal entries of a matrix are non-positive), P matrix +is equivalent to the SSM matrix (see, Theorem 3.11.10 in [1]). A natural question arises whether +the SSM matrix can be generalized for the set of matrices in the view of EHLCP and whether we +have a similar equivalence relation for the set of Z matrices. In this paper, we would like to answer +this question. +The connectedness of the solution set of LCP has a prominent role in the study of the LCP. We +say a matrix is connected if the solution set of the corresponding LCP is connected. In [19], Jones +and Gowda addressed the connectedness of the solution set of the LCP. They proved that the matrix +is connected whenever the given matrix is a P0 matrix and the solution set has a bounded connected +component. Also, they have shown that if the solution set of LCP is connected, then there is almost +one solution of LCP for all q > 0. Due to the specially structured matrices involved in the study of +the connectedness of the solution to LCP, various authors studied the connectedness of LCP, see for +example [19, 20, 21]. The main objectives of this paper are to answer the following questions: +(Q1) In LCP theory, it is a well-known result that the R0 matrix gives boundedness to the LCP +solution set. The same holds true for HLCP [17]. This motivates the question of whether or +not the notion of R0 matrix can be generalized to the set of matrices. If so, then can we expect +the same kind of outcome in the EHLCP? +(Q2) Given that a strictly semimonotone matrix guarantees the existence of the LCP solution and +its uniqueness for q ≥ 0, it is natural to wonder whether the concept of SSM matrix can be +extended to the set of matrices. If so, then whether the same result holds true for EHLCP. +(Q3) Motivated by the results of Gowda and Jones [19] regarding the connectedness of the solution +set of LCP, one can ask whether the solution set of EHLCP is connected if the set of matrices +has the column W0 property and the solution set of the corresponding EHLCP has a bounded +connected component. +The paper’s outline is as follows: We present some basic definitions and results in section 2. We +generalize the concept of R0 matrix and prove the existence result for EHLCP in section 3. In +section 4, we introduce the SSM-W property, and we then study an existence and uniqueness result +for the EHLCP when the underlying set of matrices have this property. In the last section, we give +a necessary and sufficient condition for the connectedness of the solution set of the EHLCP. +2 +Notations and Preliminaries +2.1 +Notations +Throughout this paper, we use the following notations: +(i) The n dimensional Euclidean space with the usual inner product will be denoted by Rn. The +set of all non-negative vectors (respectively, positive vectors) in Rn will be denoted by Rn ++ +2 + +(respectively, Rn +++ ). We say x ≥ 0 (respectively, > 0) if and only if x ∈ Rn ++ (respectively, +Rn +++). +(ii) The k-ary Cartesian power of Rn will be denoted by Λ(k) +n +and the k-ary Cartesian power of +Rn +++ will be denoted by Λ(k) +n,++. The bold zero ’0’ will be used for denoting the zero vector +(0, 0, ..., 0) ∈ Λ(k) +n . +(iii) The set of all n×n real matrices will be denoted by Rn×n. We use the symbol Λ(k) +n×n to denote +the k-ary Cartesian product of Rn×n. +(iv) We use [n] to denote the set {1, 2, ..., n}. +(v) Let M ∈ Rn×n. We use diag(M) to denote the vector (M11, M22, ..., Mkk) ∈ Rn, where Mii is +the iith diagonal entry of matrix M and det(M) is used to denote the determinant of matrix +M. +(vi) SOL(C, d, q) will be used for denoting the set of all solution to EHLCP(C, d, q). +We now recall some definitions and results from the LCP theory, which will be used frequently in +our paper. +Proposition 2.1 ([8]). Let V = Rn. Then, the following statements are equivalent. +(i) x ∧ y = 0. +(ii) x, y ≥ 0 and x ∗ y = 0, where ∗ is the Hadamard product. +(iii) x, y ≥ 0 and ⟨x, y⟩ = 0. +Definition 1 ([4]). Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . +Then a matrix R ∈ Rn×n is column +representative of C if +R.j ∈ +� +(C0).j, (C1).j, ..., (Ck).j +� +, ∀j ∈ [n], +where R.j is the jth column of matrix R. +Next, we define the column W-property. +Definition 2 ([4]). Let C := (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . Then we say that C has the +(i) column W-property if the determinants of all the column representative matrices of C are all +positive or all negative. +(ii) column W0-property if there exists N := (N0, N1, ..., Nk) ∈ Λ(k+1) +n×n +such that C + ǫN := (C0 + +ǫN0, C1 + ǫN1, ..., Ck + ǫNk) has the column W-property for all ǫ > 0. +Due to Gowda and Sznajder [4], we have the following result. +Theorem 2.2 ([4]). For C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n , the following are equivalent: +(i) C has the column W-property. +(ii) For arbitrary non-negative diagonal matrices D0, D1, ..., Dk ∈ Rn×n with diag(D0 +D1 +D2 + +... + Dk) > 0, +det +� +C0D0 + C1D1 + ... + CkDk +� +̸= 0. +(iii) C0 is invertible and (I, C−1 +0 C1, ..., C−1 +0 Ck) has the column W-property. +3 + +(iv) For all q ∈ Rn and d ∈ Λ(k−1) +n,++ , EHLCP(C, d, q) has a unique solution. +If k = 1 and C−1 +0 +exists, then HLCP(C0, C1, q) is equivalent to LCP(C−1 +0 C1, C−1 +0 (q)). In this +case, C−1 +0 C1 is a P matrix if and only if for all q ∈ Rn, LCP(C−1 +0 C1, C−1 +0 (q)) has a unique solution +(see, Theorem 3.3.7 in [1]). Hence we have the following theorem given the previous theorem. +Theorem 2.3 ([4]). Let (C0, C1) ∈ Λ(2) +n×n. Then the following are equivalent. +(i) (C0, C1) has the column W-property. +(ii) C0 is invertible and C−1 +0 C1 is a P matrix. +(iii) For all q ∈ Rn, HLCP(C0, C1, q) has a unique solution. +2.2 +Degree theory +We now recall the definition and some properties of a degree from [2, 3] for our discussion. +Let Ω be an open bounded set in Rn. Suppose h : ¯Ω → Rn is a continuous function and a vector +p /∈ h(∂Ω), where ∂Ω and ¯Ω denote the boundary and closure of Ω, respectively. Then the degree of +h is defined with respect to p over Ω denoted by deg(h, Ω, p). The equation h(x) = p has a solution +whenever deg(h, Ω, p) is non-zero. If h(x) = p has only one solution, say y in Rn, then the degree is +the same overall bounded open sets containing y. This common degree is denoted by deg(h, p). +2.2.1 +Properties of the degree +The following properties are used frequently here. +(D1) deg(I, Ω, ·) = 1, where I is the identity function. +(D2) Homotopy invariance: Let a homotopy Φ(x, s) : Rn ×[0, 1] → Rn be continuous. If the zero +set of Φ(x, s), X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded, then for any bounded +open set Ω in Rn containing the zero set X, we have +deg(Φ(x, 1), Ω, 0) = deg(Φ(x, 0), Ω, 0). +(D3) Nearness property: Assume deg(h1(x), Ω, p) is defined and h2 : ¯Ω → Rn is a continuous +function. If supx∈Ω∥h2(x) − h1(x)∥ < dist(p, ∂Ω), then deg(h2(x), Ω, p) is defined and equals +to deg(h1(x), Ω, p). +The following result from Facchinei and Pang [2] will be used later. +Proposition 2.4 ([2]). Let Ω be a non-empty, bounded open subset of Rn and let Φ : ¯Ω → Rn be a +continuous injective mapping. Then deg(Φ, Ω, p) ̸= 0 for all p ∈ Φ(Ω). +Note: All the degree theoretic results and concepts are also applicable over any finite dimensional +Hilbert space (like Rn or Rn × Rn × Rn etc). +3 +R0-W property +In this section, we first define the R0-W property for the set of matrices which is a natural generaliza- +tion of R0 matrix in the LCP theory. We then show that the R0-W property gives the boundedness +of the solution set of the corresponding EHLCP. +4 + +Definition 3. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . We say that C has the R0-W property if the +system +C0x0 = +k +� +i=1 +Cixi and x0 ∧ xj = 0 ∀ j ∈ [k] +has only zero solution. +It can be seen easily that the R0-W property coincides with R0 matrix when k = 1 and C0 = I. +Also it is noted (see, [8]) that if k = 1, then the R0-W property referred as R0 pair. To proceed +further, we prove the following result. +Lemma 3.1. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +and x = (x0, x1, ..., xk) ∈ SOL(C, d, q). Then x +satisfies the following system +C0x0 = q + +k +� +i=1 +Cixi and x0 ∧ xj = 0 ∀ j ∈ [k]. +Proof. As x0 ≥ 0, there exists an index set α ⊆ [n] such that (x0)i = +� +> 0 +i ∈ α +0 +i ∈ [n] \ α . Since +x0 ∧ x1 = 0, we have (x1)i = 0 for all i ∈ α. From (d1 − x1) ∧ x2 = 0, we get (d1)i(x2)i = 0 ∀i ∈ α. +This gives that (x2)i = 0 ∀i ∈ α. By substituting (x2)i = 0 ∀i ∈ α in (d2 − x2) ∧ x3 = 0, we obtain +(x3)i = 0 ∀i ∈ α. Continue the process in the similar way, one can get (x4)i = (x5)i = ... = (xk)i = +0 ∀i ∈ α. So, x0 ∧ xj = 0 ∀ j ∈ [k]. This completes the proof. +We now prove the boundedness of the solution set of EHLCP when the involved set of matrices +has the R0-W property. +Theorem 3.2. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . If C has the R0-W property then SOL(C, d, q) +is bounded for every q ∈ Rn and d ∈ Λ(k−1) +n,++ . +Proof. Suppose there exist q ∈ Rn and d = (d1, d2, ..., dk−1) ∈ Λ(k−1) +n,++ such that SOL(C, d, q) is +unbounded. Then there exists a sequence x(m) = (x(m) +0 +, x(m) +1 +, ..., x(m) +k +) in Λ(k+1) +n +such that ||x(m)|| → +∞ as m → ∞ and it satisfies +C0x(m) +0 += q + +k +� +i=1 +Cix(m) +i +x(m) +0 +∧ x(m) +1 += 0 and (dj − x(m) +j +) ∧ x(m) +j+1 = 0 ∀j ∈ [k − 1]. +(3) +From the Lemma 3.1, equation 3 gives that +C0x(m) +0 +=q + +k +� +i=1 +Cix(m) +i +and x(m) +0 +∧ x(m) +j += +0 ∀j ∈ [k]. +(4) +As +x(m) +∥x(m)∥ is a unit vector for all m, +x(m) +∥x(m)∥ converges to some vector y = (y0, y1, ..., yk) ∈ Λ(k+1) +n +with ||y|| = 1. Now first divide the equation 4 by ∥x(m)∥ and then take the limit m → ∞, we get +C0y0 = +k +� +i=1 +Ciyi and y0 ∧ yj = 0 ∀j ∈ [k]. +This implies that y must be a zero vector as C has the R0-W property, which contradicts the fact +that ||y|| = 1. Therefore SOL(C, d, q) is bounded. +5 + +3.1 +Degree of EHLCP +Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +and d = (d1, d2, ...., dk−1) ∈ Λ(k−1) +n,++ . +We define a function +F : Λ(k+1) +n +→ Λ(k+1) +n +as +F(x) = + + +C0x0 − �k +i=1 Cixi +x0 ∧ x1 +(d1 − x1) ∧ x2 +(d2 − x2) ∧ x3 +. +. +. +(dk−1 − xk−1) ∧ xk + + +. +(5) +We denote the degree of F with respect to 0 over bounded open set Ω ⊆ Λ(k+1) +n +as deg(C, Ω, 0). +It is noted that if C has the R0-W property, in view of the Lemma 3.1, F(x) = 0 ⇔ x = 0 which +implies that deg(C, Ω, 0) = deg(C, 0) for any bounded open set Ω contains the origin in Λ(k+1) +n +. We +call this degree as EHLCP-degree of C. +We now prove an existence result for EHLCP. +Theorem 3.3. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . Suppose the following hold: +(i) C has the R0-W property. +(ii) deg(C, 0) ̸= 0. +Then EHLCP(C, d, q) has non-empty compact solution for all q ∈ Rn and d ∈ Λ(k−1) +n,++ . +Proof. As the solution set of EHLCP is closed, it is enough to prove that the solution set is non-empty +and bounded. We first define a homotopy Φ : Λ(k+1) +n +× [0, 1] → Λ(k+1) +n +as +Φ(x, s) = + + +C0x0 − �k +i=1 Cixi − sq +x0 ∧ x1 +(d1 − x1) ∧ x2 +(d2 − x2) ∧ x3 +. +. +. +(dk−1 − xk−1) ∧ xk + + +. +Then, +Φ(x, 0) = F(x) and Φ(x, 1) = F(x) − ˆq, where ˆq = (q, 0, 0, ...0) ∈ Λ(k+1) +n +. +By using the similar argument as in above Theorem 3.2, we can easily show that the zero set of +homotopy, X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded. From the property of degree (D2), +we get deg(F, Ω, 0) = deg(F − ˆq, Ω, 0) for any open bounded set Ω containing X. As deg(F, Ω, 0) = +deg(C, 0) ̸= 0, we obtain deg(F − ˆq, Ω, 0) ̸= 0 which implies SOL(C, d, q) is non-empty. As C has +the R0-W property, by Theorem 3, SOL(C, d, q) is bounded. This completes the proof. +4 +SSM-W property +In this section, we first define the SSM-W property for the set of matrices which is a generalization +of the SSM matrix in the LCP theory, and we then prove that the existence and uniqueness result +for the EHLCP when the involved set of matrices have the SSM-W property. +We now recall that an n × n real matrix M is called strictly semimonotone (SSM) matrix if +[x ∈ Rn ++, x ∗ Mx ≤ 0 ⇒ x = 0]. We generalize this concept to the set of matrices. +6 + +Definition 4. We say that C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +has the SSM-W property if +{C0x0 = +k +� +i=1 +Cixi, xi ≥ 0 and x0 ∗ xi ≤ 0 ∀i ∈ [k]} ⇒ x = (x0, x1, .., xk) = 0. +We prove the following result. +Proposition 4.1. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . If C has the SSM-W property, then the +followings hold: +(i) C−1 +0 +exists and C−1 +0 Ci is a strict semimonotone matrix for all i ∈ [k]. +(ii) (I, C−1 +0 C1, ..., C−1 +0 Ck) has the SSM-W property. +(iii) (P T C0P, P T C1P, ..., P T CkP) has the SSM-W property for any permutation matrix P of order +n. +Proof. (i): Suppose there exists a vector x0 ∈ Rn such that C0x0 = 0. Then we have +C0x0 = C10 + C20 + ... + Ck0. +This gives that x0 = 0 as C has the SSM-W property. Thus C0 is invertible. +Now we prove the second part of (i). +Without loss of generality, it is enough to prove that +C−1 +0 C1 is a strictly semimonotone matrix. Suppose there exists a vector y ∈ Rn such that y ≥ 0 +and y ∗ (C−1 +0 C1)y ≤ 0. Let y0 := (C−1 +0 C1)y, y1 := y and yi := 0 for all 2 ≤ i ≤ k. Then we get +C0y0 = C1y1 + C2y2 + ... + Ciyi + .. + Ckyk, +yj ≥ 0 and y0 ∗ yj ≤ 0 ∀j ∈ [k]. +Since C has the SSM-W property, yj = 0 ∀j ∈ [k]. Thus C−1 +0 C1 is a strict semimonotone matrix. +This completes the proof. +(ii): It follows from the definition of the SSM-W property. +(iii): Let x = (x0, x1, ..., xk) ∈ Λ(k+1) +n +such that +(P T C0P)x0 = +k +� +i=1 +(P T CiP)xi, xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k]. +As P is a non-negative matrix and PP T = P T P, we can rewrite the above equation as +C0Px0 = +k +� +i=1 +CiPxi, Pxj ≥ 0 and Px0 ∗ Pxj ≤ 0 ∀j ∈ [k]. +By the SSM-W property of C, Pxj = 0 for all 0 ≤ j ≤ k which implies x = 0. This completes the +proof. +In the above Proposition 4.1, it can be seen easily that the converse of the item (ii) and (iii) are +valid. But the converse of item (i) need not be true. The following example illustrates this. +Example 4.2. Let C = (C0, C1, C2) ∈ Λ(3) +2×2, where +C0 = +� +1 +0 +0 +1 +� +, C1 = +� +1 +−2 +0 +1 +� +, C2 = +� +1 +0 +−2 +1 +� +. +It is easy to check that C−1 +0 C1 = C1 and C−1 +0 C2 = C2 are P matrix. So, C−1 +0 C1 and C−1 +0 C2 are SSM +matrix. Let x = (x0, x1, x2) = ((0, 0)T , (1, 1)T , (1, 1)T ) ∈ Λ(3) +2 . Then we can see that the non-zero x +satisfies +C0x0 = C1x1 + C2x2, x1 ≥ 0, x2 ≥ 0 and x0 ∗ x1 = 0 = x0 ∗ x2. +So C can not have the SSM-W property. +7 + +The following result is a generalization of a well-known result in matrix theory that every P +matrix is a SSM matrix. +Theorem 4.3. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . If C has the column W-property, then C has the +SSM-W property. +Proof. Suppose there exists a non-zero vector x = (x0, ..., xk) ∈ Λ(k+1) +n +such that +C0x0 = +k +� +i=1 +Cixi, xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k]. +Consider a vector y ∈ Rn whose jth component is given by +yj = + + + + + + + + + +−1 +if (x0)j > 0 +1 +if (x0)j < 0 +1 +if (x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k] +0 +if (x0)j = 0 and (xi)j = 0 for all i ∈ [k] +. +As x is a non-zero vector, y must be a non-zero vector. Consider the diagonal matrices D0, D1, ..., Dk +which are defined by +(D0)jj = + + + + + + + + + +(x0)j +if (x0)j > 0 +−(x0)j +if (x0)j < 0 +0 +if(x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k] +1 +if (x0)j = 0 and (xi)j = 0 for all i ∈ [k] +and for all i ∈ [k], +(Di)jj = +� +0 +if (x0)j > 0 +(xi)j +else +. +It is easy to verify that D0, D1, ..., Dk are non-negative diagonal matrices and diag(D0 + D1 + ... + +Dk) > 0. And also note that +x0 = −D0y and xi = Diy ∀i ∈ [k]. +(6) +By substituting the Equation 6 in C0x0 = �k +i=1 Cixi, we get +C0(−D0y) = +k +� +i=1 +CiDi(y) ⇒ +� +C0D0 + C1D1 + ... + CkDk +� +y = 0. +This implies that det(C0D0 + C1D1 + ... + CkDk +� += 0. So, C does not have the column W-property +from Theorem 2.2. Thus we get a contradiction. Therefore, C has the SSM-W property. +The following example illustrates that the converse of the above theorem is invalid. +Example 4.4. Let C = (C0, C1, C2) ∈ Λ(3) +2×2 such that +C0 = +�1 +0 +0 +1 +� +, C1 = +�1 +1 +1 +1 +� +, C2 = +�1 +1 +1 +1 +� +. +Suppose w = (x, y, z) ∈ Λ3 +2 such that +C0x = C1y + C2z and y, z ≥ 0, x ∗ y ≤ 0, x ∗ z ≤ 0. +8 + +From C0x = C1y + C2z, we get +�x1 +x2 +� += +�y1 + y2 + z1 + z2 +y1 + y2 + z1 + z2 +� +. +As x ∗ y ≤ 0, x ∗ z ≤ 0 and from the above equation, we have +y1(y1 + y2 + z1 + z2) ≤ 0 and y2(y1 + y2 + z1 + z2) ≤ 0, +z1(y1 + y2 + z1 + z2) ≤ 0 and z2(y1 + y2 + z1 + z2) ≤ 0. +(7) +Since y, z ≥ 0, from the equation 7, we get x = y = z = 0. Hence C has the SSM-W property. As +det(C1) = 0, by the definition of the column W-property, C does not have the column W-property. +We now give a characterization for SSM-W property. +Theorem 4.5. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n has the SSM-W property if and only if (C0, C1D1+ +C2D2 + ... + CkDk) ∈ Λ(2) +n×n has the SSM-W property for any set of non-negative diagonal matrix +(D1, D2, ..., Dk) ∈ Λ(k) +n×n with diag(D1 + D2 + ... + Dk) > 0. +Proof. Necessary part: Let (D1, D2..., Dk) ∈ Λ(k) +n×n be the set of non-negative diagonal matrix with +diag(D1 + D2 + ... + Dk) > 0. Suppose there exist vectors x0 ∈ Rn and y ∈ Rn ++ such that +C0x0 = +� +C1D1 + C2D2 + ... + CkDk +� +y and x0 ∗ y ≤ 0. +For each i ∈ [k], we set xi := Diy. As each Di is a non-negative diagonal matrix, from x0 ∗ y ≤ 0, +we get x0 ∗ xi ≤ 0 ∀i ∈ [k]. Then we have +C0x0 = C1x1 + C2x2 + ... + Ckxk, +xi ≥ 0, x0 ∗ xi ≤ 0 ∀i ∈ [k]. +As C has the SSM-W property of C, we must have x0 = x1 = ... = xk = 0. This implies +x1 + x2 + ... + xk = (D1 + D2 + ... + Dk)y = 0. +As diag(D1 + D2 + .... + Dk) > 0, we have +y = 0. This completes the necessary part. +Sufficiency part: Let x = (x0, x1, ..., xk) ∈ Λ(k+1) +n +such that +C0x0 = C1x1 + C2x2 + ... + Ckxk and xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k]. +(8) +We now consider an n × k matrix X whose jth column as xj for j ∈ [k]. So, X = [x1 x2 ... xk]. +Let S := {i ∈ [k] : ith row sum of X is zero}. From this, we define a vector y ∈ Rn and diagonal +matrices D1, D2, .., Dk such that +yi = +� +1 +i /∈ S +0 +i ∈ S +and (Dj)ii = +� +(xj)i +i /∈ S +1 +i ∈ S , +where (Dj)ii is the diagonal entry of Dj for all j ∈ [k]. It can be seen easily that Djy = xj for all +j ∈ [k] and each Dj is a non-negative diagonal matrix with diag(D1 + D2 + ...+ Dk) > 0. Therefore, +from equation 8, we get +C0x0 = +� +C1D1 + C2D2 + ... + CkDk +� +y, +x0 ∗ y ≤ 0. +From the hypothesis, we get x0 = 0 = y which implies x = 0. +This completes the sufficiency +part. +9 + +We now give a characterization for the column W-property. +Theorem 4.6. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +has the column-W property if and only if +(C0, C1D1 + C2D2 + ... + CkDk) ∈ Λ(2) +n×n has the column-W property for any set of non-negative +diagonal matrices D1, D2, ..., Dk of order n with diag(D1 + D2 + ... + Dk) > 0. +Proof. Necessary part: It is obvious. +Sufficiency part: Let {E0, E1, ..., Ek} be a set of non-negative diagonal matrices of order n such that +diag(E0 + E1 + ... + Ek) > 0. We claim that det(C0E0 + C1E1 + ... + CkEk) ̸= 0. +To prove this, we first construct a set of non-negative diagonal matrices D1, D2, ..., Dk and E as +follows: +(Dj)ii = +� +Ej +ii +if �k +m=1 Em +ii ̸= 0 +1 +if �k +m=1 Em +ii = 0 and Eii = +� +1 +if �k +m=1 Em +ii ̸= 0 +0 +if �k +m=1 Em +ii = 0 , +where (Dj)ii is iith diagonal entry of Dj for j ∈ [k] and Eii is iith diagonal entry of matrix E. +By an easy computation, we have DjE = Ej ∀j ∈ [k] and diag(D1 + D2 + ... + Dk) > 0. From +diag(E0 + E1 + ... + Ek) > 0, we get diag(E0 + E) > 0. As DjE = Ej ∀j ∈ [k] and (C0, C1D1 + +C2D2 + ... + CkDk) has column W-property, by Theorem 2.2, we have +det(C0E0 + C1E1 + ... + CkEk) = det(C0E0 + C1D1E + ... + CkDkE) += det(C0E0 + (C1D1 + ... + CkDk)E) ̸= 0. +Hence C has the column W-property. This completes the proof. +A well-known result in the standard LCP is that strictly semimonotone matrix and P matrix are +equivalent in the class of Z matrices (see, Theorem 3.11.10 in [1]). Analogue this result, we prove +the following theorem. +Theorem 4.7. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +such that C−1 +0 Ci be a Z matrix for all i ∈ [k]. +Then the following statements are equivalent. +(i) C has the column W-property. +(ii) C has the SSM-W property. +Proof. (i) =⇒ (ii): It follows from Theorem 4.3. +(ii) =⇒ (i): Let {D1, D2, ..., Dk} be the set of non-negative diagonal matrices of order n such +that diag(D1 + D2 + ... + Dk) > 0. In view of Theorem 4.6, it is enough to prove that (C0, C1D1 + +C2D2 + ... + CkDk) has the column W-property. +As C has the SSM-W property, by Theorem 4.5, we have (C0, C1D1 + ... + CkDk) has the +SSM-W property. So, by Proposition 4.1, +� +I, C−1 +0 +� +C1D1 + ... + CkDk +�� +has the SSM-W property +and C−1 +0 +� +C1D1 + C2D2 + ... + CkDk +� +is a strict semimonotone matrix. As C−1 +0 Ci is a Z matrix, we +get C−1 +0 +� +C1D1 + C2D2 + ... + CkDk +� +is also a Z matrix. Hence C−1 +0 +� +C1D1 + C2D2 + ... + CkDk +� +is a P matrix. So, by Theorem 2.3, (C0, C1D1 + C2D2 + ... + CkDk) has the column W-property. +Hence we have our claim. +Corollary 4.8. Let C = (C0, C1, ..., Ck) ∈ Λk+1 +n×n such that C−1 +0 Ci be a Z matrix for all i ∈ [k]. +Then the following statements are equivalent. +(i) C has the SSM-W property. +(ii) For all q ∈ Rn and d ∈ Λ(k−1) +n,++ , EHLCP(C, d, q) has a unique solution. +Proof. (i) +=⇒ (ii): It follows from Theorem 4.7 and Theorem 2.2. (ii) =⇒ (i): It follows from +Theorem 2.2 and Theorem 4.3. +10 + +In the standard LCP [3], the strictly semimonotone matrix gives the existence of a solution of +LCP. We now prove that the same result holds in EHLCP. +Theorem 4.9. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +has the SSM-W property, then SOL(C, d, q) ̸= ∅ +for all q ∈ Rn and d ∈ Λ(k+1) +n,++ . +Proof. As C has the SSM-W property, C has the R0-W property. From Theorem 4.1, it is enough +to prove that deg(C, 0) ̸= 0. To prove this, we consider a homotopy Φ : Λ(k+1) +n +× [0, 1] → Λ(k+1) +n +as +Φ(x, t) = t + + +C0x0 +x1 +x2 +x3 +. +. +. +xk + + ++ (1 − t) + + +C0x0 − �k +i=1 Cixi +x0 ∧ x1 +(d1 − x1) ∧ x2 +(d2 − x2) ∧ x3 +. +. +. +(dk−1 − xk−1) ∧ xk + + +. +Let F(x) := Φ(x, 0) and G(x) := Φ(x, 1). We first prove that the zero set X = {x : Φ(x, t) = +0 for some t ∈ [0, 1]} of homotopy Φ contains only zero. We consider the following cases. +Case 1: Suppose t = 0 or t = 1. If t = 0, then Φ(x, 0) = 0 =⇒ F(x) = 0. As C has the SSM-W +property, by Lemma 3.1, we have F(x) = 0 ⇒ x = 0. If t = 1, then Φ(x, 1) = 0 =⇒ G(x) = 0. +Again by C has the SSM-W property, C−1 +0 +exists, which implies that G is a one-one map. So, +G(x) = 0 ⇒ x = 0. +Case 2: Suppose t ∈ (0, 1). Then Φ(x, t) = 0 which gives that + + +C0x0 − �k +i=1 Cixi +x0 ∧ x1 +(d1 − x1) ∧ x2 +(d2 − x2) ∧ x3 +. +. +. +(dk−1 − xk−1) ∧ xk + + += −α + + +C0x0 +x1 +x2 +x3 +. +. +. +xk + + +, +where α = +t +1 − t > 0. +(9) +From the second row of above equation, we have +x0 ∧ x1 = −αx1 =⇒ min{x0 + αx1, (1 + α)x1} = 0. +By Proposition 2.1, we get x1 ≥ 0 and (x0 + αx1) ∗ (1 + α)x1 = 0 which implies that x0 ∗ x1 ≤ 0. +Set ∆ := {i ∈ [n] : (x1)i > 0}. So, we have +(x0)i = +� +≤ 0 +if i ∈ ∆ +≥ 0 +if i /∈ ∆ +and (x1)i = +� +> 0 if i ∈ ∆ += 0 if i /∈ ∆ +. +(10) +From third row of the equation 9, we have (d1 − x1) ∧ x2 = −αx2 which is equivalent +min{d1 − x1 + αx2, (1 + α)x2} = 0. +This gives that x2 ≥ 0 and (d1 − x1 + αx2) ∗ (1 + α)x2 = 0. As d1 > 0 and from the last term in +equation 10, we have +(x2)i = +� +≥ 0 if i ∈ ∆ += 0 if i /∈ ∆ +. +11 + +This leads that x0 ∗ x2 ≤ 0. By continuing the similar argument for the remaining rows, we get +xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k]. +From the first row of the equation 9, the vectors x = (x0, x1, ..., xk) satisfies +C0(1 + α)x0 = +k +� +i=1 +Cixi and xj ≥ 0, x0 ∗ xj ≤ 0, j ∈ [k]. +So, x = 0 as C has the SSM-W property. +From both cases, we get X contains only zero. By the homotopy invariance property of degree +(D2), we have deg(Φ(x, 0), Ω, 0) = deg +� +Φ(x, 1), Ω, 0 +� +for any bounded open set containing 0. As G +is a continuous one-one function, by Proposition 2.4, we have +deg +� +C, 0 +� += deg +� +Φ(x, 0), Ω, 0 +� += deg +� +F, Ω, 0 +� += deg +� +G, Ω, 0 +� +̸= 0. +This completes the proof. +We now recall that a matrix A ∈ Rn×n is said to be a M matrix if it is Z matrix and A−1(Rn ++) ⊆ +Rn ++. We prove a uniqueness result for EHLCP when q ≥ 0 and d ∈ Λ(k−1) +n,++ . +Theorem 4.10. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +has the SSM-W property. If C0 is a M matrix. +then for every q ∈ Rn ++ and for every d ∈ Λ(k−1) +n,++ , EHLCP(C, d, q) has a unique solution. +Proof. Let q ∈ Rn ++ and d = (d1, d2, ..., dk−1) ∈ Λ(k−1) +n,++ . We first show (C−1 +0 q, 0, ..., 0) ∈ SOL(C, d, q). +As C0 is a M matrix and q ∈ Rn ++, we have C−1 +0 q ≥ 0. If we set y = (y0, y1, ..., yk) := (C−1 +0 q, 0, ..., 0) ∈ +Λ(k+1) +n +, then we can see easily that (y0, y1, ..., yk) satisfies that +C0y0 = q + +k +� +i=1 +Ciyi, y0 ∧ y1 = 0 and (dj − yj) ∧ yj+1 = 0 ∀j ∈ [k − 1]. +Hence (C−1 +0 q, 0, ..., 0) ∈ SOL(C, d, q). +Suppose x = (x0, x1, ..., xk) ∈ Λ(k+1) +n +is an another solution to EHLCP(C, q, d). Then, +C0x0 = q + +k +� +i=1 +Cixi, x0 ∧ x1 = 0, (dj − xj) ∧ xj+1 = 0 ∀j ∈ [k − 1]. +(11) +From the Lemma 3.1, we have +C0x0 = q + +k +� +i=1 +Cixi and x0 ∧ xj = 0 ∀ j ∈ [k]. +(12) +We let z := x − y, then z = (x0 − C−1 +0 q, x1, x2, .., xk). By an easy computation, from Equation 12, +we get +C0(x0 − C−1 +0 q) = +k +� +i=1 +Cixi +and +xj ≥ 0, +(x0 − C−1 +0 q) ∗ xj = x0 ∗ xj − C−1 +0 q ∗ xj = −C−1 +0 q ∗ xj ≤ 0 ∀j ∈ [k]. +Since C has the SSM-W property, z = 0 which implies that (x0, x1, ..., xk) = (C−1 +0 q, 0, ..., 0). This +completes the proof. +12 + +5 +Connected solution set and Column W0 property +In this section, we give a necessary and sufficient condition for the connected solution set of the +EHLCP. +Definition 5. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n . We say that C is connected if SOL(C, d, q) is +connected for all q ∈ Rn and for all d ∈ Λ(k−1) +n,++ . +We now recall some definitions and results to proceed further. +Definition 6. [22] A subset of Rn is said to be a semi-algebraic set it can be represented as, +S = +s� +u=1 +ru +� +v=1 +{x ∈ Rn; fu,v(x) ∗uv 0}, +where for all u ∈ [s] and for all v ∈ [ru], ∗uv ∈ { >, =} and fu,v is in the space of all real polynomials. +Theorem 5.1 ([22]). Let S be a semi-algebraic set. Then S is connected iff S is path-connected. +Lemma 5.2. The SOL(C, d, q) is a semi-algebraic set. +Proof. It is clear from the definition of SOL(C, d, q). +The following result gives a necessary condition for a connected solution whenever C0 is a M +matrix. +Theorem 5.3. Let C0 ∈ Rn×n be a M matrix. If C = (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +is connected, then +SOL(C, d, q) = {(C−1 +0 q, 0, ..., 0)} for all q ∈ Rn +++ and for all d ∈ Λ(k−1) +n,++ . +Proof. Let q ∈ Rn +++ and d = (d1, d2, ..., dk−1) ∈ Λ(k−1) +n,++ . It can be seen from the proof of The- +orem 4.10 that x = (C−1 +0 q, 0, ..., 0) ∈ SOL(C, d, q). We now show that x is the only solution to +EHLCP(C, d, q). +Assume contrary. Suppose y is another solution to EHLCP(C, d, q). As SOL(C, d, q) is con- +nected, by Lemma 5.2 and Theorem 5.1, it is path-connected. So, there exists a path γ = (γ0, γ1, ..., γk) : +[0, 1] → SOL(C, d, q) such that +γ(0) = x, γ(1) = y and γ(t) ̸= x ∀t > 0. +Let {tm} ⊆ (0, 1) be a sequence such that tm → 0 as m → ∞. Then, by the continuity of γ, +γ(tm) → γ(0) = x as m → ∞. Since +� +γ0(tm), γ1(tm), ...γk(tm) +� +∈ SOL(C, d, q), +C0γ0(tm) = q + +k +� +i=1 +Ciγi(tm), +γ0(tm) ∧ γ1(tm) = 0 and +� +dj − γj(tm) +� +∧ γ(j+1)(tm) = 0 ∀j ∈ [k − 1]. +Now we claim that there exists a subsequence {tml} of {tm} such that +� +γj(tml) +� +i ̸= 0, for some j ∈ [k] and for some i ∈ [n]. +Suppose the claim is not true. This means that for given any subsequence {tml} of {tm}, there exists +m0 ∈ N such that for all ml ≥ m0, we have +� +γj(tml) +� +i = 0 ∀i ∈ [n] ∀j ∈ [k]. +13 + +So, γj(tm) is an eventually zero sequence for all j ∈ [k]. This implies that there exists a natural +number m0 such that +γ1(tm) = γ2(tm) = ... = γk(tm) = 0 ∀m ≥ m0. +As +� +γ0(tm), γ1(tm), ...γk(tm) +� +∈ SOL(C, d, q), we get γ0(tm) = C−1 +0 (q) +∀m ≥ m0. This gives us +that γ(tm) = x for all m ≥ m0 which contradicts the fact that γ(tm) ̸= x for all m. Therefore, our +claim is true. No loss of generality, we assume a sequence {tm} itself satisfies the condition +� +γj(tm) +� +i ̸= 0, for some j ∈ [k] and for some i ∈ [n]. +We now consider the following cases for possibilities of j. +Case 1 : If j = 1, then (γ0(tm))i(γ1(tm))i = 0 which leads to (γ0(tm))i = 0. This implies that +0 = lim +m→∞ γ0(tm)i = (C−1 +0 q)i. +But (C−1 +0 q) > 0 as C0 is a M matrix. This is not possible. So, j ̸= 1. +Case 2 : If 2 ≤ j ≤ k, then we have (dj−1 − γj−1(tm))i(γj(tm))i = 0 which gives that (dj−1 − +γj−1(tm))i = 0. By taking limit m → ∞, +0 = lim +m→∞(dj−1 − γj−1(tm))i = (dj−1)i − (γj−1(0))i = (dj−1)i > 0. +This is not possible. +From both cases, there is no such a j exists. This contradicts the fact. Hence x = (C−1 +0 q, 0, ..., 0) +is the only solution to EHLCP(C, d, q). +The following result gives a sufficient condition for a connected solution to EHLCP. +Theorem 5.4. Let C := (C0, C1, ..., Ck) ∈ Λ(k+1) +n×n +has the column W0-property. If SOL(C, d, q) has +a bounded connected component, then SOL(C, d, q) is connected. +Proof. If SOL(C, d, q) = ∅, then we have nothing to prove. Let SOL(C, d, q) ̸= ∅ and A be a con- +nected component of SOL(C, d, q). If SOL(C, d, q) = A, then we are done. Suppose SOL(C, d, q) ̸= +A. Then there exists y = (y0, y1, .., yk) ∈ SOL(C, d, q)\ A. As A is a bounded connected component +of SOL(C, d, q), we can find an open bounded set Ω ⊆ Λ(k+1) +n +which contains A and it does not +intersect with other component of SOL(C, d, q). Therefore y /∈ Ω and ∂(Ω) ∩ SOL(C, d, q) = ∅. +Since C has the column W0-property, there exists N := (N0, N1, ..., Nk) ∈ Λ(k+1) +n×n +such that +C + ǫN := (C0 + ǫN0, C1 + ǫN1, ..., Ck + ǫNk) has the column W-property for every ǫ > 0. +Let z = (z0, z1, ..., zk) ∈ A and ǫ > 0, we define functions H1, H2 and H3 as follows: +H1(x) = + + +C0x0 − �k +i=1 Cixi − q +x0 ∧ x1 +(d1 − x1) ∧ x2 +. +. +. +(dk−1 − xk−1) ∧ xk + + +, +H2(x) = + + +(C0 + ǫN0)x0 − �k +i=1(Ci + ǫNi)xi + (�k +i=1 ǫNiyi − ǫN0y0 − q) +x0 ∧ x1 +(d1 − x1) ∧ x2 +. +. +. +(dk−1 − xk−1) ∧ xk + + +, +14 + +H3(x) = + + +(C0 + ǫN0)x0 − �k +i=1(Ci + ǫNi)xi + (�k +i=1 ǫNizi − ǫN0z0 − q) +x0 ∧ x1 +(d1 − x1) ∧ x2 +. +. +. +(dk−1 − xk−1) ∧ xk + + +. +By putting x = y in H2(x), and x = z in H1(x) and H3(x), we get +H1(z) = H2(y) = H3(z) = 0. +For ǫ is near to zero, deg(H1, Ω, 0)= deg(H2, Ω, 0)= deg(H3, Ω, 0) due to the nearness property +of degree (D3). As z ∈ Ω is a solution to H3(x) = 0 and C + ǫN has the column W-property, +we get deg(H3, Ω, 0) ̸= 0 by Theorem 4.3 and 4.9. Since deg(H2, Ω, 0)= deg(H3, Ω, 0), we have +deg(H2, Ω, 0) ̸= 0. This implies that if we set q2 := q+ǫN0y0−�k +i=1 ǫNiyi, then EHLCP(C + ǫN, d, q2) +must have a solution in Ω. As C + ǫN has the column W-property, by Theorem 2.2, EHLCP(C + ǫN, d, q2) +has a unique solution which must be equal to y. So, y ∈ Ω. It gives us a contradiction. Hence +SOL(C, d, q) = A. Thus SOL(C, d, q) is connected. +6 +Conclusion +In this paper, we introduced the R0-W property and SSM-W properties and then studied the +existence and uniqueness result for EHLCP when the underlying set of matrices has these properties. +Last, we gave a necessary and sufficient condition for the connectedness of the solution set of the +EHLCP. +Declaration of Competing Interest +The authors have no competing interests. +Acknowledgements +The first author is a CSIR-SRF fellow, and he wants to thank the Council of Scientific & Industrial +Research(CSIR) for the financial support. +References +[1] R.W. Cottle, J.-S. Pang, R.E. Stone, The Linear Complementarity Problem, Classics in Applied +Mathematics, Philadelphia: SIAM ; 2009. +[2] Facchinei, F., Pang, J.S. Finite Dimensional Variational Inequalities and Complementarity +Problems. New York: Springer; 2003. +[3] M. S. Gowda.: Applications of degree theory to linear complementarity problems, Math. Oper. +Res. 18,868-879(1993) +[4] R. 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Berlin: +SpringerVerlag; (2006) +16 + diff --git a/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/load_file.txt b/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..16661fd0c1dcef1be47c204e85152806e9a70324 --- /dev/null +++ b/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/load_file.txt @@ -0,0 +1,874 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf,len=873 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='01479v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='OC] 4 Jan 2023 Generalizations of R0 and SSM properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Extended Horizontal Linear Complementarity Problem Punit Kumar Yadav Department of Mathematics Malaviya National Instiute of Technology, Jaipur, 302017, India E-mail address: punitjrf@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='com K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Palpandi Department of Mathematics Malaviya National Instiute of Technology, Jaipur, 302017, India E-mail address: kpalpandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='maths@mnit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='in Abstract In this paper, we first introduce R0-W and SSM-W property for the set of matrices which is a generalization of R0 and the strictly semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We then prove some existence results for the extended horizontal linear complementarity problem when the involved matrices have these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' With an additional condition on the set of matrices, we prove that the SSM-W property is equivalent to the unique solution for the corresponding extended horizontal linear complementarity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Finally, we give a necessary and sufficient condition for the connectedness of the solution set of the extended horizontal linear complementarity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 1 Introduction The standard linear complementarity problem (for short LCP), LCP(C, q), is to find vectors x, y such that x ∈ Rn, y = Cx + q ∈ Rn and x ∧ y = 0, (1) where C ∈ Rn×n, q ∈ Rn and ′∧′ is a min map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The LCP has numerous applications in numerous domains, such as optimization, economics, and game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Cottle and Pang’s monograph [1] is the primary reference for standard LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Various generalisations of the linear complementarity problem have been developed and discussed in the literature during the past three decades (see, [7, 10, 11, 13, 14, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The extended horizontal linear complementarity problem is one of the most important extensions of LCP, which various authors have studied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' see [4, 6, 7] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For a given ordered set of matrices C := {C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck} ⊆ Rn×n, vector q ∈ Rn and ordered set of positive vectors d := {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk} ⊆ Rn, the extended horizontal linear complementarity problem (for short EHLCP), denoted by EHCLP(C, d, q), is to find a vector x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk ∈ Rn such that C0x0 =q + k � i=1 Cixi, x0 ∧ x1 = 0 and (dj−xj) ∧ xj+1 = 0, 1 ≤ j ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (2) If k = 1, then EHLCP becomes the horizontal linear complementarity problem (for short HLCP), that is, C0x0 − C1x1 = q and x0 ∧ x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Further, HLCP reduces to the standard LCP by taking C0 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Due to its widespread applications in numerous domains, the horizontal linear complementarity problem has received substantial research attention from many academics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' see [13, 14, 16, 18] and reference therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Various writers have presented new classes of matrices for analysing the structure of LCP solution sets in recent years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' see for example, [1, 2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The classes of R0, P0, P, and strictly semimonotone 1 (SSM) matrices play a crucial role in the existence and uniqueness of the solution to LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For instance, P matrix (if [x ∈ Rn, x ∗ Ax ≤ 0 =⇒ x = 0]) gives a necessary and sufficient condition for the uniqueness of the solution for the LCP (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To get a similar type of existence and uniqueness results for the generalized LCPs, the notion of P matrix was extended for the set of matrices as the column W-property by Gowda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' They proved that column W- property gives the solvability and the uniqueness for the extended horizontal linear complementarity problem (EHLCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Also, they have generalized the concept of the P0-matrix as the column W0- property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Another class of matrix, the so-called SSM matrix, has importance in LCP theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This class of matrices provides a unique solution to LCP on Rn + and also gives the existence of the solution for the LCP (see, [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For a Z matrix (if all the off-diagonal entries of a matrix are non-positive), P matrix is equivalent to the SSM matrix (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' A natural question arises whether the SSM matrix can be generalized for the set of matrices in the view of EHLCP and whether we have a similar equivalence relation for the set of Z matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In this paper, we would like to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The connectedness of the solution set of LCP has a prominent role in the study of the LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say a matrix is connected if the solution set of the corresponding LCP is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In [19], Jones and Gowda addressed the connectedness of the solution set of the LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' They proved that the matrix is connected whenever the given matrix is a P0 matrix and the solution set has a bounded connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Also, they have shown that if the solution set of LCP is connected, then there is almost one solution of LCP for all q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Due to the specially structured matrices involved in the study of the connectedness of the solution to LCP, various authors studied the connectedness of LCP, see for example [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The main objectives of this paper are to answer the following questions: (Q1) In LCP theory, it is a well-known result that the R0 matrix gives boundedness to the LCP solution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The same holds true for HLCP [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This motivates the question of whether or not the notion of R0 matrix can be generalized to the set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If so, then can we expect the same kind of outcome in the EHLCP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (Q2) Given that a strictly semimonotone matrix guarantees the existence of the LCP solution and its uniqueness for q ≥ 0, it is natural to wonder whether the concept of SSM matrix can be extended to the set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If so, then whether the same result holds true for EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (Q3) Motivated by the results of Gowda and Jones [19] regarding the connectedness of the solution set of LCP, one can ask whether the solution set of EHLCP is connected if the set of matrices has the column W0 property and the solution set of the corresponding EHLCP has a bounded connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The paper’s outline is as follows: We present some basic definitions and results in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We generalize the concept of R0 matrix and prove the existence result for EHLCP in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In section 4, we introduce the SSM-W property, and we then study an existence and uniqueness result for the EHLCP when the underlying set of matrices have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In the last section, we give a necessary and sufficient condition for the connectedness of the solution set of the EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 2 Notations and Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 Notations Throughout this paper, we use the following notations: (i) The n dimensional Euclidean space with the usual inner product will be denoted by Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The set of all non-negative vectors (respectively, positive vectors) in Rn will be denoted by Rn + 2 (respectively, Rn ++ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say x ≥ 0 (respectively, > 0) if and only if x ∈ Rn + (respectively, Rn ++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) The k-ary Cartesian power of Rn will be denoted by Λ(k) n and the k-ary Cartesian power of Rn ++ will be denoted by Λ(k) n,++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The bold zero ’0’ will be used for denoting the zero vector (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ Λ(k) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) The set of all n×n real matrices will be denoted by Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We use the symbol Λ(k) n×n to denote the k-ary Cartesian product of Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iv) We use [n] to denote the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (v) Let M ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We use diag(M) to denote the vector (M11, M22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Mkk) ∈ Rn, where Mii is the iith diagonal entry of matrix M and det(M) is used to denote the determinant of matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (vi) SOL(C, d, q) will be used for denoting the set of all solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall some definitions and results from the LCP theory, which will be used frequently in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let V = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) x ∧ y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) x, y ≥ 0 and x ∗ y = 0, where ∗ is the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) x, y ≥ 0 and ⟨x, y⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 1 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then a matrix R ∈ Rn×n is column representative of C if R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j ∈ � (C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j, (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', (Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j � , ∀j ∈ [n], where R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j is the jth column of matrix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Next, we define the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 2 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C := (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we say that C has the (i) column W-property if the determinants of all the column representative matrices of C are all positive or all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) column W0-property if there exists N := (N0, N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Nk) ∈ Λ(k+1) n×n such that C + ǫN := (C0 + ǫN0, C1 + ǫN1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck + ǫNk) has the column W-property for all ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Due to Gowda and Sznajder [4], we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n , the following are equivalent: (i) C has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) For arbitrary non-negative diagonal matrices D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk ∈ Rn×n with diag(D0 +D1 +D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0, det � C0D0 + C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) C0 is invertible and (I, C−1 0 C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', C−1 0 Ck) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 3 (iv) For all q ∈ Rn and d ∈ Λ(k−1) n,++ , EHLCP(C, d, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If k = 1 and C−1 0 exists, then HLCP(C0, C1, q) is equivalent to LCP(C−1 0 C1, C−1 0 (q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In this case, C−1 0 C1 is a P matrix if and only if for all q ∈ Rn, LCP(C−1 0 C1, C−1 0 (q)) has a unique solution (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence we have the following theorem given the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let (C0, C1) ∈ Λ(2) n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) (C0, C1) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) C0 is invertible and C−1 0 C1 is a P matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) For all q ∈ Rn, HLCP(C0, C1, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 Degree theory We now recall the definition and some properties of a degree from [2, 3] for our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let Ω be an open bounded set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose h : ¯Ω → Rn is a continuous function and a vector p /∈ h(∂Ω), where ∂Ω and ¯Ω denote the boundary and closure of Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the degree of h is defined with respect to p over Ω denoted by deg(h, Ω, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The equation h(x) = p has a solution whenever deg(h, Ω, p) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If h(x) = p has only one solution, say y in Rn, then the degree is the same overall bounded open sets containing y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This common degree is denoted by deg(h, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 Properties of the degree The following properties are used frequently here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (D1) deg(I, Ω, ·) = 1, where I is the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (D2) Homotopy invariance: Let a homotopy Φ(x, s) : Rn ×[0, 1] → Rn be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If the zero set of Φ(x, s), X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded, then for any bounded open set Ω in Rn containing the zero set X, we have deg(Φ(x, 1), Ω, 0) = deg(Φ(x, 0), Ω, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (D3) Nearness property: Assume deg(h1(x), Ω, p) is defined and h2 : ¯Ω → Rn is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If supx∈Ω∥h2(x) − h1(x)∥ < dist(p, ∂Ω), then deg(h2(x), Ω, p) is defined and equals to deg(h1(x), Ω, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following result from Facchinei and Pang [2] will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4 ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let Ω be a non-empty, bounded open subset of Rn and let Φ : ¯Ω → Rn be a continuous injective mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then deg(Φ, Ω, p) ̸= 0 for all p ∈ Φ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Note: All the degree theoretic results and concepts are also applicable over any finite dimensional Hilbert space (like Rn or Rn × Rn × Rn etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 3 R0-W property In this section, we first define the R0-W property for the set of matrices which is a natural generaliza- tion of R0 matrix in the LCP theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We then show that the R0-W property gives the boundedness of the solution set of the corresponding EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 4 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say that C has the R0-W property if the system C0x0 = k � i=1 Cixi and x0 ∧ xj = 0 ∀ j ∈ [k] has only zero solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It can be seen easily that the R0-W property coincides with R0 matrix when k = 1 and C0 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Also it is noted (see, [8]) that if k = 1, then the R0-W property referred as R0 pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To proceed further, we prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n and x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then x satisfies the following system C0x0 = q + k � i=1 Cixi and x0 ∧ xj = 0 ∀ j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As x0 ≥ 0, there exists an index set α ⊆ [n] such that (x0)i = � > 0 i ∈ α 0 i ∈ [n] \\ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since x0 ∧ x1 = 0, we have (x1)i = 0 for all i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From (d1 − x1) ∧ x2 = 0, we get (d1)i(x2)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives that (x2)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By substituting (x2)i = 0 ∀i ∈ α in (d2 − x2) ∧ x3 = 0, we obtain (x3)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Continue the process in the similar way, one can get (x4)i = (x5)i = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' = (xk)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, x0 ∧ xj = 0 ∀ j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now prove the boundedness of the solution set of EHLCP when the involved set of matrices has the R0-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C has the R0-W property then SOL(C, d, q) is bounded for every q ∈ Rn and d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exist q ∈ Rn and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk−1) ∈ Λ(k−1) n,++ such that SOL(C, d, q) is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then there exists a sequence x(m) = (x(m) 0 , x(m) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', x(m) k ) in Λ(k+1) n such that ||x(m)|| → ∞ as m → ∞ and it satisfies C0x(m) 0 = q + k � i=1 Cix(m) i x(m) 0 ∧ x(m) 1 = 0 and (dj − x(m) j ) ∧ x(m) j+1 = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (3) From the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, equation 3 gives that C0x(m) 0 =q + k � i=1 Cix(m) i and x(m) 0 ∧ x(m) j = 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (4) As x(m) ∥x(m)∥ is a unit vector for all m, x(m) ∥x(m)∥ converges to some vector y = (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', yk) ∈ Λ(k+1) n with ||y|| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Now first divide the equation 4 by ∥x(m)∥ and then take the limit m → ∞, we get C0y0 = k � i=1 Ciyi and y0 ∧ yj = 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that y must be a zero vector as C has the R0-W property, which contradicts the fact that ||y|| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore SOL(C, d, q) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 Degree of EHLCP Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., dk−1) ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We define a function F : Λ(k+1) n → Λ(k+1) n as F(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (5) We denote the degree of F with respect to 0 over bounded open set Ω ⊆ Λ(k+1) n as deg(C, Ω, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is noted that if C has the R0-W property, in view of the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, F(x) = 0 ⇔ x = 0 which implies that deg(C, Ω, 0) = deg(C, 0) for any bounded open set Ω contains the origin in Λ(k+1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We call this degree as EHLCP-degree of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now prove an existence result for EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose the following hold: (i) C has the R0-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) deg(C, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then EHLCP(C, d, q) has non-empty compact solution for all q ∈ Rn and d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As the solution set of EHLCP is closed, it is enough to prove that the solution set is non-empty and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We first define a homotopy Φ : Λ(k+1) n × [0, 1] → Λ(k+1) n as Φ(x, s) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi − sq x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, Φ(x, 0) = F(x) and Φ(x, 1) = F(x) − ˆq, where ˆq = (q, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='0) ∈ Λ(k+1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By using the similar argument as in above Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2, we can easily show that the zero set of homotopy, X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From the property of degree (D2), we get deg(F, Ω, 0) = deg(F − ˆq, Ω, 0) for any open bounded set Ω containing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As deg(F, Ω, 0) = deg(C, 0) ̸= 0, we obtain deg(F − ˆq, Ω, 0) ̸= 0 which implies SOL(C, d, q) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the R0-W property, by Theorem 3, SOL(C, d, q) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 4 SSM-W property In this section, we first define the SSM-W property for the set of matrices which is a generalization of the SSM matrix in the LCP theory, and we then prove that the existence and uniqueness result for the EHLCP when the involved set of matrices have the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall that an n × n real matrix M is called strictly semimonotone (SSM) matrix if [x ∈ Rn +, x ∗ Mx ≤ 0 ⇒ x = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We generalize this concept to the set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 6 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say that C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property if {C0x0 = k � i=1 Cixi, xi ≥ 0 and x0 ∗ xi ≤ 0 ∀i ∈ [k]} ⇒ x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., xk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C has the SSM-W property, then the followings hold: (i) C−1 0 exists and C−1 0 Ci is a strict semimonotone matrix for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) (I, C−1 0 C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', C−1 0 Ck) has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) (P T C0P, P T C1P, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', P T CkP) has the SSM-W property for any permutation matrix P of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i): Suppose there exists a vector x0 ∈ Rn such that C0x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we have C0x0 = C10 + C20 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ck0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives that x0 = 0 as C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus C0 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Now we prove the second part of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Without loss of generality, it is enough to prove that C−1 0 C1 is a strictly semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exists a vector y ∈ Rn such that y ≥ 0 and y ∗ (C−1 0 C1)y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let y0 := (C−1 0 C1)y, y1 := y and yi := 0 for all 2 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we get C0y0 = C1y1 + C2y2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ciyi + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='. + Ckyk, yj ≥ 0 and y0 ∗ yj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since C has the SSM-W property, yj = 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus C−1 0 C1 is a strict semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii): It follows from the definition of the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii): Let x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n such that (P T C0P)x0 = k � i=1 (P T CiP)xi, xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As P is a non-negative matrix and PP T = P T P, we can rewrite the above equation as C0Px0 = k � i=1 CiPxi, Pxj ≥ 0 and Px0 ∗ Pxj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By the SSM-W property of C, Pxj = 0 for all 0 ≤ j ≤ k which implies x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In the above Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, it can be seen easily that the converse of the item (ii) and (iii) are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' But the converse of item (i) need not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following example illustrates this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, C2) ∈ Λ(3) 2×2, where C0 = � 1 0 0 1 � , C1 = � 1 −2 0 1 � , C2 = � 1 0 −2 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is easy to check that C−1 0 C1 = C1 and C−1 0 C2 = C2 are P matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, C−1 0 C1 and C−1 0 C2 are SSM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let x = (x0, x1, x2) = ((0, 0)T , (1, 1)T , (1, 1)T ) ∈ Λ(3) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we can see that the non-zero x satisfies C0x0 = C1x1 + C2x2, x1 ≥ 0, x2 ≥ 0 and x0 ∗ x1 = 0 = x0 ∗ x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So C can not have the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 7 The following result is a generalization of a well-known result in matrix theory that every P matrix is a SSM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C has the column W-property, then C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exists a non-zero vector x = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n such that C0x0 = k � i=1 Cixi, xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Consider a vector y ∈ Rn whose jth component is given by yj = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −1 if (x0)j > 0 1 if (x0)j < 0 1 if (x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k] 0 if (x0)j = 0 and (xi)j = 0 for all i ∈ [k] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As x is a non-zero vector, y must be a non-zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Consider the diagonal matrices D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk which are defined by (D0)jj = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (x0)j if (x0)j > 0 −(x0)j if (x0)j < 0 0 if(x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k] 1 if (x0)j = 0 and (xi)j = 0 for all i ∈ [k] and for all i ∈ [k], (Di)jj = � 0 if (x0)j > 0 (xi)j else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is easy to verify that D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk are non-negative diagonal matrices and diag(D0 + D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' And also note that x0 = −D0y and xi = Diy ∀i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (6) By substituting the Equation 6 in C0x0 = �k i=1 Cixi, we get C0(−D0y) = k � i=1 CiDi(y) ⇒ � C0D0 + C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that det(C0D0 + C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, C does not have the column W-property from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore, C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following example illustrates that the converse of the above theorem is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, C2) ∈ Λ(3) 2×2 such that C0 = �1 0 0 1 � , C1 = �1 1 1 1 � , C2 = �1 1 1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose w = (x, y, z) ∈ Λ3 2 such that C0x = C1y + C2z and y, z ≥ 0, x ∗ y ≤ 0, x ∗ z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 8 From C0x = C1y + C2z, we get �x1 x2 � = �y1 + y2 + z1 + z2 y1 + y2 + z1 + z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As x ∗ y ≤ 0, x ∗ z ≤ 0 and from the above equation, we have y1(y1 + y2 + z1 + z2) ≤ 0 and y2(y1 + y2 + z1 + z2) ≤ 0, z1(y1 + y2 + z1 + z2) ≤ 0 and z2(y1 + y2 + z1 + z2) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (7) Since y, z ≥ 0, from the equation 7, we get x = y = z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As det(C1) = 0, by the definition of the column W-property, C does not have the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now give a characterization for SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property if and only if (C0, C1D1+ C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) ∈ Λ(2) n×n has the SSM-W property for any set of non-negative diagonal matrix (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk) ∈ Λ(k) n×n with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Necessary part: Let (D1, D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk) ∈ Λ(k) n×n be the set of non-negative diagonal matrix with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exist vectors x0 ∈ Rn and y ∈ Rn + such that C0x0 = � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � y and x0 ∗ y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For each i ∈ [k], we set xi := Diy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As each Di is a non-negative diagonal matrix, from x0 ∗ y ≤ 0, we get x0 ∗ xi ≤ 0 ∀i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we have C0x0 = C1x1 + C2x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ckxk, xi ≥ 0, x0 ∗ xi ≤ 0 ∀i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property of C, we must have x0 = x1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' = xk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies x1 + x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + xk = (D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk)y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='. + Dk) > 0, we have y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the necessary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Sufficiency part: Let x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n such that C0x0 = C1x1 + C2x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ckxk and xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (8) We now consider an n × k matrix X whose jth column as xj for j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, X = [x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' xk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let S := {i ∈ [k] : ith row sum of X is zero}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From this, we define a vector y ∈ Rn and diagonal matrices D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., Dk such that yi = � 1 i /∈ S 0 i ∈ S and (Dj)ii = � (xj)i i /∈ S 1 i ∈ S , where (Dj)ii is the diagonal entry of Dj for all j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It can be seen easily that Djy = xj for all j ∈ [k] and each Dj is a non-negative diagonal matrix with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='+ Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore, from equation 8, we get C0x0 = � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � y, x0 ∗ y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From the hypothesis, we get x0 = 0 = y which implies x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the sufficiency part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 9 We now give a characterization for the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the column-W property if and only if (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) ∈ Λ(2) n×n has the column-W property for any set of non-negative diagonal matrices D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk of order n with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Necessary part: It is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Sufficiency part: Let {E0, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ek} be a set of non-negative diagonal matrices of order n such that diag(E0 + E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ek) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We claim that det(C0E0 + C1E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkEk) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To prove this, we first construct a set of non-negative diagonal matrices D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk and E as follows: (Dj)ii = � Ej ii if �k m=1 Em ii ̸= 0 1 if �k m=1 Em ii = 0 and Eii = � 1 if �k m=1 Em ii ̸= 0 0 if �k m=1 Em ii = 0 , where (Dj)ii is iith diagonal entry of Dj for j ∈ [k] and Eii is iith diagonal entry of matrix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By an easy computation, we have DjE = Ej ∀j ∈ [k] and diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From diag(E0 + E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ek) > 0, we get diag(E0 + E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As DjE = Ej ∀j ∈ [k] and (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has column W-property, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2, we have det(C0E0 + C1E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkEk) = det(C0E0 + C1D1E + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDkE) = det(C0E0 + (C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk)E) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence C has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' A well-known result in the standard LCP is that strictly semimonotone matrix and P matrix are equivalent in the class of Z matrices (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Analogue this result, we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n such that C−1 0 Ci be a Z matrix for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) C has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) =⇒ (ii): It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) =⇒ (i): Let {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk} be the set of non-negative diagonal matrices of order n such that diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='6, it is enough to prove that (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='5, we have (C0, C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, � I, C−1 0 � C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk �� has the SSM-W property and C−1 0 � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � is a strict semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C−1 0 Ci is a Z matrix, we get C−1 0 � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � is also a Z matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence C−1 0 � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � is a P matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3, (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence we have our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λk+1 n×n such that C−1 0 Ci be a Z matrix for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) For all q ∈ Rn and d ∈ Λ(k−1) n,++ , EHLCP(C, d, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) =⇒ (ii): It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) =⇒ (i): It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 10 In the standard LCP [3], the strictly semimonotone matrix gives the existence of a solution of LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now prove that the same result holds in EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property, then SOL(C, d, q) ̸= ∅ for all q ∈ Rn and d ∈ Λ(k+1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property, C has the R0-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, it is enough to prove that deg(C, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To prove this, we consider a homotopy Φ : Λ(k+1) n × [0, 1] → Λ(k+1) n as Φ(x, t) = t \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 x1 x2 x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + (1 − t) \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let F(x) := Φ(x, 0) and G(x) := Φ(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We first prove that the zero set X = {x : Φ(x, t) = 0 for some t ∈ [0, 1]} of homotopy Φ contains only zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 1: Suppose t = 0 or t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If t = 0, then Φ(x, 0) = 0 =⇒ F(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, we have F(x) = 0 ⇒ x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If t = 1, then Φ(x, 1) = 0 =⇒ G(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Again by C has the SSM-W property, C−1 0 exists, which implies that G is a one-one map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, G(x) = 0 ⇒ x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 2: Suppose t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then Φ(x, t) = 0 which gives that \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = −α \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 x1 x2 x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , where α = t 1 − t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (9) From the second row of above equation, we have x0 ∧ x1 = −αx1 =⇒ min{x0 + αx1, (1 + α)x1} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, we get x1 ≥ 0 and (x0 + αx1) ∗ (1 + α)x1 = 0 which implies that x0 ∗ x1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Set ∆ := {i ∈ [n] : (x1)i > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, we have (x0)i = � ≤ 0 if i ∈ ∆ ≥ 0 if i /∈ ∆ and (x1)i = � > 0 if i ∈ ∆ = 0 if i /∈ ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (10) From third row of the equation 9, we have (d1 − x1) ∧ x2 = −αx2 which is equivalent min{d1 − x1 + αx2, (1 + α)x2} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives that x2 ≥ 0 and (d1 − x1 + αx2) ∗ (1 + α)x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As d1 > 0 and from the last term in equation 10, we have (x2)i = � ≥ 0 if i ∈ ∆ = 0 if i /∈ ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 11 This leads that x0 ∗ x2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By continuing the similar argument for the remaining rows, we get xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From the first row of the equation 9, the vectors x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) satisfies C0(1 + α)x0 = k � i=1 Cixi and xj ≥ 0, x0 ∗ xj ≤ 0, j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, x = 0 as C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From both cases, we get X contains only zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By the homotopy invariance property of degree (D2), we have deg(Φ(x, 0), Ω, 0) = deg � Φ(x, 1), Ω, 0 � for any bounded open set containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As G is a continuous one-one function, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4, we have deg � C, 0 � = deg � Φ(x, 0), Ω, 0 � = deg � F, Ω, 0 � = deg � G, Ω, 0 � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall that a matrix A ∈ Rn×n is said to be a M matrix if it is Z matrix and A−1(Rn +) ⊆ Rn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We prove a uniqueness result for EHLCP when q ≥ 0 and d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C0 is a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' then for every q ∈ Rn + and for every d ∈ Λ(k−1) n,++ , EHLCP(C, d, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let q ∈ Rn + and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk−1) ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We first show (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C0 is a M matrix and q ∈ Rn +, we have C−1 0 q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If we set y = (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', yk) := (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ Λ(k+1) n , then we can see easily that (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', yk) satisfies that C0y0 = q + k � i=1 Ciyi, y0 ∧ y1 = 0 and (dj − yj) ∧ yj+1 = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n is an another solution to EHLCP(C, q, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, C0x0 = q + k � i=1 Cixi, x0 ∧ x1 = 0, (dj − xj) ∧ xj+1 = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (11) From the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, we have C0x0 = q + k � i=1 Cixi and x0 ∧ xj = 0 ∀ j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (12) We let z := x − y, then z = (x0 − C−1 0 q, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By an easy computation, from Equation 12, we get C0(x0 − C−1 0 q) = k � i=1 Cixi and xj ≥ 0, (x0 − C−1 0 q) ∗ xj = x0 ∗ xj − C−1 0 q ∗ xj = −C−1 0 q ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since C has the SSM-W property, z = 0 which implies that (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) = (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 12 5 Connected solution set and Column W0 property In this section, we give a necessary and sufficient condition for the connected solution set of the EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say that C is connected if SOL(C, d, q) is connected for all q ∈ Rn and for all d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall some definitions and results to proceed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' [22] A subset of Rn is said to be a semi-algebraic set it can be represented as, S = s� u=1 ru � v=1 {x ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' fu,v(x) ∗uv 0}, where for all u ∈ [s] and for all v ∈ [ru], ∗uv ∈ { >, =} and fu,v is in the space of all real polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let S be a semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then S is connected iff S is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The SOL(C, d, q) is a semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is clear from the definition of SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following result gives a necessary condition for a connected solution whenever C0 is a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C0 ∈ Rn×n be a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n is connected, then SOL(C, d, q) = {(C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0)} for all q ∈ Rn ++ and for all d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let q ∈ Rn ++ and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk−1) ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It can be seen from the proof of The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10 that x = (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now show that x is the only solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Assume contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose y is another solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As SOL(C, d, q) is con- nected, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, it is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, there exists a path γ = (γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', γk) : [0, 1] → SOL(C, d, q) such that γ(0) = x, γ(1) = y and γ(t) ̸= x ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let {tm} ⊆ (0, 1) be a sequence such that tm → 0 as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, by the continuity of γ, γ(tm) → γ(0) = x as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since � γ0(tm), γ1(tm), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='γk(tm) � ∈ SOL(C, d, q), C0γ0(tm) = q + k � i=1 Ciγi(tm), γ0(tm) ∧ γ1(tm) = 0 and � dj − γj(tm) � ∧ γ(j+1)(tm) = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Now we claim that there exists a subsequence {tml} of {tm} such that � γj(tml) � i ̸= 0, for some j ∈ [k] and for some i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose the claim is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This means that for given any subsequence {tml} of {tm}, there exists m0 ∈ N such that for all ml ≥ m0, we have � γj(tml) � i = 0 ∀i ∈ [n] ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 13 So, γj(tm) is an eventually zero sequence for all j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that there exists a natural number m0 such that γ1(tm) = γ2(tm) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' = γk(tm) = 0 ∀m ≥ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As � γ0(tm), γ1(tm), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='γk(tm) � ∈ SOL(C, d, q), we get γ0(tm) = C−1 0 (q) ∀m ≥ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives us that γ(tm) = x for all m ≥ m0 which contradicts the fact that γ(tm) ̸= x for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore, our claim is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' No loss of generality, we assume a sequence {tm} itself satisfies the condition � γj(tm) � i ̸= 0, for some j ∈ [k] and for some i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now consider the following cases for possibilities of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 1 : If j = 1, then (γ0(tm))i(γ1(tm))i = 0 which leads to (γ0(tm))i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that 0 = lim m→∞ γ0(tm)i = (C−1 0 q)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' But (C−1 0 q) > 0 as C0 is a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, j ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 2 : If 2 ≤ j ≤ k, then we have (dj−1 − γj−1(tm))i(γj(tm))i = 0 which gives that (dj−1 − γj−1(tm))i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By taking limit m → ∞, 0 = lim m→∞(dj−1 − γj−1(tm))i = (dj−1)i − (γj−1(0))i = (dj−1)i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From both cases, there is no such a j exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This contradicts the fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence x = (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) is the only solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following result gives a sufficient condition for a connected solution to EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C := (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the column W0-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If SOL(C, d, q) has a bounded connected component, then SOL(C, d, q) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If SOL(C, d, q) = ∅, then we have nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let SOL(C, d, q) ̸= ∅ and A be a con- nected component of SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If SOL(C, d, q) = A, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose SOL(C, d, q) ̸= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then there exists y = (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., yk) ∈ SOL(C, d, q)\\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As A is a bounded connected component of SOL(C, d, q), we can find an open bounded set Ω ⊆ Λ(k+1) n which contains A and it does not intersect with other component of SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore y /∈ Ω and ∂(Ω) ∩ SOL(C, d, q) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since C has the column W0-property, there exists N := (N0, N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Nk) ∈ Λ(k+1) n×n such that C + ǫN := (C0 + ǫN0, C1 + ǫN1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck + ǫNk) has the column W-property for every ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let z = (z0, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', zk) ∈ A and ǫ > 0, we define functions H1, H2 and H3 as follows: H1(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi − q x0 ∧ x1 (d1 − x1) ∧ x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , H2(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 (C0 + ǫN0)x0 − �k i=1(Ci + ǫNi)xi + (�k i=1 ǫNiyi − ǫN0y0 − q) x0 ∧ x1 (d1 − x1) ∧ x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , 14 H3(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 (C0 + ǫN0)x0 − �k i=1(Ci + ǫNi)xi + (�k i=1 ǫNizi − ǫN0z0 − q) x0 ∧ x1 (d1 − x1) ∧ x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By putting x = y in H2(x), and x = z in H1(x) and H3(x), we get H1(z) = H2(y) = H3(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For ǫ is near to zero, deg(H1, Ω, 0)= deg(H2, Ω, 0)= deg(H3, Ω, 0) due to the nearness property of degree (D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As z ∈ Ω is a solution to H3(x) = 0 and C + ǫN has the column W-property, we get deg(H3, Ω, 0) ̸= 0 by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since deg(H2, Ω, 0)= deg(H3, Ω, 0), we have deg(H2, Ω, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that if we set q2 := q+ǫN0y0−�k i=1 ǫNiyi, then EHLCP(C + ǫN, d, q2) must have a solution in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C + ǫN has the column W-property, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2, EHLCP(C + ǫN, d, q2) has a unique solution which must be equal to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It gives us a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence SOL(C, d, q) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus SOL(C, d, q) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 6 Conclusion In this paper, we introduced the R0-W property and SSM-W properties and then studied the existence and uniqueness result for EHLCP when the underlying set of matrices has these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Last, we gave a necessary and sufficient condition for the connectedness of the solution set of the EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Declaration of Competing Interest The authors have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Acknowledgements The first author is a CSIR-SRF fellow, and he wants to thank the Council of Scientific & Industrial Research(CSIR) for the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Cottle, J.' metadata={'source': 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Algorithms in Real Algebraic Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Berlin: SpringerVerlag;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (2006) 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} diff --git a/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf b/-tAyT4oBgHgl3EQfdfed/content/2301.00304v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9b76721bbc283490288d9f2c0ad8a5b669d44ac8 --- /dev/null +++ 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sha256:d8f9349fc2ea3e188bdcf643423e684ad7989e4aaf64cd047170ec13e01169b4 +size 158819 diff --git a/-tE1T4oBgHgl3EQfUwM8/content/tmp_files/2301.03093v1.pdf.txt b/-tE1T4oBgHgl3EQfUwM8/content/tmp_files/2301.03093v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..194fc162122702d9fc2008a78195277238961c34 --- /dev/null +++ b/-tE1T4oBgHgl3EQfUwM8/content/tmp_files/2301.03093v1.pdf.txt @@ -0,0 +1,712 @@ +2019 22nd International Conference on Computer and Information Technology (ICCIT), 18-20 December 2019 +978-1-7281-5842-6/19/$31.00 ©2019 IEEE + +Prognosis and Treatment Prediction of Type-2 +Diabetes Using Deep Neural Network and Machine +Learning Classifiers +Md. Kowsher +Dept. of Applied Mathematics +Noakhali Science and Technology +University, Noakhali-3814,Bangladesh. +ga.kowsher@gmail.com + +Mahbuba Yesmin Turaba +Dept. of Information and +Communication Technology +Comilla University +Comilla, Bangladesh +mahbuba.yesmin11@gmail.com + +M M Mahabubur Rahman +Dept. of CSTE +Noakhali Science and Technology +University Noakhali-3814, Bangladesh +toufikrahman098@gmail.com + Tanvir Sajed +Dept. of Computing Science +University of Alberta +Edmonton, Canada +tsajed@ualbarta.ca + + + Abstract—Type 2 Diabetes is a fast-growing, chronic +metabolic disorder due to imbalanced insulin activity. As lots +of people are suffering from it, access to proper treatment is +necessary to control the problem. Most patients are unaware of +health complexity, symptoms and risk factors before diabetes. +The motion of this research is a comparative study of seven +machine learning classifiers and an artificial neural network +method to prognosticate the detection and treatment of +diabetes with a high accuracy, in order to identify and treat +diabetes patients at an early age. Our training and test dataset +is an accumulation of 9483 diabetes patients’ information. The +training dataset is large enough to negate overfitting and +provide for highly accurate test performance. We use +performance measures such as accuracy and precision to find +out the best algorithm deep ANN which outperforms with +95.14% accuracy among all other tested machine learning +classifiers. We hope our high performing model can be used by +hospitals to predict diabetes and drive research into more +accurate prediction models. +Keywords—Artificial Neural Network, Type 2 diabetes, Support +Vector Machine, Decision Tree, Naive Bayes, LDA, Random +forest classifier +I. +INTRODUCTION +Diabetes Mellitus (DM) is a very common metabolic +disorder that affects millions of people worldwide. It occurs +when the concentration of blood glucose reaches excessive +level due to lack of production of insulin by the pancreas +organ (Type 1 Diabetes) or due to insulin resistance (Type 2 +Diabetes) [1]. It has been published that 422 million people +are suffering from diabetes approximately in 2014 and it is +expected to rise to 438 million in 2030[2, 3]. Among them, +90% of cases are Type 2 diabetes (T2DM) [4]. It may arise +at an early childhood because of the failure of cells to +respond to insulin appropriately [5]. So, patients have to +face excessive tiredness, visual disorders, excessive thirst, +skin infection recurrence, delayed wound healing and +frequent discharge of urine [6]. It has been pointed out by +Diabetes Research Center that 80 percent of cases of +diabetes can be prevented or delayed if it is detected early +[7]. Also, by controlling blood sugar, it is possible to lessen +the T2DM effect. A healthy diet, physical exercise, +sufficient nutrition for pregnant women, proper medication, +weight at a necessary level are crucial to maintaining a safer +sugar level. +When the diabetes is diagnosed with medical tests, it +shows significantly dangerous symptoms but these methods +do not perform well because of clinical complexity, time- +consuming process and very high expense. However, using +automated machine learning algorithms, a researcher can +predict a disease like diabetes with reduced cost and time. In +the field of Artificial Intelligence, classification is +considered a supervised technique that analyses patient data +and classifies whether or not the patient is suffering from a +disease. Researchers have created different AI and machine +learning techniques to automate prognosis of various +diseases. Machine learning techniques studies algorithm and +statistical model that has the capability for accurate +prediction by using implicit programming. In medical +science, they take the concept of the human brain as it +contains millions of neurons to complete tasks of the human +body. It is called nonlinear modelling and they are +interconnected like brain cells although the neuron creation +is done by program [8]. +In this paper, first we have discussed various procedures +and existing works about the prognosis of T2D , though we +emphasized various classification algorithms known as +Logistic Regression, KNN, Decision Tree, Naive Bayes, +SVM, Linear Discriminant Analysis and Random forest +classifier and Artificial Neural Network (ANN) for T2DM +prediction. Our selected model is an Artificial Neural +Network is found to be superior among all of them. +Feedforward neural network contains the signal in one +Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore. Restrictions apply. + +direction from the input to the output. It is used in different +medical diagnostic applications such as nephritis disease, +heart disease, myeloid leukemia etc. [ref]. +We have taken a medical dataset from Noakhali Medical +College, Bangladesh, consisting of 9483 samples and 14 +symptoms per sample. The 80% data and 20% data are +chosen to be training dataset and testing dataset +respectively. Machine learning classification algorithms are +applied to dataset and some elements may be missed. Then, +the mean and median method is applied in order to detect it. +The contributions of this paper are summarized as + +● +We have proposed a prediction model for T2D +using Artificial Neural Network machine learning +classifier +● +We have exerted seven classifier techniques and +ANN on T2D data and provided comparison of +accuracy among them. +● +The improvement systems of the model, as well as +accuracy, are mentioned in this work. + +The remaining of the discussion is organized as follows: +Section-II explains related work of various classification +techniques for prediction of diabetes, Section-III describes +the methodology and materials used, Section-IV discusses +evaluated Results and Section-V delineates the conclusion +of the research work. + +II. +RELATED WORK + +In recent years, several studies have been published using +multiple machine learning classifiers, ANN techniques and +various feature extraction methods. These have a drastic +change in potential research and some works are discussed +related to T2DM. Ebenezer et al. used the backpropagation +feature of ANN in order to diagnose diabetes. It finds out +the error by juxtaposing input and output number. Here, the +preceding round error is greater than the present error each +time by means of changing weight to minimize gradient of +errors using a technique known as gradient descent [9]. +Nongyao et al. delineated risk prediction by using various +machine learning classification algorithms such as Decision +Tree, Neural Network, Random Forest algorithms, Naïve +Bayes, Logistic Regression. All of them followed Bagging +and Boosting approaches to improve robustness except RFA +[10]. Deepti et al. proposed a model to identify diabetes at a +premature age by applying Decision Tree, SVM and Naïve +Bayes on Pima Indians Diabetes Database (PIDD) datasets. +They chose sufficient measures for accuracy including +precision, ROC, F measure, Recall but Naïve Bayes beat +them by acquiring the highest accuracy [11]. Su et al. +applied decision tree, logistic regression, neural network,and +rough sets to assess accuracy through various features like +age, right thigh circumference, left thigh circumference, +trunk volume and illustrates thigh circumference as a better +feature than BMI in anthropometrical data [12]. Al-Rubeaan +et al. has presented T2DM based on diabetic nephropathy +(DP), then defined high impact risk factors; age and diabetes +duration for microalbuminuria, macroalbuminuria and end- +stage renal disease(ESRD) classifications[13]. Vijayan V. +examines various types of preprocessing techniques which +includes PCA and discretization. It increases the accuracy of +Naïve Bayes classifier and Decision Tree algorithm but +reduces SVM accuracy [14]. +Micheal et al. proposed Multi-Layer Feed Forward +Neural Networks (MLFNN) in order to diagnose diabetes by +considering activation units, learning techniques on Pima +Indian Diabetes (PID) data set and achieved 82.5% +accuracy. It performs better than Naïve Bayes, Logistic +Regression (LR) and Random Forest (RF) classifier [15]. +Sadri et al. chose data mining algorithms like Naive Bayes, +RBF Network, and J48 to diagnose T2DM for Pima Indians +Diabetes Dataset that has 768 samples. Each sample has +nine features as the total number of Pregnancy, Plasma +Glucose Concentration, Diastolic Blood Pressure and 2- +Hour Serum Insulin. Among them, the Naive Bayes +algorithm is unbeatable and has 76.95% accuracy [16]. +Pradhan et al. devised a classifier for diabetes detection +using Genetic programming (GP) at low cost. Simplified +function pool consists of arithmetic operations that are used +in lower validation [17]. Yang Guo et al. applied Naïve +Bayes classifier by using WEKA tool in order to predict +Type2 diabetes and obtained remarkable accuracy [18]. + +Unlike these works, we have introduced diabetes‟s +medication detection system using machining learning and +deep ANN that will act like a doctor to choose the right +medication of a patient suffering from diabetes. + +III.MATERIALS AND METHODS + +In order to categorize diabetes therapy and drugs system for +patients, the whole workflow is separated into four parts +such as data collection, data preprocessing, training data via +the proposed algorithms, and predictions. We have exerted +seven machine learning classifiers and deep neural networks +into the pre-processed data set. + + +Fig.1. System Diagram of T2D analysis +Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore. Restrictions apply. + +Data +Processing Data +Training Model +T2Dpatients +-ML Classifiers:Logistic +80% Training +1.Missing Value Check +Regression, KNN +2.Handling Categorical +Decision Tree, Naive +Bayes, SVM, Linear +20%Testingset +3.Value +4.Features selection +Discriminant Analysis +5.Feature Scaling +and Random Forest +Classifier +6.Dimension Reduction +·DeepANN +D=Diet & Lifestyle +I=Insulin +10 Fold Cross +M=Bigreanides +Validation +S=Secretagogues +PredictionThe source of our data came from Noakhali Medical +College, Bangladesh and the data set is separated into two +parts such as training and test set. The training data are +manipulated to the diagnostic system and 13 factors have +been taken to determine therapy in order to apply machine +learning and multilayer ANN. The dataset is tested from the +trained machine learning classifiers and artificial neural +network. +A. Dataset +As discussed before our data set contains information about +9483 diabetes patients and formatted in comma-separated- +file (CSV). The dimension of the data set is 9483*14. It +preserves 14 different kind of information of a diabetes +patient such as „Name of patient‟, “Fasting”, “2 h after +Pressure” “BMI”, “Duration”, “Age”, “Sex”, “Blood +pressure”, “High Cholesterols”, “Heart Diseases”, “Kidney +Diseases”, and , “Medications”. The first 13 columns are +considered independent variables and the last one is the +dependent variable. It contains kinds of basic medicine +name of diabetes such as Diet and Lifestyle Modification, +Secretagogues, Biguanides, and Insulin. +When datasets consist of enough variables, it increases the +accuracy of prediction. Here, “Fasting” measures blood test +just before taking food, “2 h after glucose load” provides a +blood test after two hours of eating. “BMI” refers to the +weight and height of patients in kg/m2. “Medication” +indicates proper drugs and therapy. People who recovered +T2DM at early stage follow some features: age group 30-75 +years, diabetes of diagnosis duration is more than half years, +glucose level at fasting plasma is higher than 125 mg/dl, +creation of plasma indicates equal or greater than 1.7 mg/dl, +plasma glucose after two hours is 11. +When a patient suffers from kidney problems, it may be a +symptom of T2DM as higher sugar level may damage +nephron. Even bleary eyesight is considered as a side effect +for patients as eye‟s retina and the macula is affected. Bad +cholesterol may lead to Diabetic dyslipidemia which can +increase heart diseases and atherosclerosis. Here, we suggest +treatment for kidney and v---ision problems. In order to +categorize diabetes therapy and drugs system for patients, +we applied seven machine learning classifiers and eight +deep neural into a data system of Noakhali Medical College, +Bangladesh. Training data are manipulated to diagnostic +system and twelve factors have been taken to determine +therapy in order to apply machine learning and multilayer +ANN. For the training and testing of the systems, we +divided the data set into 80% training and 20% test set. The +training dataset is used to find out the appropriate model and +best hyper-parameters and testing data set contains unseen +data to predict the performance. + +B. Data preprocessing +Data preprocessing involves raw data converting into a +recognizable format from various sources. The well- +preprocessed data aids for the best training of algorithms. +Multi pre-processing training is held in our presented +systems. +1. Missing Value Check +Usually, +missing +values +may +occur +due +to +data +incompleteness, missing field, programming error, manual +data transfer from a database and so on. We may ignore +missing values but it causes problems in parameter +calculation and data accuracy for features such as age, +wages and fare. We need to inspect whether a dataset has +any missing value or not. There are many ways to handle +missing values such as delete rows, missing values +prediction, mean, median, mode and so on. But the most +prominent policy for missing value replacement is the mean +method and also it is used to exchange the approximate +results in the dataset [19]. Mean is written in this way in +mathematics, + (1) + +Where, denotes the mean and provides the average number +of n. + +2. Handling Categorical value +Categorical encoding identifies data type and transfers +categorical features into numerical numbers as the majority +of machine learning algorithms could not cope up with label +data directly. Then numerical values are fed into the +specific model. In our data set, there are five categorical +variable names as „Name of patients‟, “Heart Diseases”, +“Kidney Diseases”, “Sex”, and “Medications”. There are +two popular ways of transforming categorical data into +numerical data such as Integer encoding and one-hot +encoding. In the label encoder, categorical features are an +integer value and contain a natural order relationship, but +the multiclass relationship will provide different values for +various classes. One hot encoding maps categorical value +into binary vectors. Firstly, it is obvious to assign binary +value to an integer value of female and male is 0 and 1. +Then converting it to a 2 size of 2 possible integers in a +binary vector. Here, a female is encoded as 0 and +represented as [1, 0] in which index 0 has value 1 and vice +versa. It chooses this value as a feature to influence model +training [20]. + +3. Features Selection +Feature selection incorporates the identification and +reduction of unnecessary features that have no impact on the +objective function and high impact features are kept. Our +dataset contains 14 types of elements and we have checked +p-value which is a statistical process for finding out the +probability for the null hypothesis. The features are taken +out whose p-value indicates less than 0.05. +Moreover, multicollinearity refers to determine the high +correlation which exists between two or more independent +features and features that are influential to each other. It is +called redundancy when two features are highly correlated. +As we have to handle redundancy, it is essential to choose +some methods such as χ2Test and Correlation Coefficient. +Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore. Restrictions apply. + +x +KThe Correlation Coefficient can be calculated by numerical +data. Assume that A and B are two features and it can be +defined as, + + + + +(2) + +After performing both p-value and multicollinearity test, we +could come forward with seven features among thirteen +independent features. Those are “Fasting”, “2 Hours after +Glucose Load” “Duration”, “BMI”, “High Cholesterols”, +“Heart Diseases”, and “Kidney Diseases”. + + 4. Feature scaling +Most of the time, the dataset does not remain on the same +scale or even not normalized. So, feature scaling is a +fundamental data transformation method for coping the +dataset to algorithms. We need to scale value of features and +provide equal weight to all features in order to obtain the +same scale for all data. Moreover, it is possible for scaling +to change in different values for different features. There are +lots of techniques for feature scaling for example +Standardization, Mean Normalization, Min-Max Scaling, +Unit Vector and so on. +In our research work, we have taken Min-Max Scaling or +normalization process as the features are confined within a +bounded area. Minmax normalization is a z-series +normalization to transform linearly x to x‟ where maxX and +minX are the maximum and minimum value for X +respectively. + + (3) + +When x=max, then y =1 and x=min, y=1. + +The scaling range belongs between 0 and 1(positive value) +and -1 to 1(negative) and we have taken the value between 0 +and 1. + +5. Dimension Reducing +Dimensionality reduction refers to minimizing random +variables by considering the principal set of variables that +avoids overfitting. For a large number of dataset, we need to +use dimension reduction technique. In our study, we prefer +dimension reduction for dimensional graphical visualization. +There are a lot of methods for reducing dimension, for +instance, LDA, PCA, SVD, NMF, etc. In our system, we +have applied Principal Component Analysis (PCA). It is a +linear transformation based on the correlation between +features in order to identify patterns. High dimensional data +are estimated into equal or lower dimensions through +maximum variance. We have taken two components of PCA +according to their high variance so that we can graphically +visualize in Cartesian coordinate system. + +C. Training Algorithms +The training dataset for T2DM is applied to each algorithm +to find out medications and model performance is assessed +by obtaining accuracy. + +a. Machine Learning Classifier +Since we focus on the performance of treatment predictions, +we have implemented seven machine learning classifiers +such as logistic regression, KNN, SVM, Naive Bayes, +decision tree, LDA, random forest tree. +Logistic regression is based on the probability model; it is +derived from linear regression that mapped the dataset into +two categories by considering existing data. At first, features +are mapped linearly that are transferred to a sigmoid +function layer for prediction. It shows the relationship +between the dependent and independent values but output +limits the prediction range on [0, 1]. As we need to predict +the right treatment of a diabetes person, it is beneficial to +use a binary classification problem. +Linear Discriminant Analysis (LDA) belongs to a linear +classifier to find out the linear correlation between elements +in order to support binary and multiclass classification. The +chance of inserting a new dataset into every class is detected +by LDA. Then, the class that contains the dataset is detected +as output. It can calculate the mean function for each class +and it is estimated by vectors for finding group variance. +Support Vector Machine (SVM) is the most recognized +classifier to make decision boundary as hyperplane to keep +the widest distance from both sides of points. This +hyperplane refers to separating data into two groups in two- +dimensional space. It performs better with non-linear +classification by the kernel function. It is capable of +separating and classifying unsupported data. +K-nearest neighbours (KNN) works instant learning +algorithm and input labeled data that act as training instance. +Then, the output produces a group of data. When k=1, 2, 5 +then it means the class has 1, 2 or 5 neighbours of every data +point. For this system, we choose k=5 that means 5 +neighbours for every data point. We have taken Minkowski +distance to provide distance between two points in N- +dimensional vector space to run data. Suppose, points p1(x1, +y1) and p2(x2, y2) illustrates Minkowski distance as, + + (4) +Here, d denotes Minkowski distance between p1 and p2 +point. + +Naive Bayes Classifier is constructed from Bayes theorem, +in which features are independent of each other in present +class and classification that counts the total number of +observations by calculating the probability to create a +predictive model in the fastest time. It outperformed with a +huge dataset of categorical variables. The main benefits of +that it involves limited training data to estimate better +results. Naive Bayes theorem probability can be derived +from P (T), P(X) and P (X|T). Therefore, + + (5) +Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore. Restrictions apply. + +at A-(b:. +17 +noaobmmax(x). +mmtm(x71 +dp : (x,y) → Ilx -yllp +Ix - yilP +i=1The decision tree is a decision-supporting a predictive +model based on tree structure by putting logic to interpret +features. It provides a conditional control system and marks +red or green for died or alive leaves. It has three types of +nodes: root node, decision nodes and leaf nodes. The root +node is the topmost node among them and data are split into +choices to find out the decision‟s result. Decision nodes +basically comprise of decision rules to produce the output +by considering all information gain and oval shape is used to +denote it. The terminal node represents the action that needs +to be taken after getting the outcome of all decisions. + +Multiple random trees lead to the random forest to calculate +elements of molecular structure. A decision tree looks like a +tree that is the storehouse of results from the random forest +algorithm and bagging is applied to it in order to reduce +bias-variance trade-off. It can perform feature selection +directly and output represents the mode of all classes. In +Random Forest Tree, we took the total number of trees in +the forest: 10. + +b. Artificial Neural Network +An ANN is considered as a human brain due to consisting +millions of neurons to communicate with each other. It has +three layers; the input layer fed raw data to network, hidden +layer is the middle layer based on input, weight and the +relationship denoted by activity function. Output layers +value is determined by activity, weight and relationship +from the second layer. +Since we need to find out the probability of each treatment +and the objective function is not binary, so we used softmax +activation function instead of sigmoid between the hidden +layer and output layer. There is no rule of thumb to choose +hidden layer in ANN. If our data is linearly separable then +we don‟t need any hidden layer. Then the average node +between the input and output node is preferable. +In our system, we prefer six hidden layers between the input +node and the hidden layer and 25 epochs to train a neural +network. It has no gradient vanishing problem and uses +ReLU activation function to train dataset without +pretraining. + +c. Validation + +The validation is a technique of evaluating the performance +of algorithms. It cooperates to evaluate the model and +reduce overfitting. Different types of validation method +includes +Holdout +method, +K-Fold +Cross-Validation, +Stratified K-Fold Cross-Validation and Leave-P-Out Cross- +Validation. We have picked out k-fold validation dataset is +divided into k subsets in k times. One k subset act as test set +and error is estimated by average k trails. Therefore, k-1 +subsets produce training set. We prefer k=10 generally +which contains 10 folds, repeat one time and stratified +sampling as each fold has a similar amount of samples. + +IV.EXPERIMENTAL RESULT ANALYSIS + +A. Experimental tool +The whole task has been implemented in python 3.6 +programming language in Anaconda distribution. Python +library offers various facilities to implement machine +learning and deep learning. The unbeatable library for data +representation is pandas that provide huge commands and +large data management. We have used it to read and +analyze data in less writing. Afterward, scikit-learn has +features for various classification, clustering algorithms to +build models. Also, Keras combines the advantages of +theano and TensorFlow to train a neural network model. We +use to fit and evaluate function to train and assess neural +network model respectively bypassing the same input and +output, then we apply matplotlib for graphical visualization. +B. Model performance +For boosting performance, it is always a better idea to +increase data size instead of depending on prediction and +weak correlations. Also, adding a hidden layer may increase +accuracy and speed due to its tendency to make a training +dataset overfit. But partially it is dependent on the +complexity of the model. Contrarily, increasing the epochs +number ameliorate performance though it sometimes +overfits training data. It works well for the deep network +than shallow network when considering regulation factor. +Hereafter, we have added another hidden layer; choose +epoch 100 then the Deep ANN accuracy risen up to 95.14% +which is superior among all of them. + + +Fig.2. Models Performance Comparison. +C. Improving Model performance +For boosting performance, it is always a better idea to +increase data size instead of depending on prediction and +weak correlations. Also, adding a hidden layer may increase +training accuracy and speed due to its tendency to make +training dataset overfit. But partially it is dependent on the +complexity of the model. Contrarily, increasing the epochs +number ameliorate performance though it sometimes +overfits training data. It works well for the deep network +than shallow network when considering regulation factor. +Hereafter, we added another hidden layer; choose epoch 100 +then the Deep ANN accuracy of the training and test set is +risen up to 96.42% and 95.14% which is superior among all +of them. +Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore. Restrictions apply. + + +Fig.3. 2-D Graphical Visualization of Test set + +D.Final Result +After applying feature extraction to the dataset and +implementing several types of classification and deep neural +network, we found artificial neural network as better +performer with best validity and Random forest classifier +are preferable among other machine learning classifiers. + + +Fig.4. Final Result Comparison + +V. CONCLUSIONS +Type-2 diabetes can lead to a lot of complications as heart +attack, kidney damage, blurred vision, hearing problems and +Alzheimer‟s disease. The main problem is lower accuracy of +the prediction model, small datasets and inadaptability to +various datasets. In this paper, the medication and treatment +are predicted by a comparative study of seven machine +learning algorithms and deep neural networks. Artificial +neural networks play a vital role in medical science by +minimizing classification error that leads to greater +accuracy. Experiment result determines the designed ANN +system achieved higher accuracy of 94.7%. It can cooperate +with experts to detect T2DM patients at a very early age and +provide the best treatment option. +In the future, we can enhance the accuracy of early +treatment to lessen the suffering of patients. Also, we can +implement more classifiers to pick up the leading one for +record-breaking performance and extend it to automation +analysis. There is a plan to apply this designed system in +diabetes or for other diseases. It may increase the +performance of prediction of various diseases. Larger +dataset leads to the higher training set and it cooperates in +advanced accuracy. It is convenient for people to have an +application on their smartphones related to T2DM that may +have T2DM symptoms, treatment, risk factors, and health +management. +REFERENCES + +[1] +https://www.niddk.nih.gov/health- +information/diabetes/overview/what-is- +diabetes?fbclid=IwAR36jKI7GXUE4D0PhZ1Wk4zAa49kKXtn3hB7 +OqrYSoAqA925MzkXa_1u_Sk [Accessed: 24 June, 2019] +[2] +https://www.who.int/health-topics/diabetes [Accessed: 24 June, 2019] +[3] +Rawal LB, Tapp RJ, Williams ED, Chan C, Yasin S, Oldenburg B. +Prevention of type 2 diabetes and its complications in developing +countries: a review. Int J Behav Med. 2012; 19:121–133. +[4] +https://www.diabetes.org.uk/diabetes-the-basics/what-is-type-2- +diabetes [Accessed: 24 June, 2019] +[5] +https://en.m.wikipedia.org/wiki/Diabetes?fbclid=IwAR3c20p4V8Np +MvAwkTZmEK-rXxnBCZ61jhV87-ZnfPMNUJDpm9Easq9dDzA +[Accessed: 24 June, 2019] +[6] +https://idf.org/52-about-diabetes.html [Accessed: 24 June, 2019] +[7] +E. I. Mohamed, R. Linde, G. Perriello, N. Di Daniele, S. J. Pöppl and +A. De Lorenzo. "Predicting type 2 diabetes using an electronic nose- +based artificial neural network analysis," in Diabetes nutrition & +metabolism Vol.15, No.4, (2002). pp. 222-215. +[8] +R. A. Dunne, Wiley, J., Inc, S. "A Statistical Approach to Neural +Networks for Pattern Recognition", New Jersey: John Wiley & Sons +Inc; (2007). +[9] +Ebenezer Obaloluwa Olaniyi and Khashman Adnan..“Onset diabetes +diagnosis using artificial neural network”, International Journal of +Scientific and Engineering research 5.10 (2014). +[10] Nai-Arun, N., Moungmai, R. “Comparison of Classifiers for the Risk +of Diabetes Prediction”, Procedia Computer Science vol: 69, pp: 132– +142, 2015. +[11] Deepti Sisodiaa, Dilip Singh Sisodia. “Prediction of Diabetes using +Classification +Algorithms” +International +Conference +on +Computational Intelligence and Data Science, 2018 +[12] Kowsher, M., Tithi, F. S., Rabeya, T., Afrin, F., & Huda, M. N. +(2020). Type 2 Diabetics Treatment and Medication Detection with +Machine +Learning +Classifier +Algorithm. In +Proceedings +of +International Joint Conference on Computational Intelligence (pp. +519-531). Springer, Singapore. +[13] https://www.ncbi.nlm.nih.gov/pubmed/24586457 [Accessed: 24 June, +2019] +[14] Veena Vijayan V. and Anjali C. Decision support systems for +predicting diabetes mellitus –a review. Proceedings of 2015 global +conference on communication technologies (GCCT 2015). +[15] https://www.researchgate.net/publication/331352518_A_Multi- +layer_Feed_Forward_Neural_Network_Approach_for_Diagnosing_D +iabetes +[16] https://pdfs.semanticscholar.org/ab93/6e4630720cb7f7ead833222b94 +5dc3801438.pdf +[17] Pradhan, M.A., Rahman, A., Acharya, P., Gawade, R., Pateria, A. +Design of classifier for Detection of Diabetes using Genetic +Programming. In: International Conference on Computer Science and +Information Technology, Pattaya, Thailand, pp. 125–130 (2011). +[18] Yang Guo, Karlskrona, S Guohua Bai and Yan Hu. Using Bayes +Network for Prediction of Type-2 diabetes, IEEE: International +Conference on Internet Technology And Secured Transactions, pp: +471 - 472, Dec. 2012. +[19] https://www.analyticsindiamag.com/5-ways-handle-missing-values- +machine-learning-datasets/ [Accessed: 24 June, 2019] +[20] https://medium.com/@contactsunny/label-encoder-vs-one-hot- +encoder- [Accessed: 5 August, 2019] + +Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore. Restrictions apply. + +ANN (Test set) +CI \ No newline at end of file diff --git a/-tE1T4oBgHgl3EQfUwM8/content/tmp_files/load_file.txt b/-tE1T4oBgHgl3EQfUwM8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a26355120307c3d0194240bdfbce048bba1c0c6 --- /dev/null +++ b/-tE1T4oBgHgl3EQfUwM8/content/tmp_files/load_file.txt @@ -0,0 +1,390 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf,len=389 +page_content='2019 22nd International Conference on Computer and Information Technology (ICCIT), 18-20 December 2019 978-1-7281-5842-6/19/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='00 ©2019 IEEE Prognosis and Treatment Prediction of Type-2 Diabetes Using Deep Neural Network and Machine Learning Classifiers Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Kowsher Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' of Applied Mathematics Noakhali Science and Technology University, Noakhali-3814,Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='kowsher@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='com Mahbuba Yesmin Turaba Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' of Information and Communication Technology Comilla University Comilla, Bangladesh mahbuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='yesmin11@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='com M M Mahabubur Rahman Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' of CSTE Noakhali Science and Technology University Noakhali-3814, Bangladesh toufikrahman098@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='com Tanvir Sajed Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' of Computing Science University of Alberta Edmonton, Canada tsajed@ualbarta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='ca Abstract—Type 2 Diabetes is a fast-growing, chronic metabolic disorder due to imbalanced insulin activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' As lots of people are suffering from it, access to proper treatment is necessary to control the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Most patients are unaware of health complexity, symptoms and risk factors before diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The motion of this research is a comparative study of seven machine learning classifiers and an artificial neural network method to prognosticate the detection and treatment of diabetes with a high accuracy, in order to identify and treat diabetes patients at an early age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Our training and test dataset is an accumulation of 9483 diabetes patients’ information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The training dataset is large enough to negate overfitting and provide for highly accurate test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We use performance measures such as accuracy and precision to find out the best algorithm deep ANN which outperforms with 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='14% accuracy among all other tested machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We hope our high performing model can be used by hospitals to predict diabetes and drive research into more accurate prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Keywords—Artificial Neural Network, Type 2 diabetes, Support Vector Machine, Decision Tree, Naive Bayes, LDA, Random forest classifier I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' INTRODUCTION Diabetes Mellitus (DM) is a very common metabolic disorder that affects millions of people worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It occurs when the concentration of blood glucose reaches excessive level due to lack of production of insulin by the pancreas organ (Type 1 Diabetes) or due to insulin resistance (Type 2 Diabetes) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It has been published that 422 million people are suffering from diabetes approximately in 2014 and it is expected to rise to 438 million in 2030[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Among them, 90% of cases are Type 2 diabetes (T2DM) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It may arise at an early childhood because of the failure of cells to respond to insulin appropriately [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' So, patients have to face excessive tiredness, visual disorders, excessive thirst, skin infection recurrence, delayed wound healing and frequent discharge of urine [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It has been pointed out by Diabetes Research Center that 80 percent of cases of diabetes can be prevented or delayed if it is detected early [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Also, by controlling blood sugar, it is possible to lessen the T2DM effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' A healthy diet, physical exercise, sufficient nutrition for pregnant women, proper medication, weight at a necessary level are crucial to maintaining a safer sugar level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' When the diabetes is diagnosed with medical tests, it shows significantly dangerous symptoms but these methods do not perform well because of clinical complexity, time- consuming process and very high expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' However, using automated machine learning algorithms, a researcher can predict a disease like diabetes with reduced cost and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In the field of Artificial Intelligence, classification is considered a supervised technique that analyses patient data and classifies whether or not the patient is suffering from a disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Researchers have created different AI and machine learning techniques to automate prognosis of various diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Machine learning techniques studies algorithm and statistical model that has the capability for accurate prediction by using implicit programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In medical science, they take the concept of the human brain as it contains millions of neurons to complete tasks of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It is called nonlinear modelling and they are interconnected like brain cells although the neuron creation is done by program [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In this paper, first we have discussed various procedures and existing works about the prognosis of T2D , though we emphasized various classification algorithms known as Logistic Regression, KNN, Decision Tree, Naive Bayes, SVM, Linear Discriminant Analysis and Random forest classifier and Artificial Neural Network (ANN) for T2DM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Our selected model is an Artificial Neural Network is found to be superior among all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Feedforward neural network contains the signal in one Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' direction from the input to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It is used in different medical diagnostic applications such as nephritis disease, heart disease, myeloid leukemia etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' [ref].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We have taken a medical dataset from Noakhali Medical College, Bangladesh, consisting of 9483 samples and 14 symptoms per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The 80% data and 20% data are chosen to be training dataset and testing dataset respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Machine learning classification algorithms are applied to dataset and some elements may be missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Then, the mean and median method is applied in order to detect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The contributions of this paper are summarized as We have proposed a prediction model for T2D using Artificial Neural Network machine learning classifier We have exerted seven classifier techniques and ANN on T2D data and provided comparison of accuracy among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The improvement systems of the model, as well as accuracy, are mentioned in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The remaining of the discussion is organized as follows: Section-II explains related work of various classification techniques for prediction of diabetes, Section-III describes the methodology and materials used, Section-IV discusses evaluated Results and Section-V delineates the conclusion of the research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' RELATED WORK In recent years, several studies have been published using multiple machine learning classifiers, ANN techniques and various feature extraction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' These have a drastic change in potential research and some works are discussed related to T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Ebenezer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' used the backpropagation feature of ANN in order to diagnose diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It finds out the error by juxtaposing input and output number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Here, the preceding round error is greater than the present error each time by means of changing weight to minimize gradient of errors using a technique known as gradient descent [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Nongyao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' delineated risk prediction by using various machine learning classification algorithms such as Decision Tree, Neural Network, Random Forest algorithms, Naïve Bayes, Logistic Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' All of them followed Bagging and Boosting approaches to improve robustness except RFA [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Deepti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' proposed a model to identify diabetes at a premature age by applying Decision Tree, SVM and Naïve Bayes on Pima Indians Diabetes Database (PIDD) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' They chose sufficient measures for accuracy including precision, ROC, F measure, Recall but Naïve Bayes beat them by acquiring the highest accuracy [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' applied decision tree, logistic regression, neural network,and rough sets to assess accuracy through various features like age, right thigh circumference, left thigh circumference, trunk volume and illustrates thigh circumference as a better feature than BMI in anthropometrical data [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Al-Rubeaan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' has presented T2DM based on diabetic nephropathy (DP), then defined high impact risk factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' age and diabetes duration for microalbuminuria, macroalbuminuria and end- stage renal disease(ESRD) classifications[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Vijayan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' examines various types of preprocessing techniques which includes PCA and discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It increases the accuracy of Naïve Bayes classifier and Decision Tree algorithm but reduces SVM accuracy [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Micheal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' proposed Multi-Layer Feed Forward Neural Networks (MLFNN) in order to diagnose diabetes by considering activation units, learning techniques on Pima Indian Diabetes (PID) data set and achieved 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='5% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It performs better than Naïve Bayes, Logistic Regression (LR) and Random Forest (RF) classifier [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Sadri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' chose data mining algorithms like Naive Bayes, RBF Network, and J48 to diagnose T2DM for Pima Indians Diabetes Dataset that has 768 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Each sample has nine features as the total number of Pregnancy, Plasma Glucose Concentration, Diastolic Blood Pressure and 2- Hour Serum Insulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Among them, the Naive Bayes algorithm is unbeatable and has 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='95% accuracy [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Pradhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' devised a classifier for diabetes detection using Genetic programming (GP) at low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Simplified function pool consists of arithmetic operations that are used in lower validation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Yang Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' applied Naïve Bayes classifier by using WEKA tool in order to predict Type2 diabetes and obtained remarkable accuracy [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Unlike these works, we have introduced diabetes‟s medication detection system using machining learning and deep ANN that will act like a doctor to choose the right medication of a patient suffering from diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='MATERIALS AND METHODS In order to categorize diabetes therapy and drugs system for patients, the whole workflow is separated into four parts such as data collection, data preprocessing, training data via the proposed algorithms, and predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We have exerted seven machine learning classifiers and deep neural networks into the pre-processed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' System Diagram of T2D analysis Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Data Processing Data Training Model T2Dpatients ML Classifiers:Logistic 80% Training 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='Missing Value Check Regression, KNN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='Handling Categorical Decision Tree, Naive Bayes, SVM, Linear 20%Testingset 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='Value 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='Features selection Discriminant Analysis 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='Feature Scaling and Random Forest Classifier 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='Dimension Reduction DeepANN D=Diet & Lifestyle I=Insulin 10 Fold Cross M=Bigreanides Validation S=Secretagogues PredictionThe source of our data came from Noakhali Medical College, Bangladesh and the data set is separated into two parts such as training and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The training data are manipulated to the diagnostic system and 13 factors have been taken to determine therapy in order to apply machine learning and multilayer ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The dataset is tested from the trained machine learning classifiers and artificial neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Dataset As discussed before our data set contains information about 9483 diabetes patients and formatted in comma-separated- file (CSV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The dimension of the data set is 9483*14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It preserves 14 different kind of information of a diabetes patient such as „Name of patient‟, “Fasting”, “2 h after Pressure” “BMI”, “Duration”, “Age”, “Sex”, “Blood pressure”, “High Cholesterols”, “Heart Diseases”, “Kidney Diseases”, and , “Medications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The first 13 columns are considered independent variables and the last one is the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It contains kinds of basic medicine name of diabetes such as Diet and Lifestyle Modification, Secretagogues, Biguanides, and Insulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' When datasets consist of enough variables, it increases the accuracy of prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Here, “Fasting” measures blood test just before taking food, “2 h after glucose load” provides a blood test after two hours of eating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' “BMI” refers to the weight and height of patients in kg/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' “Medication” indicates proper drugs and therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' People who recovered T2DM at early stage follow some features: age group 30-75 years, diabetes of diagnosis duration is more than half years, glucose level at fasting plasma is higher than 125 mg/dl, creation of plasma indicates equal or greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='7 mg/dl, plasma glucose after two hours is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' When a patient suffers from kidney problems, it may be a symptom of T2DM as higher sugar level may damage nephron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Even bleary eyesight is considered as a side effect for patients as eye‟s retina and the macula is affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Bad cholesterol may lead to Diabetic dyslipidemia which can increase heart diseases and atherosclerosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Here, we suggest treatment for kidney and v---ision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In order to categorize diabetes therapy and drugs system for patients, we applied seven machine learning classifiers and eight deep neural into a data system of Noakhali Medical College, Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Training data are manipulated to diagnostic system and twelve factors have been taken to determine therapy in order to apply machine learning and multilayer ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' For the training and testing of the systems, we divided the data set into 80% training and 20% test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The training dataset is used to find out the appropriate model and best hyper-parameters and testing data set contains unseen data to predict the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Data preprocessing Data preprocessing involves raw data converting into a recognizable format from various sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The well- preprocessed data aids for the best training of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Multi pre-processing training is held in our presented systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Missing Value Check Usually, missing values may occur due to data incompleteness, missing field, programming error, manual data transfer from a database and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We may ignore missing values but it causes problems in parameter calculation and data accuracy for features such as age, wages and fare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We need to inspect whether a dataset has any missing value or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' There are many ways to handle missing values such as delete rows, missing values prediction, mean, median, mode and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' But the most prominent policy for missing value replacement is the mean method and also it is used to exchange the approximate results in the dataset [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Mean is written in this way in mathematics, (1) Where, denotes the mean and provides the average number of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Handling Categorical value Categorical encoding identifies data type and transfers categorical features into numerical numbers as the majority of machine learning algorithms could not cope up with label data directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Then numerical values are fed into the specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In our data set, there are five categorical variable names as „Name of patients‟, “Heart Diseases”, “Kidney Diseases”, “Sex”, and “Medications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' There are two popular ways of transforming categorical data into numerical data such as Integer encoding and one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In the label encoder, categorical features are an integer value and contain a natural order relationship, but the multiclass relationship will provide different values for various classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' One hot encoding maps categorical value into binary vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Firstly, it is obvious to assign binary value to an integer value of female and male is 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Then converting it to a 2 size of 2 possible integers in a binary vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Here, a female is encoded as 0 and represented as [1, 0] in which index 0 has value 1 and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It chooses this value as a feature to influence model training [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Features Selection Feature selection incorporates the identification and reduction of unnecessary features that have no impact on the objective function and high impact features are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Our dataset contains 14 types of elements and we have checked p-value which is a statistical process for finding out the probability for the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The features are taken out whose p-value indicates less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Moreover, multicollinearity refers to determine the high correlation which exists between two or more independent features and features that are influential to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It is called redundancy when two features are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' As we have to handle redundancy, it is essential to choose some methods such as χ2Test and Correlation Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' x KThe Correlation Coefficient can be calculated by numerical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Assume that A and B are two features and it can be defined as, (2) After performing both p-value and multicollinearity test, we could come forward with seven features among thirteen independent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Those are “Fasting”, “2 Hours after Glucose Load” “Duration”, “BMI”, “High Cholesterols”, “Heart Diseases”, and “Kidney Diseases”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Feature scaling Most of the time, the dataset does not remain on the same scale or even not normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' So, feature scaling is a fundamental data transformation method for coping the dataset to algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We need to scale value of features and provide equal weight to all features in order to obtain the same scale for all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Moreover, it is possible for scaling to change in different values for different features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' There are lots of techniques for feature scaling for example Standardization, Mean Normalization, Min-Max Scaling, Unit Vector and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In our research work, we have taken Min-Max Scaling or normalization process as the features are confined within a bounded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Minmax normalization is a z-series normalization to transform linearly x to x‟ where maxX and minX are the maximum and minimum value for X respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' (3) When x=max, then y =1 and x=min, y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The scaling range belongs between 0 and 1(positive value) and -1 to 1(negative) and we have taken the value between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Dimension Reducing Dimensionality reduction refers to minimizing random variables by considering the principal set of variables that avoids overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' For a large number of dataset, we need to use dimension reduction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In our study, we prefer dimension reduction for dimensional graphical visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' There are a lot of methods for reducing dimension, for instance, LDA, PCA, SVD, NMF, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In our system, we have applied Principal Component Analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It is a linear transformation based on the correlation between features in order to identify patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' High dimensional data are estimated into equal or lower dimensions through maximum variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We have taken two components of PCA according to their high variance so that we can graphically visualize in Cartesian coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Training Algorithms The training dataset for T2DM is applied to each algorithm to find out medications and model performance is assessed by obtaining accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Machine Learning Classifier Since we focus on the performance of treatment predictions, we have implemented seven machine learning classifiers such as logistic regression, KNN, SVM, Naive Bayes, decision tree, LDA, random forest tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Logistic regression is based on the probability model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' it is derived from linear regression that mapped the dataset into two categories by considering existing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' At first, features are mapped linearly that are transferred to a sigmoid function layer for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It shows the relationship between the dependent and independent values but output limits the prediction range on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' As we need to predict the right treatment of a diabetes person, it is beneficial to use a binary classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Linear Discriminant Analysis (LDA) belongs to a linear classifier to find out the linear correlation between elements in order to support binary and multiclass classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The chance of inserting a new dataset into every class is detected by LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Then, the class that contains the dataset is detected as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It can calculate the mean function for each class and it is estimated by vectors for finding group variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Support Vector Machine (SVM) is the most recognized classifier to make decision boundary as hyperplane to keep the widest distance from both sides of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' This hyperplane refers to separating data into two groups in two- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It performs better with non-linear classification by the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It is capable of separating and classifying unsupported data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' K-nearest neighbours (KNN) works instant learning algorithm and input labeled data that act as training instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Then, the output produces a group of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' When k=1, 2, 5 then it means the class has 1, 2 or 5 neighbours of every data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' For this system, we choose k=5 that means 5 neighbours for every data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We have taken Minkowski distance to provide distance between two points in N- dimensional vector space to run data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Suppose, points p1(x1, y1) and p2(x2, y2) illustrates Minkowski distance as, (4) Here, d denotes Minkowski distance between p1 and p2 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Naive Bayes Classifier is constructed from Bayes theorem, in which features are independent of each other in present class and classification that counts the total number of observations by calculating the probability to create a predictive model in the fastest time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It outperformed with a huge dataset of categorical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The main benefits of that it involves limited training data to estimate better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Naive Bayes theorem probability can be derived from P (T), P(X) and P (X|T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Therefore, (5) Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' at A-(b:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' 17 noaobmmax(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' mmtm(x71 dp : (x,y) → Ilx -yllp Ix - yilP i=1The decision tree is a decision-supporting a predictive model based on tree structure by putting logic to interpret features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It provides a conditional control system and marks red or green for died or alive leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It has three types of nodes: root node, decision nodes and leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The root node is the topmost node among them and data are split into choices to find out the decision‟s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Decision nodes basically comprise of decision rules to produce the output by considering all information gain and oval shape is used to denote it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The terminal node represents the action that needs to be taken after getting the outcome of all decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Multiple random trees lead to the random forest to calculate elements of molecular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' A decision tree looks like a tree that is the storehouse of results from the random forest algorithm and bagging is applied to it in order to reduce bias-variance trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It can perform feature selection directly and output represents the mode of all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In Random Forest Tree, we took the total number of trees in the forest: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Artificial Neural Network An ANN is considered as a human brain due to consisting millions of neurons to communicate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It has three layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' the input layer fed raw data to network, hidden layer is the middle layer based on input, weight and the relationship denoted by activity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Output layers value is determined by activity, weight and relationship from the second layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Since we need to find out the probability of each treatment and the objective function is not binary, so we used softmax activation function instead of sigmoid between the hidden layer and output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' There is no rule of thumb to choose hidden layer in ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' If our data is linearly separable then we don‟t need any hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Then the average node between the input and output node is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In our system, we prefer six hidden layers between the input node and the hidden layer and 25 epochs to train a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It has no gradient vanishing problem and uses ReLU activation function to train dataset without pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Validation The validation is a technique of evaluating the performance of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It cooperates to evaluate the model and reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Different types of validation method includes Holdout method, K-Fold Cross-Validation, Stratified K-Fold Cross-Validation and Leave-P-Out Cross- Validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We have picked out k-fold validation dataset is divided into k subsets in k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' One k subset act as test set and error is estimated by average k trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Therefore, k-1 subsets produce training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We prefer k=10 generally which contains 10 folds, repeat one time and stratified sampling as each fold has a similar amount of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='EXPERIMENTAL RESULT ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Experimental tool The whole task has been implemented in python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='6 programming language in Anaconda distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Python library offers various facilities to implement machine learning and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The unbeatable library for data representation is pandas that provide huge commands and large data management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We have used it to read and analyze data in less writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Afterward, scikit-learn has features for various classification, clustering algorithms to build models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Also, Keras combines the advantages of theano and TensorFlow to train a neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' We use to fit and evaluate function to train and assess neural network model respectively bypassing the same input and output, then we apply matplotlib for graphical visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Model performance For boosting performance, it is always a better idea to increase data size instead of depending on prediction and weak correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Also, adding a hidden layer may increase accuracy and speed due to its tendency to make a training dataset overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' But partially it is dependent on the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Contrarily, increasing the epochs number ameliorate performance though it sometimes overfits training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It works well for the deep network than shallow network when considering regulation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Hereafter, we have added another hidden layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' choose epoch 100 then the Deep ANN accuracy risen up to 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='14% which is superior among all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Models Performance Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Improving Model performance For boosting performance, it is always a better idea to increase data size instead of depending on prediction and weak correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Also, adding a hidden layer may increase training accuracy and speed due to its tendency to make training dataset overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' But partially it is dependent on the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Contrarily, increasing the epochs number ameliorate performance though it sometimes overfits training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It works well for the deep network than shallow network when considering regulation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Hereafter, we added another hidden layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' choose epoch 100 then the Deep ANN accuracy of the training and test set is risen up to 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='42% and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='14% which is superior among all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' 2-D Graphical Visualization of Test set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='Final Result After applying feature extraction to the dataset and implementing several types of classification and deep neural network, we found artificial neural network as better performer with best validity and Random forest classifier are preferable among other machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Final Result Comparison V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' CONCLUSIONS Type-2 diabetes can lead to a lot of complications as heart attack, kidney damage, blurred vision, hearing problems and Alzheimer‟s disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' The main problem is lower accuracy of the prediction model, small datasets and inadaptability to various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In this paper, the medication and treatment are predicted by a comparative study of seven machine learning algorithms and deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Artificial neural networks play a vital role in medical science by minimizing classification error that leads to greater accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Experiment result determines the designed ANN system achieved higher accuracy of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It can cooperate with experts to detect T2DM patients at a very early age and provide the best treatment option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' In the future, we can enhance the accuracy of early treatment to lessen the suffering of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Also, we can implement more classifiers to pick up the leading one for record-breaking performance and extend it to automation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' There is a plan to apply this designed system in diabetes or for other diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It may increase the performance of prediction of various diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' Larger dataset leads to the higher training set and it cooperates in advanced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' It is convenient for people to have an application on their smartphones related to T2DM that may have T2DM symptoms, treatment, risk factors, and health management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content=' REFERENCES [1] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='niddk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'} +page_content='gov/health- information/diabetes/overview/what-is- diabetes?' metadata={'source': 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Q. Ye,1 T. Le,1 H. Su,1 Y. N. Zhang,1 S. S. Luo,1 M. J. Gutmann,2 H. Q. Yuan,1, 3, 4, 5 and M. Smidman1, 3, ∗ +1Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China +2ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot Oxon OX11 0QX, United Kingdom +3Zhejiang Province Key Laboratory of Quantum Technology and Device, +Department of Physics, Zhejiang University, Hangzhou 310058, China +4State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310058, China +5Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China +(Dated: January 5, 2023) +We report the magnetic properties of the layered heavy fermion antiferromagnet CePdGa6, and +their evolution upon tuning with the application of magnetic field and pressure. CePdGa6 orders +antiferromagnetically below TN = 5.2 K, where there is evidence for heavy fermion behavior from an +enhanced Sommerfeld coefficient. Our results are best explained by a magnetic ground state of fer- +romagnetically coupled layers of Ce 4f-moments orientated along the c-axis, with antiferromagnetic +coupling between layers. At low temperatures we observe two metamagnetic transitions for fields +applied along the c-axis corresponding to spin-flip transitions, where the lower transition is to a dif- +ferent magnetic phase with a magnetization one-third of the saturated value. From our analysis of +the magnetic susceptibility, we propose a CEF level scheme which accounts for the Ising anisotropy +at low temperatures, and we find that the evolution of the magnetic ground state can be explained +considering both antiferromagnetic exchange between nearest neighbor and next nearest neighbor +layers, indicating the influence of long-range interactions. Meanwhile we find little change of TN +upon applying hydrostatic pressures up to 2.2 GPa, suggesting that significantly higher pressures +are required to examine for possible quantum critical behaviors. +I. +INTRODUCTION +Heavy fermion compounds are prototypical examples +of strongly correlated electron systems, and have been +found to host a range of emergent phenomena including +unconventional superconductivity, complex magnetic or- +der and strange metal behavior [1–3]. Ce-based heavy +fermions contain a Kondo lattice of Ce-ions with an un- +paired 4f electron, which can both couple to other 4f mo- +ments via the Ruderman-Kittel-Kasuya-Yosida (RKKY) +interaction and undergo the Kondo interaction due to +hybridization with the conduction electrons. +Here the +RKKY interaction gives rise to long-range magnetic or- +der, while the Kondo interaction favors a non-magnetic +Fermi-liquid ground state with greatly enhanced quasi- +particle masses. Due to the small energy scales, the rel- +ative strengths of these competing interactions can often +be tuned by non-thermal parameters such as pressure, +magnetic fields and chemical doping [4], and in many +cases the magnetic ordering can be continuously sup- +pressed to zero temperature at a quantum critical point +(QCP). +A major question for heavy fermion systems is the +relationship between quantum criticality, and the dome +of unconventional superconductivity sometimes found to +encompass the QCP. CeIn3 is a canonical example of this +phenomenon, which at ambient pressure orders antiferro- +magnetically below TN = 10.1 K, but exhibits a pressure- +induced QCP around 2.6 GPa, which is surrounded by +a superconducting dome with a maximum Tc of 0.2 K +[5]. The layered CeMIn5 (M= transition metal) com- +pounds consist of alternating layers of MIn2 and CeIn3 +along the c-axis [6], and among the remarkable proper- +ties is a significantly enhanced superconducting Tc for +the M= Rh and Co systems, reaching over 2 K [7, 8], +giving a strong indication that quasi-two-dimensionality +is important for promoting heavy fermion superconduc- +tivity. Meanwhile the Ce2MIn8 compounds correspond +to a stacked arrangement of two units of CeIn3, and one +of MIn2 [9], and are expected to have an intermediate +degree of two dimensionality relative to CeMIn5. Cor- +respondingly, the superconducting phases have lower Tc +values of 0.4 and 0.68 K for Ce2CoIn8 [10] and Ce2PdIn8 +[11] at ambient pressure, and a maximum of Tc = 2 K +at 2.3 GPa for Ce2RhIn8 [12]. On the other hand, these +different series of related Ce-based heavy fermion sys- +tems also exhibit different magnetic ground states and +crystalline electric field (CEF) level schemes [13–17] and +therefore it is challenging to disentangle the role of these +factors from that of the reduced dimensionality. +The +elucidation of the interplay between these different as- +pects requires examining additional families of layered +Ce-based heavy fermion systems for quantum critical be- +haviors, as well as detailed characterizations of the mag- +netic ground states and exchange interactions. +The properties of layered Ce-based heavy fermion gal- +lides have been less studied than the indium-based sys- +tems. CeGa6 has a layered tetragonal structure (space +group P4/nbm), with four Ga-layers between each Ce +layer [18]. +This compound orders magnetically below +TN = 1.7 K, and there is evidence for the build-up of +magnetic correlations at significantly higher tempera- +tures [19]. A more layered structure is realized in the +Ce2MGa12 (M= Cu, Ni, Rh, Pd, Ir, Pt) series, where the +Ce-layers are alternately separated by four Ga-layers, and +units of MGa6, leading to a larger interlayer separation + +2 +of the Ce-atoms [20, 21]. Several members of this series +show evidence for both antiferromagnetism and heavy +fermion behavior [20–25], where pressure can readily sup- +press the antiferromagnetic transitions of Ce2NiGa12 and +Ce2PdGa12 [26, 27], while evidence for field-induced crit- +ical fluctuations is revealed in Ce2IrGa12 [25]. +CePdGa6 has a different layered tetragonal structure +(space group P4/mmm) displayed in Fig. 1(a), consist- +ing of square layers of Ce-atoms, with each Ce con- +tained in a CeGa4 prism, separated by PdGa2 layers +[28]. Correspondingly, there is a distance between Ce- +layers of 7.92 ˚A, while the nearest neighbor in-plane Ce- +Ce separation is 4.34 ˚A, compared to respective values +of 7.54 ˚A +and 4.65 ˚A in CeRhIn5 [29]. CePdGa6 or- +ders antiferromagnetically below TN = 5.2 K, and heavy +fermion behavior is evidenced by an enhanced Sommer- +feld coefficient [20, 28]. As such, CePdGa6 is a good can- +didate to look for novel behaviors arising in quasi-two- +dimensional heavy fermion systems, but there is both a +lack of detailed characterizations of the magnetic ground +state, and no reports of the evolution under pressure. In +addition, most measurements of CePdGa6 are reported +in Ref. 28, where the results are affected by the inclu- +sion of an extrinsic antiferromagnetic phase Ce2PdGa12, +which can be eliminated using a modified crystal growth +procedure [20]. +In this article we report detailed measurements of the +magnetic properties of single crystals of CePdGa6, in- +cluding their evolution upon applying magnetic fields and +hydrostatic pressure. We find that CePdGa6 orders an- +tiferromagnetically in zero-field, where the Ce-moments +are orientated along the c-axis and align ferromagneti- +cally within the ab-plane, but there is antiferromagnetic +coupling between layers. At low temperatures, two meta- +magnetic transitions are observed for fields along the c- +axis, the lower of which corresponds to a spin-flip transi- +tion to a phase with magnetization one-third of the sat- +urated value. From our analysis of the magnetic suscep- +tibility, we propose a CEF level scheme which can ex- +plain the low temperature Ising anisotropy, and we find +that from considering interactions between the nearest- +neighbor and next nearest neighbor Ce-layers, the field +evolution of the magnetic state can be well accounted for. +II. +EXPERIMENTAL DETAILS +Single crystals of CePdGa6 were grown using a Ga self- +flux method with a molar ratio of Ce:Pd:Ga of 1:1.5:15 +[20]. +Starting materials of Ce ingot (99.9%), Pd pow- +der (99.99%) and Ga pieces (99.99%) were loaded into +an alumina crucible which was sealed in an evacuated +quartz tube. The tube was heated to 1150 ◦C and held +at this temperature for two hours, before being rapidly +cooled to 500 ◦C at a rate of 150 K/h and then cooled +more slowly to 400 +◦C at 8 K/h. +After being held +at 400 ◦C for two weeks, the tube was removed from +the furnace, and centrifuged to remove excess Ga. The +2 +0 +1 +FIG. 1. +(Color online) (a) Crystal structure of CePdGa6 +where the red, blue and green atoms correspond to Ce, Pd +and Ga, respectively. +J0 represents magnetic exchange in- +teractions between nearest neighbor Ce atoms within the ab- +plane, J1 is between nearest neighboring layers and J2 is +between next nearest layers. +An image of a typical single +crystal of CePdGa6 is also displayed, where each square in +the background is 2 mm × 2 mm. (b) X-ray diffraction pat- +tern measured on a single crystal of CePdGa6. The red dashes +correspond to the positions of the (00l) Bragg peaks, indicat- +ing that the [001] direction is perpendicular to the large face +of the plate-like samples. +obtained crystals are plate-like with typical dimensions +2 × 1.5 × 0.3 mm3. Note that when slower cooling rates +of 6 K/h or 4 K/h were used, the resulting crystals were +significantly smaller. Single crystals of the non-magnetic +analog LaPdGa6 were also obtained using a similar pro- +cedure. The composition was confirmed using a cold field +emission scanning electron microscope (SEM) equipped +with an energy dispersive x-ray spectrometer. The phase +of the crystals were checked using both a PANalytical +X’Pert MRD powder diffractometer using Cu-Kα radi- +ation, and a Rigaku-Oxford diffraction Xtalab synergy +single crystal diffractometer equipped with a HyPix hy- +brid pixel array detector using Mo-Kα radiation. The ob- +tained lattice parameters from the single crystal diffrac- +tion data of a = 4.3446(3) ˚A and c = 7.9173(10) ˚A are + +pg +c3 +0 +100 +200 +300 +2 +3 +4 +5 +4 +8 +1.5 +2.0 + + + + ( +cm) +T (K) +T +N + ~ 5.2 K + + + + ( +cm) +T (K) +FIG. 2. +(Color online) Temperature dependence of the re- +sistivity ρ(T ) of CePdGa6 between 1.8 and 300 K. The inset +displays the low temperature resistivity, where there is a sharp +anomaly at the antiferromagnetic transition. +in excellent agreement with previous reports [28]. Mea- +surements of a crystal using the powder diffractometer +are displayed in Fig. 1(b), where all the Bragg peaks are +well-indexed by the (00l) reflections of CePdGa6, demon- +strating that the c-axis is perpendicular to the large face +of the crystals. +Resistivity and specific heat measure- +ments were performed in applied fields up to 14 T using +a Quantum Design Physical Property Measurement Sys- +tem (PPMS-14) down to 1.8 K, and to 0.3 K using a 3He +insert. +Resistivity measurements were performed after +spot welding four Pt wires to the surface, with the exci- +tation current in the ab-plane. Magnetization measure- +ments were performed in the range 1.8 - 300 K in applied +fields up to 5 T using a Quantum Design Magnetic Prop- +erty Measurement System (MPMS) SQUID magnetome- +ter. +Heat capacity measurements under pressure were +carried out in a piston cylinder cell, using an ac calori- +metric method. +III. +RESULTS +A. +Antiferromagnetic transition and CEF +excitations of CePdGa6 +Figure 2 displays the temperature dependence of +the resistivity ρ(T ) of CePdGa6 between 1.8 and 300 +K, which has a residual resistivity ratio [RRR = +ρ(300 K)/ρ(2 K)] = 3.8. +A broad shoulder is ob- +served at around 50 K, which likely arises due to both +the Kondo effect, and as a consequence of CEF excita- +tions. +At higher temperatures, quasilinear behavior is +observed, which could be due to electron-phonon cou- +pling. +As shown in the inset, there is an anomaly at +around TN = 5.2 K, below which ρ(T ) decreases more +rapidly with decreasing temperature, which corresponds +C +m +/T (J mol +-1 + K +-2 +) +T (K) +Rln2 + + + S +m + (J mol +-1 + K +-1 +) + + +C +m + (J mol +-1 +K +-1 +) +T (K) +(b) + CePdGa +6 + LaPdGa +6 + + +C (J mol +-1 + K +-1 +) +T (K) +(a) +FIG. 3. (Color online) (a) Magnetic contribution to the spe- +cific heat Cm at low temperatures, where the red solid line +shows the results from fitting with Eq. 1. The inset shows +the total specific heat C of CePdGa6 and the non-magnetic +analog LaPdGa6. (b) Temperature dependence of Cm/T and +the magnetic entropy Sm of CePdGa6. The pink dotted line +displays the low temperature contribution to the specific heat +calculated from the CEF scheme deduced from the analysis +of χ(T ). +to the antiferromagnetic transition reported previously +[20], while no signature of the spurious transition at +higher temperatures is detected [28]. +The total spe- +cific heat of CePdGa6 and nonmagnetic isostructural +LaPdGa6 are shown in the inset of Fig. 3(a). The tem- +perature dependence of the magnetic contribution to the +specific heat Cm was estimated by subtracting the data +of LaPdGa6, which is shown in Fig. 3(a), while the spe- +cific heat coefficient Cm/T and the magnetic entropy Sm +of CePdGa6 are displayed in Fig. 3(b). A pronounced +λ-like anomaly is observed at TN = 5.2 K, as is typi- +cal for a second-order magnetic phase transition. +For +T > TN, Cm/T increases with decreasing temperature, +and extrapolates to a relatively large zero temperature +value of 250 mJ/mol K2. As discussed below, the analysis +of the magnetic susceptibility χ(T ) suggests the presence +of a low lying CEF level, which could contribute to Cm/T +in this temperature range. The dotted line in Fig. 3(b) +shows the calculated Cm/T for the CEF level scheme de- + +8030412JS12 +300 +e0 +0S0 +04 +0 +10 +20 +0.0 +0.1 +0.2 +0 +150 +300 +0 +200 +400 +(b) + + + (emu/mol) +T (K) +H = 0.1 T + H // c + H // ab +(a) + +H = 0.5 T + H // c + H // ab +1/( +) (mol/emu) +T (K) + +FIG. 4. +(Color online) (a) Low temperature magnetic sus- +ceptibility χ(T ) of CePdGa6, with an applied field of µ0H = +0.1 T both parallel to the c-axis and within the ab-plane. (b) +Temperature dependence of 1/(χ-χ0) up to 300 K for 0.5 T +applied along the two field directions, where the dashed and +solid lines show the results from fitting with the CEF model +described in the text. +scribed below, which has a sizeable value in the vicinity +of the transition. Subtracting the contribution from the +CEF at TN yields an estimate of γ ∼ 121.4 mJ/mol K2 +associated with the ground state doublet, and such an +enhanced value could arise both due to heavy fermion +behavior, as well as the presence of short range magnetic +correlations, as inferred in CeRhIn5[30, 31]. +The data +below TN were analyzed using [32]: +Cm = γT + c∆7/2 +SW +√ +T exp +�−∆SW +T +� +× +� +1 + +39T +20∆SW ++ 51 +32 +� +T +∆SW +�2� +(1) +where the first term corresponds to the electronic con- +tribution and the second term arises due to antiferro- +magnetic spin-waves. Here the coefficient c is related to +the spinwave stiffness D via c ∝ D−3, while ∆SW +is +the spin-wave gap. The results from fitting the zero-field +data are displayed in the main panel of Fig. 3(a), where +γ = 121.4 mJ/mol K2 was fixed, yielding ∆SW = 2.3 K +and c = 23 mJ/mol K2. The moderate value of ∆SW +is smaller than TN, unlike the layered heavy fermions +gallides Ce2PdGa12 and Ce2IrGa12 where ∆SW +> TN +[24, 25], likely reflecting the weaker magnetocrystalline +anisotropy in CePdGa6. The temperature dependence of +the magnetic entropy Sm of CePdGa6 is also displayed +in Fig. 3(b), obtained by integrating Cm/T , where Cm/T +was linearly extrapolated below 0.4 K. At TN, Sm reaches +0.76R ln 2, which together with the expected sizeable con- +tribution from the excited CEF level discussed above, +suggests a reduced entropy corresponding to the ground +state doublet due to Kondo screening. +Figure 4(a) displays the temperature dependence of +the magnetic susceptibility χ(T ) of CePdGa6 at low tem- +peratures, with an applied field of µ0H = 0.1 T along +the c-axis and within the ab-plane, which both exhibit +an anomaly at TN. +At low temperatures, χ(T ) is sig- +nificantly larger for fields along the c-axis than in the +ab-plane, demonstrating that the c-axis is the easy-axis +of magnetization. +At TN, there is a peak in χ(T ) for +H ∥ c, while for H ∥ ab χ(T ) weakly increases below TN, +indicating that this corresponds to an antiferromagnetic +transition with moments ordered along the easy c-axis. +At higher temperatures, the data above 100 K can +be analyzed using the Curie-Weiss law: χ=χ0+C/(T − +θCW), where χ0 is a temperature-independent term, C +is the Curie constant and θCW is the Curie-Weiss tem- +perature, yielding θc +CW = −11.7(3) K and an effective +moment of µc +eff = 2.35µB/Ce for H ∥ c, as well as +θab +CW = −12.9(8) K and µab +eff = 2.49µB/Ce for H ∥ ab. +The obtained values of µeff for both directions are close +to the full value of 2.54 µB for the J = 5 +2 ground state +multiplet of Ce3+. At lower temperatures, there is a devi- +ation of χ(T ) from Curie-Weiss behavior, due to the split- +ting of the ground state multiplet by crystalline-electric +fields. To analyze the CEF level scheme, we considered +the following Hamiltonian for a Ce3+ ion in a tetragonal +CEF [33] +HCF = B0 +2O0 +2 + B0 +4O0 +4 + B4 +4O4 +4 +(2) +where Om +l +and Bm +l +are Stevens operator equivalents and +parameters, respectively. The B0 +2 parameter can be es- +timated from the high temperature susceptibility using +[34] +B0 +2 = 10kB +� +θab +CW − θc +CW +� +3(2J − 1)(2J + 3) , +(3) +where J = +5 +2 for the ground state multiplet of Ce3+, +yielding B0 +2 = -0.01077 meV. χ(T ) along both directions +was analyzed taking into account the contribution from +the CEF χi +CEF, as well as molecular field parameters λi +using +χi = χi +0 + +χi +CEF +1 − λiχi +CEF +, +(4) +where the superscript i denotes the c-axis or ab-plane. +With B0 +2 fixed from Eq. 3, values of B0 +4 = -0.0746 +meV and |B4 +4| = 0.496 meV were obtained, together +with molecular field parameters of λc = -3.55 mol/emu +and λab = 8.15 mol/emu, χc +0 = 2.2 × 10−4emu/mol +and χab +0 += −2.3 × 10−3emu/mol, and the fitted re- +sults are shown in Fig. 4(b). +These parameters yield +a CEF scheme with a Γ7 ground state Kramer’s doublet +��ψ± +1 +� += 0.883 +��± 5 +2 +� +− 0.469 +��∓ 3 +2 +� +(for positive B4 +4), and +excitations to Γ6 and Γ7 levels of ∆1 = 2.8 meV and ∆2 = +32.1 meV, respectively. At high temperatures, the small + +5 +4 +8 +0.5 +1.0 +1.5 +2.0 + 0 + 4 + 6 + 8 + 10 + 12 +4 +8 +(b) +(a) + + +C +P +/T (J/mol K +2 +) +T (K) + 0 + 1 + 1.5 + 2 + 2.5 + 3 +H // c +0 +H (T) + + +T (K) +0 +H (T) +H // ab +FIG. 5. (Color online) Temperature dependence of the spe- +cific heat of CePdGa6 in various applied magnetic fields (a) +parallel to the c-axis, and (b) within the ab-plane. +4 +8 +0.1 +0.2 +0.3 +4 +8 +0.04 +0.06 +0.08 +4 +8 +(a) + + +(emu/mol) +T (K) + 0.1 + 0.5 + 1 +H // c +0 +H (T) +0 +H (T) +H // ab + + + 1 + 2 + 4 + 6 + 8 +(emu/mol) +T (K) +(c) +(b) + + +T (K) + 1.5 + 2 + 3 +H // c +0 +H (T) +FIG. 6. (Color online) Temperature dependence of the mag- +netic susceptibility χ(T ) of CePdGa6 in different magnetic +fields parallel to the c-axis for fields (a) below, and (b) above +1 T. The vertical arrows mark the position of the antiferro- +magnetic transition. Panel (c) shows χ(T ) for various fields +applied within the ab-plane, where the dashed line shows the +evolution of TN with field. +negative B0 +2 leads to a nearly isotropic χ(T ), while at low +temperatures, the negative B0 +4 leads to the observed Ising +anisotropy with an easy c-axis. The predicted moment +along the c-axis is given by ⟨µz⟩ = +� +ψ± +1 |gJJz| ψ± +1 +� += +1.4 µB/Ce, which is larger than the value obtained from +the saturated magnetization. The positive value of λab +is consistent with ferromagnetic coupling between spins +within the basal plane, while the smaller negative λc +is consistent with weaker antiferromagnetic coupling be- +tween Ce layers. +B. +Field dependence of the magnetic properties +In order to determine the behavior of the magnetic +ground state in magnetic fields, and to map the field- +temperature phase diagrams, measurements of the spe- +H // ab +H // c +(b) +H // c +T (K) + + + 2 + 3 + 4 +M ( +/Ce) +0 +H (T) +H // c +(a) + + +M ( +/Ce) +0 +H (T) + + +M ( +/Ce) +0 +H (T) +5 K +3 K +0.3 K + + +( +cm) +0 +H (T) +1.8 K +FIG. 7. (Color online) (a) Isothermal field dependence of the +magnetization M(H) of CePdGa6 for fields along the c-axis, +at three temperatures below TN. The lower inset displays the +low field region of the data in the main panel, demonstrating +hysteresis about the metamagnetic transition, while the up- +per inset shows M(H) at 2 K for fields within the ab-plane. +(b) Field dependence of the resistivity ρ(H) of CePdGa6 at +several temperatures for fields along the c-axis. The dashed +lines show the evolution of the two metamagnetic transitions. +cific heat and magnetization were performed in differ- +ent applied fields. Figure 5(a) displays the low tempera- +ture specific heat of CePdGa6 with different fields applied +along the c-axis. It can be seen that TN is gradually sup- +pressed with increasing field, and at fields greater than +2 T, no magnetic transition is observed. Instead, there +is a broad hump in C/T , which shifts to higher tempera- +ture with increasing field, corresponding to the Schottky +anomaly from the splitting of the ground state doublet +in the applied field. In Fig. 5(b), C/T is displayed for +fields within the ab-plane, where the antiferromagnetic +transition is more robust than for fields along the c-axis, +and the broad Schottky anomaly is only clearly resolved +in a field of 12 T. The differences in the field dependence +for the two different field directions is consistent with the +low temperature Ising anisotropy in CePdGa6, where a +smaller field along the easy c-axis can bring the system +to the spin-polarized state. +The low temperature χ(T ) in different applied fields + +744 +00.0 +己.0 +T0 +-0'288ST +088880.0 +2.00 +V2.0--4806 +2 +4 +6 +8 +0.0 +0.5 +1.0 + + +C +ac +/T (a.u.) +T (K) + 0.20GPa + 0.85GPa + 1.60GPa + 2.20GPa +P (GPa) +FIG. 8. (Color online) Temperature dependence of the ac heat +capacity of CePdGa6 at various hydrostatic pressures up to +2.2 GPa. The vertical dashed line shows the position of the +ambient pressure TN, which remains nearly unchanged with +pressure. +are displayed in Fig. 6. For fields along the c-axis dis- +tinctly different behaviors are observed for different field +ranges. In a field of 0.1 T, there is a sharp peak at TN, +corresponding to entering the antiferromagnetic ground +state. At a larger field of 0.5 T, only a small hump is +observed at TN, while at low temperatures there is an +increase in χ(T ), and at higher fields there is broad peak +which is gradually suppressed with field. Meanwhile for +fields within the ab-plane up to at least 8 T, there is a +gradual suppression of TN, in line with the specific heat +results. +The isothermal magnetization as a function of field +along the c-axis at three temperatures below TN is dis- +played in Fig. 7(a), measured upon both sweeping the +field up and down. +In zero-field there is no remanent +magnetization, consistent with a purely antiferromag- +netic ground state. At 2 K, there are two metamagnetic +transitions at Hm1 = 0.4 T and Hm2 = 2.1 T, where +hysteresis is also observed indicating a first-order nature, +whereas otherwise the magnetization plateaus, with only +a weak change of the magnetization with field. This is +consistent with Hm1 and Hm2 corresponding to spin-flip +transitions, with the spins remaining orientated along the +c-axis. For fields above Hm2, no magnetic transition is +observed in the specific heat, and therefore this likely cor- +responds to the system reaching the spin polarized state, +with a saturation magnetization of Ms = 1.1 µB/Ce. On +the other hand, above Hm1 the magnetization reaches +a value of 0.35 µB/Ce, corresponding to ≈ Ms/3, in- +dicating a change of magnetic structure with a ferro- +magnetic component. While there is little change in the +field-dependence of the magnetization at 3 K, the curves +at 4 K are drastically different. Instead of there being +abrupt step-like metamagnetic transitions, the magneti- +0 +1 +2 +3 +4 +0 +2 +4 +6 +0 +1 +2 +0.0 +0.5 +1.0 +1.5 +H +m2 +H +m1 +H // c + + +T (K) +H (T) + C (T) + + (T) + + (T) + M (H) + + (H) + + +M ( +/Ce) +0 +H (T) + 2 K +FIG. 9. (Color online) Temperature-field phase diagram of +CePdGa6 at ambient pressure for fields along the easy c-axis, +from measurements of the resistivity, magnetization, and spe- +cific heat. +The solid line shows the evolution of TN, while +the dashed lines show the positions of the low temperature +metamagnetic transitions. +The magnetic structures at low +temperature are also illustrated by the orange arrows, where +in zero-field there is an antiferromagnetic ground state, while +upon applying a field the system passes through an interme- +diate ↑↑↓ phase, before entering the spin polarized state. The +inset shows the field dependence of the magnetization based +on mean-field calculations of the magnetic ground state cal- +culated using the McPhase software package [35], with the +parameters described in the text. +zation smoothly increases with field, reaching a very sim- +ilar saturation value. This suggests that at higher tem- +peratures, the spins continuously rotate upon increasing +the applied field, rather than undergoing abrupt spin flip +transitions. The field dependent magnetization at 2 K +for fields in the ab-plane is also shown in the inset of +Fig. 7(a), which smoothly changes with field, with no +sign of saturation up to at least 5 T, consistent with this +being the hard direction of magnetization. The metam- +agnetic transitions are also revealed in the field depen- +dence of the resistivity ρ(H), as displayed in Fig. 7(b) for +fields along the c-axis. At 0.3 K, two abrupt anomalies +are observed corresponding to Hm1 and Hm2, which are +also detected at 1.8 K and 3 K. Above these transitions, +there is a decrease of ρ(H), consistent with the reduced +spin-flip scattering arising from a larger ferromagnetic +component to the magnetism. +On the other hand, no +metamagnetic transitions are detected at 5 K, where in- +stead there is a broad peak in ρ(H), again consistent +with a more gradual reorientation of the spins with field +at higher temperatures. + +7 +C. +Magnetism of CePdGa6 under pressure +To determine the evolution of the magnetic order un- +der pressure, the temperature dependence of the ac spe- +cific heat of CePdGa6 was measured at several differ- +ent hydrostatic pressures up to 2.2 GPa, which are dis- +played in Fig. 8. +It can be seen from the dotted line +that there is little change of TN with pressure indicat- +ing the robustness of magnetic order. +In the case of +the layered Ce2MGa12 compounds, the TN of Ce2NiGa12 +and Ce2PdGa12 decrease with pressure, and antiferro- +magnetism is suppressed entirely above 5.5 and 7 GPa, +respectively [26, 27]. +On the other hand the TN of +Ce2IrGa12 undergoes a moderate enhancement from 3.1 +to 3.7 K for pressures up to 2.3 GPa, indicating that +this compound is located on the left side of the Doniach +phase diagram [25]. In the case of CePdGa6, the robust- +ness of TN suggests that measurements to higher pres- +sures are required to situate this compound within the +framework of the Doniach phase diagram and to exam- +ine whether there is pressure-induced quantum criticality +in CePdGa6. +IV. +DISCUSSION +Our measurements of the resistivity, magnetic sus- +ceptibility and specific heat show that CePdGa6 orders +antiferromagnetically below TN = 5.2 K, with the mo- +ments orientated along the c-axis. Figure 9 displays the +temperature-field phase diagram for magnetic fields ap- +plied along the c-axis. The phase boundaries obtained +from different measurements are highly consistent, show- +ing that TN shifts to lower temperatures with field, before +abruptly disappearing in a field of 2 T. At low temper- +atures, there are two step-like metamagnetic transitions +shown by the dashed lines, where the second transition is +to the spin polarized state, while the lower transition cor- +responds to a change of magnetic state to a phase with +a magnetization of 0.35 µB/Ce, about one-third of the +saturated value. Such step-like changes in the magne- +tization suggest that the spins are strongly constrained +along the c-axis, and therefore there are abrupt spin- +flip transitions for fields applied along the ordering di- +rection. On the other hand, at 4 K the magnetization +changes smoothly with field, reaching the same saturated +magnetization, indicating that at this temperature the +spins continuously rotate in the applied field. +Such a +change with temperature may be a consequence of only +a moderate magnetocrystalline anisotropy, as also evi- +denced by the relatively small value of the spin-wave gap +∆SW /TN ≈ 0.4, as compared to the other heavy fermion +gallides Ce2IrGa12 and Ce2PdGa12 which have ∆SW /TN +of 1.5 and 2.8, respectively [24, 25]. +From the analysis of the magnetic susceptibility includ- +ing the CEF contribution, the molecular field parameter +is positive in the ab-plane (λab), while a smaller negative +value is obtained along the c-axis (λc). Together with the +fact that only a relatively small field along the c-axis is re- +quired to reach the spin polarized state, this suggests that +the antiferromagnetic ground state consists of ferromag- +netically ordered Ce-layers coupled antiferromagnetically +along the c-axis. The simplest model for such a system +would consist of ferromagnetic Heisenberg exchange in- +teractions between nearest neighbor Ce atoms within the +ab-plane J0 > 0, and antiferromagnetic exchange inter- +actions J1 < 0 between nearest neighboring layers, as +well as a sufficiently strong Ising anisotropy. This yields +an A-type antiferromagnetic ground state consisting of +ferromagnetic layers with moments orientated along the +c-axis, where the moment direction alternates between +adjacent layers, “↑↓↑↓”. This model however cannot ac- +count for the field induced phase with one-third magneti- +zation, since for fields along the c-axis, only a metamag- +netic transition directly from the ↑↓↑↓ phase to the spin +polarized state is anticipated. +In order to realize the intermediate field-induced phase, +it is necessary to consider an antiferromagnetic exchange +J2 between next nearest neighboring layers. In this case, +from considering the classical ground state energies with +sufficiently strong Ising anisotropy, the same ↑↓↑↓ ground +state is realized for J1/J2 > 2, while a ↑↑↓↓ state oc- +curs for J1/J2 < 2 [36]. Upon applying a magnetic field +along the c-axis, there is a metamagnetic transition at +a field Hm1 to an ↑↑↓ state with a net magnetization +one-third of the saturated value, and another at Hm2 +to the spin polarized state, where Hm2/Hm1 is deter- +mined by J1/J2. We performed mean-field calculations +of the magnetic ground state and magnetization using +the McPhase software package [35], which determines +the most stable magnetic structure at a given temper- +ature and magnetic field from considering multiple ran- +dom starting moment configurations. +These took into +account the Heisenberg exchange interactions described +above, as well as the CEF Hamiltonian HCF with our de- +duced values of the Stevens parameters. As shown in the +inset of Fig. 9, the observed values of Hm1 = 0.4 T and +Hm2 = 2.1 T, from the midpoints of the metamagnetic +transitions at 2 K, are well reproduced from the mean- +field calculations at 2 K with J1 = −0.023 meV and +J2 = −0.0085 meV, where for Hm1 < H < Hm2 the ↑↑↓ +ground state has the lowest energy. Keeping these values +fixed, we find that a nearest neighbor in-plane ferromag- +netic interaction J0 = 0.034 meV can yield the observed +value of TN = 5.2 K. Therefore our analysis suggests +stronger in-plane ferromagnetic interactions, where the +value of 4J0/(2J1 + 2J2) = 2.16 is close to our fitted +value of λab/λc = 2.3. Note that here we have assumed +a ↑↓↑↓ ground state with J1/J2 > 2. Although a ↑↑↓↓ +phase has been reported in CeCoGe3 [37], such a scenario +is less likely in CePdGa6 due to the larger interlayer dis- +tances. +Compared to the layered heavy fermion antiferromag- +net CeRhIn5, the magnetism in CePdGa6 appears to +have a much more three dimensional character, whereas +it is rather two-dimensional in the former, with J1/J0 = + +8 +0.13 deduced from inelastic neutron scattering [38]. In +addition, in CeRhIn5 the easy plane anisotropy and pres- +ence of in-plane antiferromagnetic interactions give rise +to spiral magnetic order which is incommensurate along +the c-axis [13, 14], and these features may be important +factors for realizing the unconventional quantum critical- +ity and superconductivity. On the other hand, the TN of +CePdGa6 is much more robust with pressure, remaining +almost unchanged at pressures up to 2.2 GPa. Therefore +an understanding of the relationship between the mag- +netism and any quantum critical behaviors will require +measurements at considerably higher pressures. +In addition, despite the layered arrangement of Ce +atoms, the local environment of the Ce atoms is rel- +atively three dimensional, as evidenced by the derived +CEF parameters being close to that for a cubic sys- +tem (where B0 +2 = 0 and |B4 +4| = 5|B0 +4|). +This CEF +scheme can correctly predict the low-temperature Ising +anisotropy, but the predicted moment along the c-axis +is larger than that observed. While such a reduced mo- +ment compared to that predicted from the CEF level- +scheme is often observed in heavy fermion antiferromag- +nets due to screening of the moments by the Kondo effect +[14, 16, 37, 39, 40], confirming whether such a scenario is +applicable to CePdGa6 requires a more precise determi- +nation of the CEF parameters, by measurements such as +inelastic neutron scattering. +V. +CONCLUSION +In summary, we have characterized the magnetic prop- +erties of the heavy fermion antiferromagnet CePdGa6, +and their +evolution upon the application of mag- +netic fields and pressure. +We have constructed the +temperature-field phase diagram for fields along the c- +axis, where at low temperatures there are two abrupt +metamagnetic transitions corresponding to spin-flip tran- +sitions. From the analysis of the magnetic susceptibility, +we propose a CEF level scheme for the splitting of the +ground state J = 5/2 multiplet, indicating that the Ising +anisotropy at low temperatures is driven by the sizeable +B0 +4 parameter. Moreover, our results are consistent with +an antiferromagnetic ground state consisting of ferromag- +netically coupled Ce-layers, with antiferromagnetic cou- +pling between layers. We have proposed a model for the +exchange interactions which can explain the evolution of +the magnetic ordering with applied magnetic field, which +has sizeable nearest neighbor and next-nearest neighbor +layer interactions, indicating the presence of significant +long-range magnetic interactions. Despite evidence for +heavy fermion behavior, there is negligible change of TN +upon applying pressures up 2.2 GPa, and hence measure- +ments at much higher pressures are necessary to look for +evidence of quantum criticality. +VI. +ACKNOWLEDGMENTS +We are grateful to Martin Rotter for advice with the +McPhase software. This work was supported by the Na- +tional Key R&D Program of China (2017YFA0303100), +the Key R&D Program of Zhejiang Province, China +(2021C01002), and the National Natural Science Foun- +dation of China (12174332, 12034017 and 11974306). +∗ msmidman@zju.edu.cn +[1] Z. +Weng, +M. +Smidman, +L. +Jiao, +X. +Lu, +and +H. +Q. +Yuan, +Multiple +quantum +phase +transitions +and +superconductivity in +Ce-based +heavy +fermions, +Rep. Prog. Phys. 79, 094503 (2016). +[2] Q. Si and F. 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1Center for Correlated Matter and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Hangzhou 310058,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' China 2ISIS Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Rutherford Appleton Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Chilton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Didcot Oxon OX11 0QX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' United Kingdom 3Zhejiang Province Key Laboratory of Quantum Technology and Device,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Hangzhou 310058,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' China 4State Key Laboratory of Silicon Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Hangzhou 310058,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' China 5Collaborative Innovation Center of Advanced Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Nanjing 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' China (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 2023) We report the magnetic properties of the layered heavy fermion antiferromagnet CePdGa6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' and their evolution upon tuning with the application of magnetic field and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' CePdGa6 orders antiferromagnetically below TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K, where there is evidence for heavy fermion behavior from an enhanced Sommerfeld coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Our results are best explained by a magnetic ground state of fer- romagnetically coupled layers of Ce 4f-moments orientated along the c-axis, with antiferromagnetic coupling between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At low temperatures we observe two metamagnetic transitions for fields applied along the c-axis corresponding to spin-flip transitions, where the lower transition is to a dif- ferent magnetic phase with a magnetization one-third of the saturated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' From our analysis of the magnetic susceptibility, we propose a CEF level scheme which accounts for the Ising anisotropy at low temperatures, and we find that the evolution of the magnetic ground state can be explained considering both antiferromagnetic exchange between nearest neighbor and next nearest neighbor layers, indicating the influence of long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Meanwhile we find little change of TN upon applying hydrostatic pressures up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 GPa, suggesting that significantly higher pressures are required to examine for possible quantum critical behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' INTRODUCTION Heavy fermion compounds are prototypical examples of strongly correlated electron systems, and have been found to host a range of emergent phenomena including unconventional superconductivity, complex magnetic or- der and strange metal behavior [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Ce-based heavy fermions contain a Kondo lattice of Ce-ions with an un- paired 4f electron, which can both couple to other 4f mo- ments via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and undergo the Kondo interaction due to hybridization with the conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Here the RKKY interaction gives rise to long-range magnetic or- der, while the Kondo interaction favors a non-magnetic Fermi-liquid ground state with greatly enhanced quasi- particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Due to the small energy scales, the rel- ative strengths of these competing interactions can often be tuned by non-thermal parameters such as pressure, magnetic fields and chemical doping [4], and in many cases the magnetic ordering can be continuously sup- pressed to zero temperature at a quantum critical point (QCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' A major question for heavy fermion systems is the relationship between quantum criticality, and the dome of unconventional superconductivity sometimes found to encompass the QCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' CeIn3 is a canonical example of this phenomenon, which at ambient pressure orders antiferro- magnetically below TN = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 K, but exhibits a pressure- induced QCP around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='6 GPa, which is surrounded by a superconducting dome with a maximum Tc of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The layered CeMIn5 (M= transition metal) com- pounds consist of alternating layers of MIn2 and CeIn3 along the c-axis [6], and among the remarkable proper- ties is a significantly enhanced superconducting Tc for the M= Rh and Co systems, reaching over 2 K [7, 8], giving a strong indication that quasi-two-dimensionality is important for promoting heavy fermion superconduc- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Meanwhile the Ce2MIn8 compounds correspond to a stacked arrangement of two units of CeIn3, and one of MIn2 [9], and are expected to have an intermediate degree of two dimensionality relative to CeMIn5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Cor- respondingly, the superconducting phases have lower Tc values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='68 K for Ce2CoIn8 [10] and Ce2PdIn8 [11] at ambient pressure, and a maximum of Tc = 2 K at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 GPa for Ce2RhIn8 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' On the other hand, these different series of related Ce-based heavy fermion sys- tems also exhibit different magnetic ground states and crystalline electric field (CEF) level schemes [13–17] and therefore it is challenging to disentangle the role of these factors from that of the reduced dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The elucidation of the interplay between these different as- pects requires examining additional families of layered Ce-based heavy fermion systems for quantum critical be- haviors, as well as detailed characterizations of the mag- netic ground states and exchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The properties of layered Ce-based heavy fermion gal- lides have been less studied than the indium-based sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' CeGa6 has a layered tetragonal structure (space group P4/nbm), with four Ga-layers between each Ce layer [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' This compound orders magnetically below TN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='7 K, and there is evidence for the build-up of magnetic correlations at significantly higher tempera- tures [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' A more layered structure is realized in the Ce2MGa12 (M= Cu, Ni, Rh, Pd, Ir, Pt) series, where the Ce-layers are alternately separated by four Ga-layers, and units of MGa6, leading to a larger interlayer separation 2 of the Ce-atoms [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Several members of this series show evidence for both antiferromagnetism and heavy fermion behavior [20–25], where pressure can readily sup- press the antiferromagnetic transitions of Ce2NiGa12 and Ce2PdGa12 [26, 27], while evidence for field-induced crit- ical fluctuations is revealed in Ce2IrGa12 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' CePdGa6 has a different layered tetragonal structure (space group P4/mmm) displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 1(a), consist- ing of square layers of Ce-atoms, with each Ce con- tained in a CeGa4 prism, separated by PdGa2 layers [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Correspondingly, there is a distance between Ce- layers of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='92 ˚A, while the nearest neighbor in-plane Ce- Ce separation is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='34 ˚A, compared to respective values of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='54 ˚A and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='65 ˚A in CeRhIn5 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' CePdGa6 or- ders antiferromagnetically below TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K, and heavy fermion behavior is evidenced by an enhanced Sommer- feld coefficient [20, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' As such, CePdGa6 is a good can- didate to look for novel behaviors arising in quasi-two- dimensional heavy fermion systems, but there is both a lack of detailed characterizations of the magnetic ground state, and no reports of the evolution under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In addition, most measurements of CePdGa6 are reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 28, where the results are affected by the inclu- sion of an extrinsic antiferromagnetic phase Ce2PdGa12, which can be eliminated using a modified crystal growth procedure [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In this article we report detailed measurements of the magnetic properties of single crystals of CePdGa6, in- cluding their evolution upon applying magnetic fields and hydrostatic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' We find that CePdGa6 orders an- tiferromagnetically in zero-field, where the Ce-moments are orientated along the c-axis and align ferromagneti- cally within the ab-plane, but there is antiferromagnetic coupling between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At low temperatures, two meta- magnetic transitions are observed for fields along the c- axis, the lower of which corresponds to a spin-flip transi- tion to a phase with magnetization one-third of the sat- urated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' From our analysis of the magnetic suscep- tibility, we propose a CEF level scheme which can ex- plain the low temperature Ising anisotropy, and we find that from considering interactions between the nearest- neighbor and next nearest neighbor Ce-layers, the field evolution of the magnetic state can be well accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' EXPERIMENTAL DETAILS Single crystals of CePdGa6 were grown using a Ga self- flux method with a molar ratio of Ce:Pd:Ga of 1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5:15 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Starting materials of Ce ingot (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='9%), Pd pow- der (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='99%) and Ga pieces (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='99%) were loaded into an alumina crucible which was sealed in an evacuated quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The tube was heated to 1150 ◦C and held at this temperature for two hours, before being rapidly cooled to 500 ◦C at a rate of 150 K/h and then cooled more slowly to 400 C at 8 K/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' After being held at 400 ◦C for two weeks, the tube was removed from the furnace, and centrifuged to remove excess Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The 2 0 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) (a) Crystal structure of CePdGa6 where the red, blue and green atoms correspond to Ce, Pd and Ga, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' J0 represents magnetic exchange in- teractions between nearest neighbor Ce atoms within the ab- plane, J1 is between nearest neighboring layers and J2 is between next nearest layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' An image of a typical single crystal of CePdGa6 is also displayed, where each square in the background is 2 mm × 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (b) X-ray diffraction pat- tern measured on a single crystal of CePdGa6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The red dashes correspond to the positions of the (00l) Bragg peaks, indicat- ing that the [001] direction is perpendicular to the large face of the plate-like samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' obtained crystals are plate-like with typical dimensions 2 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Note that when slower cooling rates of 6 K/h or 4 K/h were used, the resulting crystals were significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Single crystals of the non-magnetic analog LaPdGa6 were also obtained using a similar pro- cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The composition was confirmed using a cold field emission scanning electron microscope (SEM) equipped with an energy dispersive x-ray spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The phase of the crystals were checked using both a PANalytical X’Pert MRD powder diffractometer using Cu-Kα radi- ation, and a Rigaku-Oxford diffraction Xtalab synergy single crystal diffractometer equipped with a HyPix hy- brid pixel array detector using Mo-Kα radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The ob- tained lattice parameters from the single crystal diffrac- tion data of a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3446(3) ˚A and c = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='9173(10) ˚A are pg c3 0 100 200 300 2 3 4 5 4 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 ( cm) T (K) T N ~ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K ( cm) T (K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) Temperature dependence of the re- sistivity ρ(T ) of CePdGa6 between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8 and 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The inset displays the low temperature resistivity, where there is a sharp anomaly at the antiferromagnetic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' in excellent agreement with previous reports [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Mea- surements of a crystal using the powder diffractometer are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 1(b), where all the Bragg peaks are well-indexed by the (00l) reflections of CePdGa6, demon- strating that the c-axis is perpendicular to the large face of the crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Resistivity and specific heat measure- ments were performed in applied fields up to 14 T using a Quantum Design Physical Property Measurement Sys- tem (PPMS-14) down to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8 K, and to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 K using a 3He insert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Resistivity measurements were performed after spot welding four Pt wires to the surface, with the exci- tation current in the ab-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Magnetization measure- ments were performed in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8 - 300 K in applied fields up to 5 T using a Quantum Design Magnetic Prop- erty Measurement System (MPMS) SQUID magnetome- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Heat capacity measurements under pressure were carried out in a piston cylinder cell, using an ac calori- metric method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Antiferromagnetic transition and CEF excitations of CePdGa6 Figure 2 displays the temperature dependence of the resistivity ρ(T ) of CePdGa6 between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8 and 300 K, which has a residual resistivity ratio [RRR = ρ(300 K)/ρ(2 K)] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' A broad shoulder is ob- served at around 50 K, which likely arises due to both the Kondo effect, and as a consequence of CEF excita- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At higher temperatures, quasilinear behavior is observed, which could be due to electron-phonon cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' As shown in the inset, there is an anomaly at around TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K, below which ρ(T ) decreases more rapidly with decreasing temperature, which corresponds C m /T (J mol 1 K 2 ) T (K) Rln2 S m (J mol 1 K 1 ) C m (J mol 1 K 1 ) T (K) (b) CePdGa 6 LaPdGa 6 C (J mol 1 K 1 ) T (K) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) (a) Magnetic contribution to the spe- cific heat Cm at low temperatures, where the red solid line shows the results from fitting with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The inset shows the total specific heat C of CePdGa6 and the non-magnetic analog LaPdGa6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (b) Temperature dependence of Cm/T and the magnetic entropy Sm of CePdGa6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The pink dotted line displays the low temperature contribution to the specific heat calculated from the CEF scheme deduced from the analysis of χ(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' to the antiferromagnetic transition reported previously [20], while no signature of the spurious transition at higher temperatures is detected [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The total spe- cific heat of CePdGa6 and nonmagnetic isostructural LaPdGa6 are shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The tem- perature dependence of the magnetic contribution to the specific heat Cm was estimated by subtracting the data of LaPdGa6, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3(a), while the spe- cific heat coefficient Cm/T and the magnetic entropy Sm of CePdGa6 are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' A pronounced λ-like anomaly is observed at TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K, as is typi- cal for a second-order magnetic phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' For T > TN, Cm/T increases with decreasing temperature, and extrapolates to a relatively large zero temperature value of 250 mJ/mol K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' As discussed below, the analysis of the magnetic susceptibility χ(T ) suggests the presence of a low lying CEF level, which could contribute to Cm/T in this temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The dotted line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3(b) shows the calculated Cm/T for the CEF level scheme de- 8030412JS12 300 e0 0S0 04 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 0 150 300 0 200 400 (b) (emu/mol) T (K) H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 T H // c H // ab (a) H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 T H // c H // ab 1/( ) (mol/emu) T (K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) (a) Low temperature magnetic sus- ceptibility χ(T ) of CePdGa6, with an applied field of µ0H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 T both parallel to the c-axis and within the ab-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (b) Temperature dependence of 1/(χ-χ0) up to 300 K for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 T applied along the two field directions, where the dashed and solid lines show the results from fitting with the CEF model described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' scribed below, which has a sizeable value in the vicinity of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Subtracting the contribution from the CEF at TN yields an estimate of γ ∼ 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4 mJ/mol K2 associated with the ground state doublet, and such an enhanced value could arise both due to heavy fermion behavior, as well as the presence of short range magnetic correlations, as inferred in CeRhIn5[30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The data below TN were analyzed using [32]: Cm = γT + c∆7/2 SW √ T exp �−∆SW T � × � 1 + 39T 20∆SW + 51 32 � T ∆SW �2� (1) where the first term corresponds to the electronic con- tribution and the second term arises due to antiferro- magnetic spin-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Here the coefficient c is related to the spinwave stiffness D via c ∝ D−3, while ∆SW is the spin-wave gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The results from fitting the zero-field data are displayed in the main panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3(a), where γ = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4 mJ/mol K2 was fixed, yielding ∆SW = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 K and c = 23 mJ/mol K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The moderate value of ∆SW is smaller than TN, unlike the layered heavy fermions gallides Ce2PdGa12 and Ce2IrGa12 where ∆SW > TN [24, 25], likely reflecting the weaker magnetocrystalline anisotropy in CePdGa6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The temperature dependence of the magnetic entropy Sm of CePdGa6 is also displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3(b), obtained by integrating Cm/T , where Cm/T was linearly extrapolated below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At TN, Sm reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='76R ln 2, which together with the expected sizeable con- tribution from the excited CEF level discussed above, suggests a reduced entropy corresponding to the ground state doublet due to Kondo screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Figure 4(a) displays the temperature dependence of the magnetic susceptibility χ(T ) of CePdGa6 at low tem- peratures, with an applied field of µ0H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 T along the c-axis and within the ab-plane, which both exhibit an anomaly at TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At low temperatures, χ(T ) is sig- nificantly larger for fields along the c-axis than in the ab-plane, demonstrating that the c-axis is the easy-axis of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At TN, there is a peak in χ(T ) for H ∥ c, while for H ∥ ab χ(T ) weakly increases below TN, indicating that this corresponds to an antiferromagnetic transition with moments ordered along the easy c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At higher temperatures, the data above 100 K can be analyzed using the Curie-Weiss law: χ=χ0+C/(T − θCW), where χ0 is a temperature-independent term, C is the Curie constant and θCW is the Curie-Weiss tem- perature, yielding θc CW = −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='7(3) K and an effective moment of µc eff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='35µB/Ce for H ∥ c, as well as θab CW = −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='9(8) K and µab eff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='49µB/Ce for H ∥ ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The obtained values of µeff for both directions are close to the full value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='54 µB for the J = 5 2 ground state multiplet of Ce3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At lower temperatures, there is a devi- ation of χ(T ) from Curie-Weiss behavior, due to the split- ting of the ground state multiplet by crystalline-electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' To analyze the CEF level scheme, we considered the following Hamiltonian for a Ce3+ ion in a tetragonal CEF [33] HCF = B0 2O0 2 + B0 4O0 4 + B4 4O4 4 (2) where Om l and Bm l are Stevens operator equivalents and parameters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The B0 2 parameter can be es- timated from the high temperature susceptibility using [34] B0 2 = 10kB � θab CW − θc CW � 3(2J − 1)(2J + 3) , (3) where J = 5 2 for the ground state multiplet of Ce3+, yielding B0 2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='01077 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' χ(T ) along both directions was analyzed taking into account the contribution from the CEF χi CEF, as well as molecular field parameters λi using χi = χi 0 + χi CEF 1 − λiχi CEF , (4) where the superscript i denotes the c-axis or ab-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' With B0 2 fixed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 3, values of B0 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0746 meV and |B4 4| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='496 meV were obtained, together with molecular field parameters of λc = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='55 mol/emu and λab = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='15 mol/emu, χc 0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 × 10−4emu/mol and χab 0 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 × 10−3emu/mol, and the fitted re- sults are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' These parameters yield a CEF scheme with a Γ7 ground state Kramer’s doublet ��ψ± 1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='883 ��± 5 2 � − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='469 ��∓ 3 2 � (for positive B4 4), and excitations to Γ6 and Γ7 levels of ∆1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8 meV and ∆2 = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At high temperatures, the small 5 4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 0 4 6 8 10 12 4 8 (b) (a) C P /T (J/mol K 2 ) T (K) 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 3 H // c 0 H (T) T (K) 0 H (T) H // ab FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) Temperature dependence of the spe- cific heat of CePdGa6 in various applied magnetic fields (a) parallel to the c-axis, and (b) within the ab-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='08 4 8 (a) (emu/mol) T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 1 H // c 0 H (T) 0 H (T) H // ab 1 2 4 6 8 (emu/mol) T (K) (c) (b) T (K) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 2 3 H // c 0 H (T) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) Temperature dependence of the mag- netic susceptibility χ(T ) of CePdGa6 in different magnetic fields parallel to the c-axis for fields (a) below, and (b) above 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The vertical arrows mark the position of the antiferro- magnetic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Panel (c) shows χ(T ) for various fields applied within the ab-plane, where the dashed line shows the evolution of TN with field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' negative B0 2 leads to a nearly isotropic χ(T ), while at low temperatures, the negative B0 4 leads to the observed Ising anisotropy with an easy c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The predicted moment along the c-axis is given by ⟨µz⟩ = � ψ± 1 |gJJz| ψ± 1 � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4 µB/Ce, which is larger than the value obtained from the saturated magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The positive value of λab is consistent with ferromagnetic coupling between spins within the basal plane, while the smaller negative λc is consistent with weaker antiferromagnetic coupling be- tween Ce layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Field dependence of the magnetic properties In order to determine the behavior of the magnetic ground state in magnetic fields, and to map the field- temperature phase diagrams, measurements of the spe- H // ab H // c (b) H // c T (K) 2 3 4 M ( /Ce) 0 H (T) H // c (a) M ( /Ce) 0 H (T) M ( /Ce) 0 H (T) 5 K 3 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 K ( cm) 0 H (T) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) (a) Isothermal field dependence of the magnetization M(H) of CePdGa6 for fields along the c-axis, at three temperatures below TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The lower inset displays the low field region of the data in the main panel, demonstrating hysteresis about the metamagnetic transition, while the up- per inset shows M(H) at 2 K for fields within the ab-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (b) Field dependence of the resistivity ρ(H) of CePdGa6 at several temperatures for fields along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The dashed lines show the evolution of the two metamagnetic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' cific heat and magnetization were performed in differ- ent applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Figure 5(a) displays the low tempera- ture specific heat of CePdGa6 with different fields applied along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' It can be seen that TN is gradually sup- pressed with increasing field, and at fields greater than 2 T, no magnetic transition is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Instead, there is a broad hump in C/T , which shifts to higher tempera- ture with increasing field, corresponding to the Schottky anomaly from the splitting of the ground state doublet in the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 5(b), C/T is displayed for fields within the ab-plane, where the antiferromagnetic transition is more robust than for fields along the c-axis, and the broad Schottky anomaly is only clearly resolved in a field of 12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The differences in the field dependence for the two different field directions is consistent with the low temperature Ising anisotropy in CePdGa6, where a smaller field along the easy c-axis can bring the system to the spin-polarized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The low temperature χ(T ) in different applied fields 744 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 己.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content="0 T0 0'288ST 088880." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='00 V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0--4806 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 C ac /T (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=') T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='20GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='85GPa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='60GPa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='20GPa P (GPa) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) Temperature dependence of the ac heat capacity of CePdGa6 at various hydrostatic pressures up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The vertical dashed line shows the position of the ambient pressure TN, which remains nearly unchanged with pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' For fields along the c-axis dis- tinctly different behaviors are observed for different field ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In a field of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 T, there is a sharp peak at TN, corresponding to entering the antiferromagnetic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At a larger field of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 T, only a small hump is observed at TN, while at low temperatures there is an increase in χ(T ), and at higher fields there is broad peak which is gradually suppressed with field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Meanwhile for fields within the ab-plane up to at least 8 T, there is a gradual suppression of TN, in line with the specific heat results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The isothermal magnetization as a function of field along the c-axis at three temperatures below TN is dis- played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 7(a), measured upon both sweeping the field up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In zero-field there is no remanent magnetization, consistent with a purely antiferromag- netic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At 2 K, there are two metamagnetic transitions at Hm1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4 T and Hm2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 T, where hysteresis is also observed indicating a first-order nature, whereas otherwise the magnetization plateaus, with only a weak change of the magnetization with field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' This is consistent with Hm1 and Hm2 corresponding to spin-flip transitions, with the spins remaining orientated along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' For fields above Hm2, no magnetic transition is observed in the specific heat, and therefore this likely cor- responds to the system reaching the spin polarized state, with a saturation magnetization of Ms = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 µB/Ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' On the other hand, above Hm1 the magnetization reaches a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='35 µB/Ce, corresponding to ≈ Ms/3, in- dicating a change of magnetic structure with a ferro- magnetic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' While there is little change in the field-dependence of the magnetization at 3 K, the curves at 4 K are drastically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Instead of there being abrupt step-like metamagnetic transitions, the magneti- 0 1 2 3 4 0 2 4 6 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 H m2 H m1 H // c T (K) H (T) C (T) (T) (T) M (H) (H) M ( /Ce) 0 H (T) 2 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' (Color online) Temperature-field phase diagram of CePdGa6 at ambient pressure for fields along the easy c-axis, from measurements of the resistivity, magnetization, and spe- cific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The solid line shows the evolution of TN, while the dashed lines show the positions of the low temperature metamagnetic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The magnetic structures at low temperature are also illustrated by the orange arrows, where in zero-field there is an antiferromagnetic ground state, while upon applying a field the system passes through an interme- diate ↑↑↓ phase, before entering the spin polarized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The inset shows the field dependence of the magnetization based on mean-field calculations of the magnetic ground state cal- culated using the McPhase software package [35], with the parameters described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' zation smoothly increases with field, reaching a very sim- ilar saturation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' This suggests that at higher tem- peratures, the spins continuously rotate upon increasing the applied field, rather than undergoing abrupt spin flip transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The field dependent magnetization at 2 K for fields in the ab-plane is also shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 7(a), which smoothly changes with field, with no sign of saturation up to at least 5 T, consistent with this being the hard direction of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The metam- agnetic transitions are also revealed in the field depen- dence of the resistivity ρ(H), as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 7(b) for fields along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 K, two abrupt anomalies are observed corresponding to Hm1 and Hm2, which are also detected at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8 K and 3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Above these transitions, there is a decrease of ρ(H), consistent with the reduced spin-flip scattering arising from a larger ferromagnetic component to the magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' On the other hand, no metamagnetic transitions are detected at 5 K, where in- stead there is a broad peak in ρ(H), again consistent with a more gradual reorientation of the spins with field at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Magnetism of CePdGa6 under pressure To determine the evolution of the magnetic order un- der pressure, the temperature dependence of the ac spe- cific heat of CePdGa6 was measured at several differ- ent hydrostatic pressures up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 GPa, which are dis- played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' It can be seen from the dotted line that there is little change of TN with pressure indicat- ing the robustness of magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In the case of the layered Ce2MGa12 compounds, the TN of Ce2NiGa12 and Ce2PdGa12 decrease with pressure, and antiferro- magnetism is suppressed entirely above 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 and 7 GPa, respectively [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' On the other hand the TN of Ce2IrGa12 undergoes a moderate enhancement from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='7 K for pressures up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3 GPa, indicating that this compound is located on the left side of the Doniach phase diagram [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In the case of CePdGa6, the robust- ness of TN suggests that measurements to higher pres- sures are required to situate this compound within the framework of the Doniach phase diagram and to exam- ine whether there is pressure-induced quantum criticality in CePdGa6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' DISCUSSION Our measurements of the resistivity, magnetic sus- ceptibility and specific heat show that CePdGa6 orders antiferromagnetically below TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K, with the mo- ments orientated along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Figure 9 displays the temperature-field phase diagram for magnetic fields ap- plied along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The phase boundaries obtained from different measurements are highly consistent, show- ing that TN shifts to lower temperatures with field, before abruptly disappearing in a field of 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' At low temper- atures, there are two step-like metamagnetic transitions shown by the dashed lines, where the second transition is to the spin polarized state, while the lower transition cor- responds to a change of magnetic state to a phase with a magnetization of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='35 µB/Ce, about one-third of the saturated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Such step-like changes in the magne- tization suggest that the spins are strongly constrained along the c-axis, and therefore there are abrupt spin- flip transitions for fields applied along the ordering di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' On the other hand, at 4 K the magnetization changes smoothly with field, reaching the same saturated magnetization, indicating that at this temperature the spins continuously rotate in the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Such a change with temperature may be a consequence of only a moderate magnetocrystalline anisotropy, as also evi- denced by the relatively small value of the spin-wave gap ∆SW /TN ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4, as compared to the other heavy fermion gallides Ce2IrGa12 and Ce2PdGa12 which have ∆SW /TN of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='8, respectively [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' From the analysis of the magnetic susceptibility includ- ing the CEF contribution, the molecular field parameter is positive in the ab-plane (λab), while a smaller negative value is obtained along the c-axis (λc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Together with the fact that only a relatively small field along the c-axis is re- quired to reach the spin polarized state, this suggests that the antiferromagnetic ground state consists of ferromag- netically ordered Ce-layers coupled antiferromagnetically along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' The simplest model for such a system would consist of ferromagnetic Heisenberg exchange in- teractions between nearest neighbor Ce atoms within the ab-plane J0 > 0, and antiferromagnetic exchange inter- actions J1 < 0 between nearest neighboring layers, as well as a sufficiently strong Ising anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' This yields an A-type antiferromagnetic ground state consisting of ferromagnetic layers with moments orientated along the c-axis, where the moment direction alternates between adjacent layers, “↑↓↑↓”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' This model however cannot ac- count for the field induced phase with one-third magneti- zation, since for fields along the c-axis, only a metamag- netic transition directly from the ↑↓↑↓ phase to the spin polarized state is anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In order to realize the intermediate field-induced phase, it is necessary to consider an antiferromagnetic exchange J2 between next nearest neighboring layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In this case, from considering the classical ground state energies with sufficiently strong Ising anisotropy, the same ↑↓↑↓ ground state is realized for J1/J2 > 2, while a ↑↑↓↓ state oc- curs for J1/J2 < 2 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Upon applying a magnetic field along the c-axis, there is a metamagnetic transition at a field Hm1 to an ↑↑↓ state with a net magnetization one-third of the saturated value, and another at Hm2 to the spin polarized state, where Hm2/Hm1 is deter- mined by J1/J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' We performed mean-field calculations of the magnetic ground state and magnetization using the McPhase software package [35], which determines the most stable magnetic structure at a given temper- ature and magnetic field from considering multiple ran- dom starting moment configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' These took into account the Heisenberg exchange interactions described above, as well as the CEF Hamiltonian HCF with our de- duced values of the Stevens parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' As shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' 9, the observed values of Hm1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='4 T and Hm2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='1 T, from the midpoints of the metamagnetic transitions at 2 K, are well reproduced from the mean- field calculations at 2 K with J1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='023 meV and J2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='0085 meV, where for Hm1 < H < Hm2 the ↑↑↓ ground state has the lowest energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Keeping these values fixed, we find that a nearest neighbor in-plane ferromag- netic interaction J0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='034 meV can yield the observed value of TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Therefore our analysis suggests stronger in-plane ferromagnetic interactions, where the value of 4J0/(2J1 + 2J2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='16 is close to our fitted value of λab/λc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Note that here we have assumed a ↑↓↑↓ ground state with J1/J2 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Although a ↑↑↓↓ phase has been reported in CeCoGe3 [37], such a scenario is less likely in CePdGa6 due to the larger interlayer dis- tances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Compared to the layered heavy fermion antiferromag- net CeRhIn5, the magnetism in CePdGa6 appears to have a much more three dimensional character, whereas it is rather two-dimensional in the former, with J1/J0 = 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='13 deduced from inelastic neutron scattering [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In addition, in CeRhIn5 the easy plane anisotropy and pres- ence of in-plane antiferromagnetic interactions give rise to spiral magnetic order which is incommensurate along the c-axis [13, 14], and these features may be important factors for realizing the unconventional quantum critical- ity and superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' On the other hand, the TN of CePdGa6 is much more robust with pressure, remaining almost unchanged at pressures up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Therefore an understanding of the relationship between the mag- netism and any quantum critical behaviors will require measurements at considerably higher pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' In addition, despite the layered arrangement of Ce atoms, the local environment of the Ce atoms is rel- atively three dimensional, as evidenced by the derived CEF parameters being close to that for a cubic sys- tem (where B0 2 = 0 and |B4 4| = 5|B0 4|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' This CEF scheme can correctly predict the low-temperature Ising anisotropy, but the predicted moment along the c-axis is larger than that observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' While such a reduced mo- ment compared to that predicted from the CEF level- scheme is often observed in heavy fermion antiferromag- nets due to screening of the moments by the Kondo effect [14, 16, 37, 39, 40], confirming whether such a scenario is applicable to CePdGa6 requires a more precise determi- nation of the CEF parameters, by measurements such as inelastic neutron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' CONCLUSION In summary, we have characterized the magnetic prop- erties of the heavy fermion antiferromagnet CePdGa6, and their evolution upon the application of mag- netic fields and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' We have constructed the temperature-field phase diagram for fields along the c- axis, where at low temperatures there are two abrupt metamagnetic transitions corresponding to spin-flip tran- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' From the analysis of the magnetic susceptibility, we propose a CEF level scheme for the splitting of the ground state J = 5/2 multiplet, indicating that the Ising anisotropy at low temperatures is driven by the sizeable B0 4 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Moreover, our results are consistent with an antiferromagnetic ground state consisting of ferromag- netically coupled Ce-layers, with antiferromagnetic cou- pling between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' We have proposed a model for the exchange interactions which can explain the evolution of the magnetic ordering with applied magnetic field, which has sizeable nearest neighbor and next-nearest neighbor layer interactions, indicating the presence of significant long-range magnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Despite evidence for heavy fermion behavior, there is negligible change of TN upon applying pressures up 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='2 GPa, and hence measure- ments at much higher pressures are necessary to look for evidence of quantum criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to Martin Rotter for advice with the McPhase software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' This work was supported by the Na- tional Key R&D Program of China (2017YFA0303100), the Key R&D Program of Zhejiang Province, China (2021C01002), and the National Natural Science Foun- dation of China (12174332, 12034017 and 11974306).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' ∗ msmidman@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content='edu.' 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+page_content=' Goremychkin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Osborn, Crystal-field excita- tions in CeCu2Si2, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} +page_content=' B 47, 14280 (1993) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfe_wu/content/2301.01444v1.pdf'} diff --git a/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/2301.04192v1.pdf.txt b/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/2301.04192v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2ad6e779278bcbc7ac0cd94086180f01a39325f --- /dev/null +++ b/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/2301.04192v1.pdf.txt @@ -0,0 +1,1706 @@ +arXiv:2301.04192v1 [math.AG] 10 Jan 2023 +QUANTIZATIONS OF LOCAL CALABI–YAU THREEFOLDS +AND THEIR MODULI OF VECTOR BUNDLES +E. BALLICO, E. GASPARIM, F. RUBILAR, B. SUZUKI +Abstract. We describe the geometry of noncommutative deformations of local Calabi–Yau +threefolds, showing that the choice of Poisson structure strongly influences the geometry of the +quantum moduli space. +Contents +1. +Introduction +1 +2. +Noncommutative deformations +2 +3. +Vector bundles on noncommutative deformations +3 +4. +Moduli of bundles on noncommutative deformations +6 +5. +Quantum moduli of bundles on W1 +8 +6. +Quantum moduli of bundles on W2 +11 +Appendix A. +Computations of H1 +15 +References +16 +1. Introduction +We discuss moduli of vector bundles on those noncommutative local Calabi–Yau threefolds that +occur in noncommutative crepant resolutions of the generalised conifolds xy − znwm = 0. Such +crepant resolutions require lines of type (−1, −1) and (−2, 0), that is, those locally modelled by +W1 := Tot(OP1(−1) ⊕ OP1(−1)) +or +W2 := Tot(OP1(−2) ⊕ OP1(0)). +Their appearance is balanced in a precise sense described in [GKMR] so that no particular +configuration of such lines is more likely to occur in a crepant resolution than any other. +Our results show that the structure of the quantum moduli space (Def. 4.3) of vector bundles +over a noncommutative deformation varies drastically depending on the choice of a Poisson +structure. +In the 2-dimensional case, [BG] described the geometry of noncommutative deformations of the +local surfaces Zk := Tot(OP1(−k)), showing that the quantum moduli space of instantons over +a noncommutative deformation (Zk, σ) can be viewed as the ´etale space of a constructible sheaf +over the classical moduli space of instantons on Zk. While in 2 dimensions vector bundles occur +as mathematical representations of instantons, in the 3-dimensional case vector bundles occur +as mathematical descriptions of BPS states, with W1 and W2 appearing as building blocks, as +described in [GKMR, GSTV, OSY]. +1 + +QUANTIZATION OF CALABI–YAU THREEFOLDS +2 +In this work, we describe the geometry of noncommutative deformations W of a Calabi–Yau +threefold W, showing that the quantum moduli space of vector bundles on together with the +map taking a vector bundle on W to its classical limit +Mℏ +j(W, σ) +Mj(W) +has the structure of a constructible sheaf, whose rank and singularity set depend explicitly on +the choice of noncommutative deformation. In particular, we describe the geometry of noncom- +mutative deformations of some crepant resolutions. It is at this point yet unclear how these +compare with Van den Bergh’s noncommutative crepant resolutions [V]. +To each Poisson structure σ on Wk, with k = 1 or k = 2 there corresponds a noncommutative +deformation (Wk, Aσ) with Aσ = (O[[ℏ]], ⋆σ) where ⋆σ is the star product corresponding to σ. All +of these Poisson structures were described in [BGKS] in terms of generators over global functions; +when σ is one of such generators, we refer to it as a basic Poisson structure. There exist Poisson +structures for which all brackets vanish on the first formal neighbourhood of P1 ⊂ Wk; we call +them extremal Poisson structures, they behave very differently from the basic ones. Our main +results are: +Theorem (5.4,6.4). Let k = 1 or 2. If σ is an extremal Poisson structure on Wk, then the +quantum moduli space Mℏ +j(Wk, σ) can be viewed as the ´etale space of a constructible sheaf Ek of +generic rank 2j − k − 1 over the classical moduli space Mj(Wk) with singular stalks of all ranks +up to 4j − k − 4. +If σ′ is another Poisson structure on Wk, then the corresponding sheaf E′ +k is a subsheaf of Ek, +with the smallest possible sheaf occurring for basic Poisson structures. +Theorem (5.2,6.2). Let k = 1 or 2. If σ is a basic Poisson structure on Wk, then the quantum +moduli space Mℏ +j(Wk, σ) and its classical limit are isomorphic: +Mℏ +j(Wk, σ) ≃ Mj(Wk) ≃ P4j−5. +Therefore, comparing these results, we see that the choice of Poisson structure has a strong +influence on the geometry of the quantum moduli space. +2. Noncommutative deformations +A holomorphic Poisson structure on a complex manifold (or smooth complex algebraic variety) +X is given by a holomorphic bivector field σ ∈ H0(X, Λ2TX) whose Schouten–Nijenhuis bracket +[σ, σ] ∈ H0(X, Λ3TX) is zero. The associated Poisson bracket is then given by the pairing ⟨ · , · ⟩ +between vector fields and forms {f, g}σ = ⟨σ, df ∧ dg⟩. +To obtain a noncommutative deformation of X one must first promote the Poisson structure to +a +star product on X, that is, a C[[ℏ]]-bilinear associative product ⋆: OX[[ℏ]] × OX[[ℏ]] → OX[[ℏ]] +which is of the form f ⋆ g = fg + �∞ +n=1 Bn(f, g) ℏn where the Bn are bidifferential operators. +The pair (X, ⋆σ) is called a deformation quantization of (X, σ) when the star product on X +satisfies B1(f, g) = {f, g}σ. +For a holomorphic Poisson manifold (X, σ) with associated Poisson bracket { · , · }σ, the sheaf +of formal functions with holomorphic coefficients on the quantization (X, ⋆σ) is +Aσ := (O[[ℏ]], ⋆σ). + +QUANTIZATION OF CALABI–YAU THREEFOLDS +3 +We call Wk(σ) = (Wk, Aσ) a noncommutative deformation of Wk, and a vector bundle on a +noncommutative deformation is by definition a locally free sheaf of Aσ-modules. These vector +bundles and their moduli are our objects of study here. +When we work with a fixed Poisson structure, we use the abbreviated notations A, { · , · } and +⋆. We also use the cut to order n represented as A(n) = O[[ℏ]]/ℏn+1. +The existence of star products on Poisson manifolds was proven in the seminal papers of Kontse- +vich [Ko1, Ko2]. For a complex algebraic variety X with structure sheaf OX, if both H1(X, OX) +and H2(X, OX) vanish, then there is a bijection +{Poisson deformations of OX}/∼ ↔ {associative deformations of OX}/∼ +where ∼ denotes gauge equivalence [Y, Cor. 11.2]. These cohomological hypothesis are verified +in the cases of W1 and W2 (but not for W3, see App. A). We now recall the basic properties of +Poisson structures on Wk or k = 1, 2. All Poisson structures on Wk may be described by giving +their generators over global functions. This is a consequence of the following result. +Lemma 2.1. [BGKS, Prop. 1] Let X be a smooth complex threefold and σ a Poisson structure +on X, then fσ is integrable for all f ∈ O(X). +Local Calabi–Yau threefolds. For k ≥ 1, we set +Wk = Tot(OP1(−k) ⊕ OP1(k − 2)). +The canonical charts for the complex manifold structure of Wk is obtained by gluing the open +sets +U = C3 +{z,u1,u2} +and +V = C3 +{ξ,v1,v2} +by the relation +(ξ, v1, v2) = (z−1, zku1, z−k+2u2). +All Poisson structures on W1 can be obtained using the following generators [BGKS, Thm. 3.2] +σ1 = ∂z ∧ ∂u1, +σ2 = ∂z ∧ ∂u2, +σ3 = u1∂u1 ∧ ∂u2 − z∂z ∧ ∂u2, +σ4 = u2∂u1 ∧ ∂u2 + z∂z ∧ ∂u1. +The W1-Poisson structures σ1, σ2, σ3, σ4 are pairwise isomorphic. +All Poisson structures on W2 can be obtained using the following generators [BGKS, Lem. 3] +σ1 = ∂z ∧ ∂u1, +σ2 = ∂z ∧ ∂u2, +σ3 = z∂z ∧ ∂u2, +σ4 = u1∂u1 ∧ ∂u2, +σ5 = 2zu1∂u1 ∧ ∂u2 − z2∂z ∧ ∂u2. +The W2-Poisson structures σ2 and σ5 on are isomorphic. +Moreover, the Poisson structures +σ1, σ2, σ3, σ4 on W2 are pairwise inequivalent, giving 4 distinct Poisson manifolds. +3. Vector bundles on noncommutative deformations +To discuss moduli of vector bundles on noncommutative deformations of Wk, for k = 1 or 2 we +will consider those bundles that are formally algebraic. +Definition 3.1. We say that p = � pnℏn ∈ O[[ℏ]] is formally algebraic if pn is a polynomial +for every n. We say that a vector bundle over (Wk, σ) is formally algebraic if it is isomorphic to +a vector bundle given by formally algebraic transition functions. In addition, if there exists N +such that pn = 0 for all n > N, we then say that p is algebraic. +Lemma 3.2. Let A be a deformation quantization of O. Then an A-module S is acyclic if and +only if S = S/ℏS is acyclic. + +QUANTIZATION OF CALABI–YAU THREEFOLDS +4 +Proof. Consider the short exact sequence +0 −→ S +ℏ +−→ S −→ S −→ 0. +It gives, for j > 0 surjections +Hj(X, S) +ℏ +−→ Hj(X, S) −→ 0. +This immediately implies that Hj(X, S) = 0 for j > 0. The converse is immediate. +□ +Notation 3.3. Let Wk be a noncommutative deformation of Wk. Denote by A(j) the line +bundle over Wk with transition function z−j, hence the pull back of O(j) on P1. +Proposition 3.4. For k = 1, 2 any line bundle on Wk is isomorphic to A(j) for some j ∈ Z, +i.e., Pic(Wk) = Z when k = 1, 2. +Proof. Let f = f0 + �∞ +n=1 �fn ℏn ∈ A∗(U ∩ V ) be the transition function for the line bundle L. +Then there exist functions a0 ∈ O∗(U) and α0 ∈ O∗(V ) such that α0f0a0 = z−j and viewing +a0 resp. α0 as elements in A∗(U) resp. A∗(V ) one has α0 ⋆ f ⋆ a0 = z−j + �∞ +n=1 fnℏn for some +fn ∈ O(U ∩ V ). We may thus assume that the transition function of L is z−j + �∞ +n=1 fnℏn. +To give an isomorphism L ≃ A(j) it suffices to define functions an ∈ O(U) and αn ∈ O(V ) +satisfying +�1 + �∞ +n=1 αnℏn� ⋆ +�z−j + �∞ +n=1 fnℏn� ⋆ +�1 + �∞ +n=1 anℏn� = z−j. +(3.5) +Collecting terms by powers of ℏ, (3.5) is equivalent to the system of equations +Sn + z−jan + z−jαn = 0 +n = 1, 2, . . . +where Sn is a finite sum involving fi, Bi for i ≤ n, but only ai, αi for i < n. The first terms are +S1 = f1 +S2 = f2 + α1f1 + a1f1 + B1 +�α1, z−j� + B1 +�z−j, a1 +� + α1z−ja1 +S3 = f3 + B2 +�α1, z−j� + B2 +�z−j, a1 +� + B1 +�α2, z−j� ++ B1 +�z−j, a2 +� + B1 +�α1, f1 +� + B1 +�α1, z−ja1 +� + B1 +�z−j, a1 +� ++ α2f1 + α2z−ja1 + α1f2 + α1f1a1 + α1z−ja2 + f2a1 + f1a2 +Since by Lem. A.1 we have H1(Wk, O) = 0 when k = 1, 2, we can solve these equations recur- +sively, by defining an to cancel out all terms of zjSn having positive powers of z and setting +αn = zjSn − an. +□ +Note that this is essentially the same proof as [BG, Prop. 6.7], and it does not work for k ≥ 3, +in fact Pic(W3) is much larger, see Lem. A.3. +We now consider vector bundles of higher rank. +Theorem 3.6. For k = 1, 2, vector bundles over Wk(σ) are filtrable. +Proof. This is a generalisation of Ballico–Gasparim–K¨oppe [BGK1, Thm. 3.2] to the noncom- +mutative case. Let E be a sheaf of A-modules. Lem. 3.2 gives that the classical limit E0 = E/ℏE +is acyclic as a sheaf of A-modules (and equivalently as a sheaf of O-modules) if and only if E is +acyclic as a sheaf of A-modules. +Filtrability for a bundle E over Wk, for k = 1, 2 was proved in [K] and is obtained from the +vanishing of cohomology groups Hi(Wk, E ⊗ SymnN ∗) for i = 1, 2, where N ∗ is the conormal + +QUANTIZATION OF CALABI–YAU THREEFOLDS +5 +bundle of ℓ ⊂ Wk and n > 0 are integers, the proof proceeds by induction on n. +In the +noncommutative case, let S denote the kernel of the projection A(n) → A(n−1). By construction +we have that S/ℏS = SymnN ∗ and the required vanishing of cohomologies is guaranteed by +Lem. 3.2. +□ +The analogous proof does not work for W3, see [K, Rem. 3.13]. It is unknown whether bundles +on Wk are filtrable when k ≥ 3. +Remark 3.7. There are also some particular features happening only when k = 1. +Every +holomorphic vector bundle on W1 is algebraic [K, Thm. 3.10], and W1 is formally rigid [GKRS, +Thm. 11]. In contrast, if k > 1, then Wk has as infinite-dimensional family of deformations. In +particular, a deformation family for W2 can be given by (ξ, v1, v2) = +� +z−1, z2u1 + z � +j>0 tjuj +2, u2 +� +[GKRS, Thm. 13] and this family contains infinitely many distinct manifolds [BGS, Thm. 1.13]. +Furthermore, for k > q > 0, Wk can be deformed to Wq [BGS, Thm. 1.28]. +For each Poisson manifold (Wk, σ), we want to study moduli spaces of vector bundles over +(Wk, ⋆) where ⋆ is the corresponding star product. +[K, Prop. 3.1] showed that a rank 2 bundle E on Wk with first Chern class c1(E) = 0 is deter- +mined by a canonical transition matrix +�zj +p +0 +z−j +� +where, using ǫ = 0, 1 we have: +p = +2j−2 +� +s=ǫ +2j−2−s +� +i=1−ǫ +j−1 +� +l=i+s−j+1 +pliszlui +1us +2 +for +k = 1, +(3.8) +and +p = +∞ +� +s=ǫ +j−1 +� +i=1−ǫ +j−1 +� +l=2i−j+1 +pliszlui +1us +2 +for +k = 2. +(3.9) +Accordingly, for a noncommutative deformation (Wk, σ) we define the notion of canonical tran- +sition matrix as: +T = +�zj +p +0 +z−j +� +with +p = +∞ +� +n=0 +pnℏn ∈ Ext1(A(j), A(−j)). +(3.10) +Where we have that each pn can be given the same canonical form of the classical case, which +can be seen using: +Lemma 3.11. Let A be a deformation quantization of OWk with k = 1 or 2. There is an +injective map of C-vector spaces +Ext1 +A(A(j), A(−j)) +∞ +� +n=0 +Ext1 +O(O(j), O(−j))ℏn ≃ Ext1 +O(O(j), O(−j))[[ℏ]] +p = p0 + +∞ +� +n=1 +pnℏn +(p0, p1ℏ, p2ℏ2, . . . ) +where pi ∈ Ext1(O(j), O(−j)). +Proof. Ext1 +A(A(j), A(−j)) is the quotient of Ext1 +O(O(j), O(−j))[[ℏ]] by the relations +qn ≃ qn + � pipn−i. +□ + +QUANTIZATION OF CALABI–YAU THREEFOLDS +6 +We wish to describe the structure of moduli spaces of vector bundles on Wk. Using the results +of this section, we may proceed analogously to the classical (commutative) setup, to extract +moduli spaces out of extension groups of line bundles, by considering extension classes up to +bundle isomorphism. +4. Moduli of bundles on noncommutative deformations +We recall the notion of isomorphism of vector bundles on a noncommutative deformation of Wk. +Definition 4.1. Let E and E′ be vector bundles over (Wk, σ) defined by transition matrices T +and T ′ respectively. An isomorphism between E and E′ is given by a pair of matrices AU and +AV with entries in Aσ(U) and Aσ(V ), respectively, which are invertible with respect to ⋆ and +such that +T ′ = AV ⋆ T ⋆ AU. +Notation 4.2. Denoting by Ext1 +Alg(A(j), A(−j)) the subset of formally algebraic extension +classes, we denote by Mj(Wk) the quotient +Mj(Wk) := Ext1 +Alg(A(j), A(−j))/∼ +consisting of those classes of formally algebraic vector bundles (Def. 3.1), whose classical limit is a +stable vector bundle of charge j. Here ∼ denotes bundle isomorphism as in Def. 4.1 and following +[BGK2] stability means that the classical limit does not split on the 0-th formal neighbourhood. +We denote by Mℏn +j (Wk, σ) the moduli of bundles obtained by imposing the cut-off ℏn+1 = 0, +that is, the superscript ℏn means quantised to level n. +Note that Mj(Wk, σ) := Mℏ0 +j (Wk, σ) = Mj(Wk) recovers the classical moduli space obtained +when ℏ = 0, while Mℏ +j(Wk, σ) denotes the moduli on the first order quantization, which will be +the focus of this work. Accordingly: +Definition 4.3. We call Mj(Wk, σ) the classical moduli space and Mℏ +j(Wk, σ) the quantum +moduli space of bundles on Wk. +Lemma 4.4. [BGS, Thm. 2.7] The classical moduli spaces of vector bundles of rank 2 and +splitting type j on Wk has dimension 4j − 5. +Definition 4.5. The splitting type of a vector bundle E on (Wk, σ) is the one of its classical +limit [BG, Def. 5.2]. Hence, when the classical limit is an SL(2, C) bundle, the splitting type of +E is the smallest integer j such that E can be written as an extension of A(j) by A(−j). +We fix a splitting type j and look at rank 2 bundles on the first formal neighbourhood ℓ(1) of +ℓ ≃ P1 ⊂ W1 together with their extensions up to first order in ℏ. We now calculate isomorphism +classes. Let p + p′ℏ and q + q′ℏ be two extension classes in Ext1 +A(A(j), A(−j)) which are of +splitting type j, i.e. in canonical U-coordinates p, p′, q, q′ are multiples of u1, u2. +According to Def. 4.1 bundles defined by p + p′ℏ and q + q′ℏ are isomorphic, if there exist +invertible matrices +�a + a′ℏ +b + b′ℏ +c + c′ℏ +d + d′ℏ +� +and +�α + α′ℏ +β + β′ℏ +γ + γ′ℏ +δ + δ′ℏ +� +whose entries are holomorphic on U and V , respectively, such that +�α + α′ℏ +β + β′ℏ +γ + γ′ℏ +δ + δ′ℏ +� +⋆ +�zj +q + q′ℏ +0 +z−j +� += +�zj +p + p′ℏ +0 +z−j +� +⋆ +�a + a′ℏ +b + b′ℏ +c + c′ℏ +d + d′ℏ +� +. +(4.6) + +QUANTIZATION OF CALABI–YAU THREEFOLDS +7 +We wish to determine the constraints such an isomorphism imposes on the coefficients of q and +q′. This is more conveniently rewritten by multiplying by the right-inverse of +� +zj q+q′ℏ +0 +z−j +� +, which +(modulo ℏ2) is +�z−j +−q − q′ℏ + 2z−j{zj, q}ℏ +0 +zj +� +. +We have that the zero section ℓ ≃ P1 is cut out inside Wk by u1 = u2 = 0. Hence, the n-th +formal neighbourhood of ℓ is by definition ℓ(n) = OW1 +In+1 where I =< u1, u2 >. So, on ℓ(1) we +have that u2 +1 = u2 +2 = u1u2 = 0 and therefore we may write +a = a0 + a1 +1u1 + a2 +1u2, +α = α0 + α1 +1u1 + α2 +1u2, +etc., where ai +1, αi +1, etc. are holomorphic functions of z. +Following the details of the proof of [G, Prop. 3.3] we assume in (4.6) that a0 = α0, d0 = δ0 are +constant and b = β = 0. Since we already know that on the classical limit the only equivalence +on ℓ(1) is projectivization [G, Prop. 3.2], we assume p = q, keeping in mind a projectivization to +be done in the end. We may also assume that the determinants of the changes of coordinates +on the classical limit are 1. Accordingly, we rewrite (4.6) as: +�α + α′ℏ +β′ℏ +γ + γ′ℏ +δ + δ′ℏ +� += +�zj +p + p′ℏ +0 +z−j +� +⋆ +�a + a′ℏ +b′ℏ +c + c′ℏ +d + d′ℏ +� +⋆ +�z−j +−p − q′ℏ + 2{zj, p}z−jℏ +0 +zj +� +(4.7) +where a0 = d0 = α0 = δ0 = 1. +Since we already know the moduli in the classical limit, we only need to study terms containing +ℏ, which after multiplying are: +(1, 1) = a′ + {zja, z−j} + {zj, a}z−j + {pc, z−j} + {p, c}z−j + (pc′ + p′c)z−j +(1, 2) = {p, d}zj − {a, p}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + z2jb′ +− (pa′ + q′a)zj + (pd′ + p′d)zj − (pc′ + p′c + q′c)p +(2, 1) = z−2jc′ +(2, 2) = d′ + {z−jd, zj} + {z−j, d}zj − {z−jc, p} − {z−j, c}p − (pc′ + q′c)z−j + 2{zj, p}z−2jc. +All four terms must be adjusted using the free variables to only contain expressions which are +holomorphic on V to satisfy (4.7). For example, in the (2, 1) term this condition is satisfied +precisely when c′ is a section of O(2j). Computing Poisson brackets, we see that the (1, 1) and +(2, 2) terms can always be made holomorphic on V by appropriate choices of c and d′, leaving +the coefficients of a′ free. We will need to use these free coefficients for the next step. +It remains to analyse the (1, 2) term. Because we are working on the first formal neighbourhood +of ℓ, terms in u2 +1, u1u2, u2 +2 or higher vanish (recall that we assume that p, p′, q′ are multiples of u1 +or u2). Since z2jb′ is there to cancel out any possible terms having power of z greater or equal +to 2j, we remove it from the expression, keeping in mind that we only need to cancel out the +coefficients of the monomials ziu1 and ziu2 with i ≤ 2j − 1 in the expression: +(1, 2) = {p, d+a}zj −{zj, a}p+{zj, p}a+{pd, zj}+2z−j{zj, p}pc+p(d′−a′)zj +(p′−q′)zj. (�) +To determine the quantum moduli spaces, we must verify what restrictions are imposed on q′ so +that p′ and q′ define isomorphic bundles. Since this requires computing brackets, the analysis +must be carried out separately for each noncommutative deformation. + +QUANTIZATION OF CALABI–YAU THREEFOLDS +8 +5. Quantum moduli of bundles on W1 +The Calabi–Yau threefold we consider in this section is the crepant resolution of the conifold +singularity xy − zw = 0, that is, +W1 := Tot(OP1(−1) ⊕ OP1(−1)). +We will carry out calculations using the canonical coordinates W1 = U ∪ V where U ≃ C3 ≃ V +with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates on U ∩ V ≃ C∗ × C × C given +by +�ξ = z−1 , +v1 = zu1 , +v2 = zu2 +� . +Consequently, global functions on W1 are generated over C by the monomials 1, u1, zu1, u2, zu2. +For each specific noncommutative deformation (W1, Aσ), we wish to compare the quantum and +classical moduli spaces of vector bundles, see Def. 4.3. +This is part of the general quest to +understand how deformations of a variety affect moduli of bundles on it, and it is worth noting +that no commutative deformation of W1 is known to exits. +For a rank 2 bundle E on a noncommutative deformation W1 with a canonical matrix +� +zj +p +0 +z−j +� +as in (3.10) where p = �∞ +n=0 pnℏn, expression (3.8) gives us the general form of the coefficients +pn. In particular, on the first formal neighbourhood, we have: +p = +j−1 +� +l=−j+2 +pl10zlu1 + +j−1 +� +l=−j+2 +pl01zlu2, +(5.1) +where p = 0 if j = 1. +Each noncommutative deformation comes from some Poisson structure which determines the first +order terms of the corresponding star product, see Sec. 2. The most basic Poisson structures σ +on W1 are those which generate all others over global functions. We call these generators the +basic Poisson structures. +Theorem 5.2. If σ is a basic Poisson structure on W1, then the quantum moduli space Mℏ +j(Wk, σ) +and its classical limit are isomorphic: +Mℏ +j(W1, σ) ≃ Mj(W1) ≃ P4j−5. +Proof. We perform the computations using the bracket σ1 = ∂z ∧ ∂u1; the choice of such a +generator is irrelevant, since all the 4 generators give pairwise isomorphic Poisson manifolds. To +obtain an isomorphism, we need to cancel out all coefficients of the terms +z2u1, . . . , z2j−1u1 +and +z2u2, . . . , z2j−1u2 +appearing in expression �. Calculating σ1 brackets, we have {zj, f} = jzj−1 ∂f +∂u1 +, and following +expressions for a and d coming from the classical part +a = 1 + a1 +1u1 + a2 +1u2, +d = 1 − a1 +1u1 − a2 +1u2, +where ai +1 and di +1 are functions of z, gives ∂a +∂u1 += a1 +1, +∂d +∂u1 += −a1 +1, so that {p, d} − {a, p}zj = +−2 +�∂p +∂za1 +1 − ∂a +∂z +∂p +∂u1 +� +zj. Therefore, expression � becomes +� = −2 +�∂p +∂z a1 +1 − ∂a +∂z +∂p +∂u1 +� +zj + 2j +� ∂p +∂u1 +(a1 +1u1 + a2 +1u2 + pc) +� +zj−1 + p(d′ − a′)zj + (p′ − q′)zj. +Now we need to cancel out separately the coefficients of each monomial ziu1 and ziu2 for 2 ≤ +i ≤ 2j − 1, that is, all those terms potentially giving nonholomorphic functions. To determine + +QUANTIZATION OF CALABI–YAU THREEFOLDS +9 +the classes in the moduli space we need to verify what constraints are imposed on q′. Take for +instance the monomial ziu1 in (p′d − q′a)zj. Since a′ remains free we can always choose its +corresponding coefficient in order to cancel out the term in ziu1 in the entire expression of (1, 2). +Indeed, notice that the expressions p(d′ − a′)zj and (p′d − q′a)zj contain monomials of the same +orders, all of which may be adjusted to zero by choosing a′. Moreover the first three summands +in � also contain the same list of monomials, hence may also be absorbed by the appropriate +choices of coefficients of a, a′ and c. +Since this process can be independently carried out for each monomial, we then conclude that +the expression � can be made holomorphic on V for any choice of q′. +Hence, there are no +restrictions on q′. Thus, we obtain an equivalence p + p′ℏ ∼ p + q′ℏ for all q′ and the projection +onto the classical limit (the first coordinate) +π1 : Mℏ +j(W1, σ) → Mj(W1) +taking (p, p′) to p is an isomorphism. The isomorphism type of the moduli space is given in +[BGS, Lem. 6.2] as P4j−5. +□ +We now calculate the quantum moduli space for the particular choice of splitting type j = 2 and +for a different choice of Poisson structure on W1. We use the notation p ∈ Mj(W1) to refer to +a point in the classical moduli space, that is, a rank 2 bundle is labelled by its extension class. +Example 5.3 (j = 2 and σ = u1σ1). Here we write +p = p0zu1 + p1u1 + p2zu2 + p3u2, +p′ = p′ +0zu1 + p′ +1u1 + p′ +2zu2 + p′ +3u2. +for the first order part of the extension class, where we have renamed the coefficients to simplify +notation ( p0 := p110, p1 := p010, p2 := p101, p3 := p001). Lem. 2.1 implies that σ = u1σ1 is also +a Poisson structure on W1. With this choice, all brackets acquire an extra u1 in comparison +to the bracket σ1 used in the proof of Thm. 5.2, so that in the first formal neighbourhood the +(1, 2)-term described in � simplifies to just: +� = z2p(d′ − a′) + z2(p′ − q′). +Here a′ = a′ +0 + a′1u1 + a′2u2, +d′ = d′ +0 − d′1u1 − d′2u2, so that +d′ − a′ = (d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2. +Hence, the total expression of � is +� += +(p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)((d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2)) ++(p′ +0 − q′ +0)z3u1 + (p′ +1 − q′ +1)z2u1 + (p′ +2 − q′ +2)z3u2 + (p′ +3 − q′ +3)z2u2, +where we canceled out all the monomials containing u2 +1, u1u2, and u2 +2, since we work on the first +formal neighbourhood. We rename (d′ − a′)0(z) = λ0 + λ1z + λ2z2 + . . . to simplify notation, +and since all terms in (1, 2) having powers of z equal to 4 and higher can be cancelled out by +the appropriate choice of the z2jb′, it suffices to analyse the expression +� += +(p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)(λ0 + λ1z) ++(q′ +0 − p′ +0)z3u1 + (q′ +1 − p′ +1)z2u1 + (q′ +2 − p′ +2)z3u2 + (q′ +3 − p′ +3)z2u2. +To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z3u1, z2u1, z3u2, z2u2 +in � with appropriate choices of λi. Consequently, q′ ∼ p′ if and only if the following equality +holds for some choice of λ0 and λ1: + + + + +q′ +0 − p′ +0 +q′ +1 − p′ +1 +q′ +2 − p′ +2 +q′ +3 − p′ +3 + + + + = λ0 + + + + +p0 +p1 +p2 +p3 + + + + + λ1 + + + + +p1 +0 +p3 +0 + + + + . + +QUANTIZATION OF CALABI–YAU THREEFOLDS +10 +When the vectors v1 = (p0, p1, p2, p3) and v2 = (0, p1, 0, p3) are linearly independent, the point +q′ belongs to the plane that passes through the point p′ with v1 and v2 as direction vectors. +Therefore, whenever v1 and v2 are linearly independent vectors, the fibre over p = (p0, p1, p2, p3) +is a copy of C4 foliated by 2-planes. The leaf containing a point p′ forms the equivalence class +of p′. Thus, the moduli space over the fibre over p is parametrised by the 2-plane through the +origin in the direction perpendicular to v1, v2 over the point p, except when p1 = p3 = 0. +In contrast, the fibre over a point p = (p0, 0, p2, 0) is a copy of C4 foliated by lines in the +direction of v1 = (p0, 0, p2, 0). In this case, the moduli space over p is parametrised by a copy of +C3 perpendicular to v1. +We conclude that Mℏ +2(W1, σ) → M2(W1) ≃ P3 (where the isomorphism is given by Lem. 4.4) is +the ´etale space of a constructible sheaf, whose stalks have +• dimension 2 over the Zariski open set (p1, p3) ̸= (0, 0), and +• dimension 3 over the P1 cut out by p1 = p3 = 0 in P3. +The same techniques readily generalise to give a description of the quantum moduli spaces for +other choices of noncommutative deformations. +Theorem 5.4. If σ is an extremal Poisson structure on W1, then the quantum moduli space +Mℏ +j(W1, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 2 over +the classical moduli space Mj(W1) with singular stalks up to rank 4j − 5. +Proof. We give the details of the case j = 3, for an extremal Poisson structure, that is, the case +when all brackets vanish on the first formal neighbourhood. The general case is clear from these +calculations, just notationally more complicated. +When j = 3 and σ = u1σ1, expression � becomes: +� = p(d′ − a′)z3 + (p′d − q′a)z3, +and we get a system of equations: +� += +� +p0z5u1 + p1z4u1 + p2z3u1 + p3z2u1 + p4z5u2 + p5z4u2 + p6z3u2 + p7z2u2 +� +·(λ0 + λ1z + λ2z2 + λ3z3 + λ4z4) + ++(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1 + (p′ − q′)3z2u1 ++(p′ − q′)4z5u2 + (p′ − q′)5z4u2 + (p′ − q′)6z3u2 + (p′ − q′)7z2u2. +To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z5u1, z4u1, z3u1, z2u1, +z5u2, z4u2, z3u2, z2u2 in � with appropriate choices of λi. Consequently, q′ ∼ p′ if and only if +the following equality holds for some choice of λ0, λ1, λ2, λ3: + + + + + + + + + + + + + +q′ +0 − p′ +0 +q′ +1 − p′ +1 +q′ +2 − p′ +2 +q′ +3 − p′ +3 +q′ +4 − p′ +4 +q′ +5 − p′ +5 +q′ +6 − p′ +6 +q′ +7 − p′ +7 + + + + + + + + + + + + + += λ0 + + + + + + + + + + + + + +p0 +p1 +p2 +p3 +p4 +p5 +p6 +p7 + + + + + + + + + + + + + ++ λ1 + + + + + + + + + + + + + +p1 +p2 +p3 +0 +p5 +p6 +p7 +0 + + + + + + + + + + + + + ++ λ2 + + + + + + + + + + + + + +p2 +p3 +0 +0 +p6 +p7 +0 +0 + + + + + + + + + + + + + ++ λ3 + + + + + + + + + + + + + +p3 +0 +0 +0 +p7 +0 +0 +0 + + + + + + + + + + + + + +. + +QUANTIZATION OF CALABI–YAU THREEFOLDS +11 +Consider now the family U of vector spaces over M2(W1) ≃ P7 whose fibre at p is given by +Up = + + + + + + + + + + + + + +p0 +p1 +p2 +p3 +p1 +p2 +p3 +0 +p2 +p3 +0 +0 +p3 +0 +0 +0 +p4 +p5 +p6 +p7 +p5 +p6 +p7 +0 +p6 +p7 +0 +0 +p7 +0 +0 +0 + + + + + + + + + + + + + +. +Now, the quantum moduli space is obtained from this family after dividing by the equivalence +relation ∼ over each point p. Hence +Mℏ +2(W1, σ) = U/ ∼ . +We conclude that Mℏ +2(W1, σ) → M2(W1) ≃ P7 (where the isomorphism is given by Lem. 4.4) is +the ´etale space of a constructible sheaf or rank 4, with stalk at p having dimension equal to the +corank of Up, in this case +4 ≤ dim Mℏ +2(W1, σ)p = 8 − rk Up ≤ 7. +In the general case we have +2j − 2 ≤ dim Mℏ +j(W1, σ)p = corank Up =≤ 4j − 5. +□ +6. Quantum moduli of bundles on W2 +The Calabi–Yau threefold we consider in this section is a crepant resolution of the singularity +xy − w2 = 0 in C4, that is +W2 := Tot(OP1(−2) ⊕ OP1) = Z2 × C. +Similarly to what we did for W1, we will carry out calculations using the canonical coordinates +W2 = U ∪V where U ≃ C3 ≃ V with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates +on U ∩ V ≃ C∗ × C × C given by +�ξ = z−1 , +v1 = z2u1 , +v2 = u2 +� . +Consequently, global holomorphic functions on W2 are generated by 1, u1, zu1, z2u1, u2. +For each specific noncommutative deformation (W2, Aσ), we wish to compare the quantum and +classical moduli spaces of vector bundles, see Def. 4.3. +For a rank 2 bundle E on a noncommutative deformation W2 with a canonical matrix +� +zj +p +0 +z−j +� +as in (3.10) where p = �∞ +n=0 pnℏn, expression (3.8) gives us the general form of the coefficients +pn. In particular, on the first formal neighbourhood, we have: +p = +j−1 +� +l=−j+3 +pl10zlu1 + +j−1 +� +l=−j+1 +pl01zlu2 +(6.1) +where in case j = 1 we have only p001u2. +To describe the quantum moduli for Poisson structures on W2, we consider the expression �: +� = {p, d + a}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + p(d′ − a′)zj + (p′d − q′a)zj, +where we need to cancel out the coefficients of z3u1, . . . , z2j−1u1 +and +zu2, . . . , z2j−1u2. + +QUANTIZATION OF CALABI–YAU THREEFOLDS +12 +Each noncommutative deformation comes from some Poisson structure. The most basic Poisson +structures σ on W2 are those which generate all others over global functions. We call these +generators the basic Poisson structures. Now, we compute the quantum moduli of bundles for +them. +Remark. We observe that the 4 Poisson manifolds (W2, σi) for i = 1, 2, 3, 4, are pairwise +nonisomorphic. This can be verified by the table of their degeneracy loci: +W2 Poisson structures +bracket +degeneracy +σ1 +σ2 +∅ +σ3 +σ4 +∪ +Nevertheless, the 4 quantum moduli spaces defined by these basic Poisson structures turn out to +be all isomorphic. +Theorem 6.2. If σ is a basic Poisson structure on W2, then the quantum moduli space Mℏ +j(Wk, σ) +and its classical limit are isomorphic: +Mℏ +j(W2, σ) ≃ Mj(W2) ≃ P4j−5. +Proof. We carry out calculations for the basic bracket σ4 = u1∂u1 ∧ ∂u2. It does turn out that +the result is the same for the the basic brackets. The calculation for σ4 is shorter, since any +of the brackets having one entry equal to zj vanishes. Because we work on the first formal +neighbourhood, we also remove the expressions that are quadratic in the ui variables. +So, the expression � that remains to be analysed simplifies to: +� = {p, d + a}zj + p(d′ − a′)zj + (p′d − q′a)zj, +where we must cancel out the coefficients of the monomials z3u1, . . . , z2j−1u1 and zu2, . . . , z2j−1u2. +On the first formal neighbourhood, we write +a = 1 + a1(z)u1 + a2(z)u2, +d = 1 + d1(z)u1 + d2(z)u2, +and +a′ = a′ +0(z) + a′ +1(z)u1 + a′ +2(z)u2, +d′ = d′ +0(z) + d′ +1(z)u1 + d′ +2(z)u2, +so that the partials are +∂uia = ai(z) +∂uid = di(z) +and +∂u2a = a2(z) +∂u2d = d2(z). +The extension class given in (3.9) becomes p = +j−1 +� +l=3−j +pl10zlu1 + +j−1 +� +l=1−j +pl01zlu2, and computing +the bracket gives +{p, d + a}zj = + + +j−1 +� +l=3−j +pl10zl + + (d2(z) + a2(z))zju1 + + + +j−1 +� +l=1−j +pl01zl + + (d1(z) + a1(z))zju1. +To work with a simpler notation, we present details of � when j = 2, in which case we can +express the extension class as +p = p0zu1 + p1zu2 + p2u2 + p3z−1u2, + +QUANTIZATION OF CALABI–YAU THREEFOLDS +13 +having renamed the coefficients for simplicity (making p0 := p110, p1 := p101, p2 := p001, p3 := +p−101). We will point out the steps for generalising to higher j. +Assuming j = 2, we have +{p, d + a}z2 = p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1. +To obtain equivalence between q′ and p′, we must cancel out coefficients of z3u1, zu2, z2u2, z3u2 +in the expression of �, which becomes +� += +p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1 ++(p0z3u1 + p1z3u2 + p2z2u2 + p3zu2)(d′ +0(z) − a′ +0(z)) ++(p′ +0z3u1 + p′ +1z3u2 + p′ +2z2u2 + p′ +3zu2) +−(q′ +0z3u1 + q′ +1z3u2 + q′ +2z2u2 + q′ +3zu2). +Since the highest power of z to be considered is 3, we observe that d2(z) + a2(z) may be chosen +conveniently, we cancel out all terms in z3u1. We may also choose d1(z) + a1(z) = 0, leaving +� += +(p1z3u2 + p2z2u2 + p3zu2)(d′ +0(z) − a′ +0(z)) ++(p′ +1z3u2 + p′ +2z2u2 + p′ +3zu2) +−(q′ +1z3u2 + q′ +2z2u2 + q′ +3zu2). +Now we may choose d′ +0 − a′ +0 appropriately to cancel out all terms in u2. We conclude that there +are no conditions imposed on q′. In other words, here p + p′ℏ is equivalent to p + q′ℏ for any +choice of q′. Hence, the quantum and classical moduli spaces are isomorphic. +The generalisation to higher j works out similarly, we can first choose di + ai for i > 0 to cancel +out the coefficients of u1 and then choose d′ +0 −a′ +0 to take care of the coefficients of u2. So, for all +j using the bracket σ4 we conclude that the quantum and classical moduli spaces are isomorphic +Mℏ +j(W2, σ4) ≃ Mj(W2) ≃ P4j−5 +where the second isomorphism is proven in [K, Prop. 3.24]. +□ +Example 6.3. Now choose any Poisson structure of W2 for which all brackets in � vanish on +neighbourhood 1, for example σ = u1σ4 = u2 +1∂u1 ∧ ∂u2 works. In such a case, the expression for +� reduces to: +� = p(d′ − a′)zj + (p′d − q′a)zj. +Now, consider the case of j = 2, when we have: +� += +(p0z3u1 + p1z3u2 + p2z2u2 + p3zu2) + (d′ +0(z) − a′ +0(z)) ++(p′ +0z3u1 + p′ +1z3u2 + p′ +2z2u2 + p′ +3zu2) +−(q′ +0z3u1 + q′ +1z3u2 + q′ +2z2u2 + q′ +3zu2). +Setting +d′ +0(z) − a′ +0(z) = λ0 + λ1z + λ2z2, +we get a system of equations: + + + + +q′ +0 − p′ +0 +q′ +1 − p′ +1 +q′ +2 − p′ +2 +q′ +3 − p′ +3 + + + + = + + + + +λ0 +0 +0 +0 +0 +λ0 +λ1 +λ2 +0 +0 +λ0 +λ1 +0 +0 +0 +λ0 + + + + + + + + +p0 +p1 +p2 +p3 + + + + . +Since we can choose λ1 and λ2 to solve the second and third equations, we see that q′ +1 and q′ +2 +are free. Hence (q′ +0, q′ +1, q′ +2, q′ +3) ∼ λ0(q′ +0, ∗, ∗, q′ +3), and our system of equations reduces to +�q′ +0 − p′ +0 +q′ +3 − p′ +3 +� += λ0 +�p0 +p3 +� +, + +QUANTIZATION OF CALABI–YAU THREEFOLDS +14 +which is the parametric equation of a line in the (q′ +0, q′ +3)-plane whenever (p0, p3) ̸= (0, 0). The +entire question of moduli now reduces to the 2-dimensional case, disregarding p1, p2 coordinates. +If (p0, p3) ̸= (0, 0), then the equivalence class of q′ in the fibre over the point p is the 1-dimensional +subspace L directed by the vector (p0, p3) and passing through (q′ +0, q′ +3) in the (p′ +0, p′ +3)-plane. +If p0 = p3 = 0, then we must have the equality (q′ +0, q′ +3) = (p′ +0, p′ +3). So, its the equivalence class +consists of a single point. +Accordingly, the set of equivalence classes over p can be represented either by the line L⊥ by +the origin perpendicular to L (directed by (−p3, p0) when (p0, p3) ̸= (0, 0) or else by the entire +(p′ +0, p′ +3)-plane over (0, 0). +We conclude that Mℏ +2(W2, σ) → M2(W2) ≃ P3 (where the isomorphism is given by Lem. 4.4) is +the ´etale space of a constructible sheaf, whose stalks have +• dimension 1 over the Zariski open set (p0, p3) ̸= (0, 0), and +• dimension 2 over the P1 cut out by p0 = p3 = 0 in P3. +In fact, we could express this moduli space as a sheaf given by an extension of OP3(+1) by a +torsion sheaf. +Theorem 6.4. If σ is an extremal Poisson structure on W2, then the quantum moduli space +Mℏ +j(W2, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 3 over +the classical moduli space Mj(W2) with singular stalks up to rank 4j − 6. +Proof. Now, for j = 3, we write down the extremal example when the brackets vanish on the +first formal neighbourhood. The generalisation of the extremal cases to all j becomes clear from +this example. Where, assuming all brackets vanish on the first formal neighbourhood, we need +to cancel out the coefficients of z3u1, . . . , z2j−1u1 +and +zu2, . . . , z2j−1u2 in +� = p(d′ − a′)zj + (p′d − q′a)zj. +For j = 3 we have +p = +2 +� +l=0 +pl10zlu1 + +2 +� +l=−2 +pl01zlu2, +which we rewrite as +p = p0z2u1 + p1zu1 + p2u1 + p3z2u2 + p4z1u2 + p5u2 + p6z–1u2 + p7z−2u2. +Setting +d′ +0(z) − a′ +0(z) = λ0 + λ1z + λ2z2 + λ3z3 + λ4z4, +expression +� = p(d′ − a′)z3 + (p′d − q′a)z3 +becomes +� += +� +p0z5u1 + p1z4u1 + p2z3u1 + p3z5u2 + p4z4u2 + p5z3u2 + p6z2u2 + p7zu2 +� +·(λ0 + λ1z + λ2z2 + λ3z3 + λ4z4) ++(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1 ++(p′ − q′)3z5u2 + (p′ − q′)4z4u2 + (p′ − q′)5z3u2 + (p′ − q′)6z2u2 + (p′ − q′)7zu2. +To start with we notice that λ3 and λ4 can always be chosen to solve the equations involving q′ +3 +and q′ +4 so that these 2 coordinates can take any value, that is, there are isomorphisms +(q′ +0, q′ +1, q′ +2, q′ +3, q′ +4, q′ +5, q′ +6, q′ +7) ∼ (q′ +0, q′ +1, q′ +2, ∗, ∗, q′ +5, q′ +6, q′ +7). + +QUANTIZATION OF CALABI–YAU THREEFOLDS +15 +Consequently, we may remove q′ +3, q′ +4 and rewrite the reduced system as: + + + + + + + + + +q′ +0 − p′ +0 +q′ +1 − p′ +1 +q′ +2 − p′ +2 +q′ +5 − p′ +5 +q′ +6 − p′ +6 +q′ +7 − p′ +7 + + + + + + + + + += λ0 + + + + + + + + + +p0 +p1 +p2 +p5 +p6 +p7 + + + + + + + + + ++ λ1 + + + + + + + + + +p1 +p2 +0 +p6 +p7 +0 + + + + + + + + + ++ λ2 + + + + + + + + + +p2 +0 +0 +p7 +0 +0 + + + + + + + + + +. +Here q′ ∼ p′ if and only if the equality holds for some choice of λ0, λ1, λ2. Consider now the +family U of vector spaces over M2(W2) ≃ P7 whose fibre at p is given by +Up = + + + + + + + + + +p0 +p1 +p2 +p1 +p2 +0 +p2 +0 +0 +p5 +p6 +p7 +p6 +p7 +0 +p7 +0 +0 + + + + + + + + + +. +Now, the quantum moduli space is obtained from this family after dividing by the equivalence +relation ∼ over each point p. Hence +Mℏ +2(W2, σ) = U/ ∼ . +We conclude that Mℏ +2(W2, σ) → M2(W2) ≃ P7 (where the isomorphism is given by Lem. 4.4) is +the ´etale space of a constructible sheaf, with stalk at p having dimension equal to the corank of +Up, in this case +3 ≤ dim Mℏ +2(W2, σ)p = corank Up = 6 − rk Up ≤ 6. +In the general case we then have +2j − 3 ≤ dim Mℏ +j(W2, σ)p = corank Up = 2j − rk Up ≤ 4j − 6. +□ +Appendix A. Computations of H1 +Lemma A.1. H1(W1, O) = H1(W2, O) = 0. +Proof. A 1-cocycle τ ∈ O(U ∩ V ) may be written in the form +τU = +∞ +� +l=−∞ +∞ +� +i=0 +∞ +� +s=0 +τliszlui +1us +2. +Since terms containing only positive powers of z are holomorphic on the U-chart +τU ∼ +−1 +� +l=−∞ +∞ +� +i=0 +∞ +� +s=0 +τliszlui +1us +2, +where ∼ denotes cohomological equivalence. Changing to V coordinates we have +τV = +−1 +� +l=−∞ +∞ +� +i=0 +∞ +� +s=0 +τlisξ−l+ki+(−k+2)svi +1vs +2, +(A.2) +where, for k = 1, 2 exponents of ξ are non-negative. +Thus, τV is holomorphic on V , and +τ ∼ 0. +□ +Lemma A.3. H1(W3, O) is infinite dimensional over C. + +QUANTIZATION OF CALABI–YAU THREEFOLDS +16 +Proof. As in the proof of Lem. A.1 we arrive at the expression (A.2) for the 1-cocycle τ on the +V -chart, which in the case k = 3, gives +τV ∼ +−1 +� +l=−∞ +∞ +� +i=0 +∞ +� +s=0 +τlisξ−l+3i−svi +1vs +2. +The terms that are not holomorphic on V are all of those satisfying −l + 3i − s < 0. +We conclude that all terms having s > 3i − l, namely all of +−1 +� +l=−∞ +∞ +� +i=0 +∞ +� +s=3i−l+1 +τliszlui +1us +2 +are nontrivial in first cohomology, so that dim H1(W3, O) = ∞. +□ +Acknowledgements. E. Ballico is a member of GNSAGA of INdAM (Italy). E. Gasparim +acknowledges support of Vicerrector´ıa de Investigaci´on y Desarrollo Tecnol´ogico, UCN Chile. +F. Rubilar acknowledges support of ANID-FAPESP cooperation 2019/13204-0. B. Suzuki was +supported by Grant 2021/11750-7 S˜ao Paulo Research Foundation - FAPESP. +References +[BG] +S. Barmeier, E. Gasparim, Quantization of local surfaces and rebel instantons, J. Noncommut. Geom. +16 (2022) 311–351. +[BGK1] +E. Ballico, E. Gasparim, T. K¨oppe, Local moduli of holomorphic bundles, J. Pure Appl. Algebra 213 +n.4 (2009) 397–408. +[BGK2] +E. Ballico, E. Gasparim, T. K¨oppe, Vector bundles near negative curves: moduli and local Euler char- +acteristic. Comm. Algebra 37 n.8 (2009) 2688–2713. +[BGKS] +E. Ballico, E. Gasparim, T. K¨oppe, B. Suzuki, Poisson structures on the conifold and local Calabi-Yau +threefolds, Rep. Math. Phys. 90 n.3 (2022) 299–324. +[BGS] +E. Ballico, E. Gasparim, B. Suzuki, Infinite dimensional families of Calabi–Yau threefolds and moduli +of vector bundles, J. Pure Appl. Algebra 225 n.4 (2021) 106554, 24 pp.. +[G] +E. Gasparim, Rank two bundles on the blow-up of C2, J. Algebra 199 n.2 (1998) 581–590. +[GKMR] E. Gasparim, T. K¨oppe, P. Majumdar, K. Ray, BPS state counting on singular varieties, J. Phys. A 45 +n. 26 (2012) 265401 20pp.. +[GKRS] +E. Gasparim, T. K¨oppe, F. Rubilar, and B. Suzuki., Deformations of noncompact Calabi–Yau threefolds, +Rev. Colombiana Mat. 52 n.1 (2018)41–57. +[GSTV] +E. Gasparim, B. Suzuki, A. Torres-Gomez, C. Varea, Topological String Partition Function on Gener- +alised Conifolds, Journal of Mathematical Physics, 58 (2017) 1–16. +[Ko1] +M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 n.3 (2003) 157– +216. +[Ko2] +M. Kontsevich, Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 n.3 (2001) 271– +294. +[K] +T. K¨oppe, Moduli of bundles on local surfaces and threefolds, PhD thesis, The University of Edinburgh +(2010). +[OSY] +H. Ooguri, P. Su�lkowski, M. Yamazaki, Wall Crossing as Seen by Matrix Models, Commun. Math. Phys. +307 (2011) 429–462. +[S] +B. Szendr˝oi, Non-commutative Donaldson–Thomas invariants and the conifold, Geom. Topol. 12 (2008) +1171–1202. +[V] +M. Van den Bergh, Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel. Springer, +Berlin (2004) 749–770. +[Y] +A. Yekutieli, Twisted deformation quantization of algebraic varieties, Adv. Math. 268 (2015) 271–294. +Ballico - Dept. Mathematics, Univ. of Trento, Povo Italy; ballico@science.unitn.it, +Gasparim - Depto. Matem´aticas, Univ. Cat´olica del Norte, Chile; etgasparim@gmail.com, +Rubilar - Depto. Matem´aticas, Univ. Sant. Concepci´on, Chile; francisco.rubilar.arriagada@gmail.com, +Suzuki - Depto. Matem´atica, Univ. de S˜ao Paulo, Brazil; obrunosuzuki@gmail.com. + diff --git a/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/load_file.txt b/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f66863584c1af4f593026ea2745f0e43d7d92c1 --- /dev/null +++ b/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/load_file.txt @@ -0,0 +1,780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf,len=779 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='04192v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='AG] 10 Jan 2023 QUANTIZATIONS OF LOCAL CALABI–YAU THREEFOLDS AND THEIR MODULI OF VECTOR BUNDLES E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' BALLICO, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' GASPARIM, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' RUBILAR, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' SUZUKI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We describe the geometry of noncommutative deformations of local Calabi–Yau threefolds, showing that the choice of Poisson structure strongly influences the geometry of the quantum moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Noncommutative deformations 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Vector bundles on noncommutative deformations 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moduli of bundles on noncommutative deformations 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W1 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W2 11 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Computations of H1 15 References 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Introduction We discuss moduli of vector bundles on those noncommutative local Calabi–Yau threefolds that occur in noncommutative crepant resolutions of the generalised conifolds xy − znwm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Such crepant resolutions require lines of type (−1, −1) and (−2, 0), that is, those locally modelled by W1 := Tot(OP1(−1) ⊕ OP1(−1)) or W2 := Tot(OP1(−2) ⊕ OP1(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Their appearance is balanced in a precise sense described in [GKMR] so that no particular configuration of such lines is more likely to occur in a crepant resolution than any other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Our results show that the structure of the quantum moduli space (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3) of vector bundles over a noncommutative deformation varies drastically depending on the choice of a Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the 2-dimensional case, [BG] described the geometry of noncommutative deformations of the local surfaces Zk := Tot(OP1(−k)), showing that the quantum moduli space of instantons over a noncommutative deformation (Zk, σ) can be viewed as the ´etale space of a constructible sheaf over the classical moduli space of instantons on Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' While in 2 dimensions vector bundles occur as mathematical representations of instantons, in the 3-dimensional case vector bundles occur as mathematical descriptions of BPS states, with W1 and W2 appearing as building blocks, as described in [GKMR, GSTV, OSY].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1 QUANTIZATION OF CALABI–YAU THREEFOLDS 2 In this work, we describe the geometry of noncommutative deformations W of a Calabi–Yau threefold W, showing that the quantum moduli space of vector bundles on together with the map taking a vector bundle on W to its classical limit Mℏ j(W, σ) Mj(W) has the structure of a constructible sheaf, whose rank and singularity set depend explicitly on the choice of noncommutative deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, we describe the geometry of noncom- mutative deformations of some crepant resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It is at this point yet unclear how these compare with Van den Bergh’s noncommutative crepant resolutions [V].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To each Poisson structure σ on Wk, with k = 1 or k = 2 there corresponds a noncommutative deformation (Wk, Aσ) with Aσ = (O[[ℏ]], ⋆σ) where ⋆σ is the star product corresponding to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All of these Poisson structures were described in [BGKS] in terms of generators over global functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' when σ is one of such generators, we refer to it as a basic Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' There exist Poisson structures for which all brackets vanish on the first formal neighbourhood of P1 ⊂ Wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' we call them extremal Poisson structures, they behave very differently from the basic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Our main results are: Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let k = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is an extremal Poisson structure on Wk, then the quantum moduli space Mℏ j(Wk, σ) can be viewed as the ´etale space of a constructible sheaf Ek of generic rank 2j − k − 1 over the classical moduli space Mj(Wk) with singular stalks of all ranks up to 4j − k − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ′ is another Poisson structure on Wk, then the corresponding sheaf E′ k is a subsheaf of Ek, with the smallest possible sheaf occurring for basic Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let k = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is a basic Poisson structure on Wk, then the quantum moduli space Mℏ j(Wk, σ) and its classical limit are isomorphic: Mℏ j(Wk, σ) ≃ Mj(Wk) ≃ P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Therefore, comparing these results, we see that the choice of Poisson structure has a strong influence on the geometry of the quantum moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Noncommutative deformations A holomorphic Poisson structure on a complex manifold (or smooth complex algebraic variety) X is given by a holomorphic bivector field σ ∈ H0(X, Λ2TX) whose Schouten–Nijenhuis bracket [σ, σ] ∈ H0(X, Λ3TX) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The associated Poisson bracket is then given by the pairing ⟨ · , · ⟩ between vector fields and forms {f, g}σ = ⟨σ, df ∧ dg⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To obtain a noncommutative deformation of X one must first promote the Poisson structure to a star product on X, that is, a C[[ℏ]]-bilinear associative product ⋆: OX[[ℏ]] × OX[[ℏ]] → OX[[ℏ]] which is of the form f ⋆ g = fg + �∞ n=1 Bn(f, g) ℏn where the Bn are bidifferential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The pair (X, ⋆σ) is called a deformation quantization of (X, σ) when the star product on X satisfies B1(f, g) = {f, g}σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a holomorphic Poisson manifold (X, σ) with associated Poisson bracket { · , · }σ, the sheaf of formal functions with holomorphic coefficients on the quantization (X, ⋆σ) is Aσ := (O[[ℏ]], ⋆σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 3 We call Wk(σ) = (Wk, Aσ) a noncommutative deformation of Wk, and a vector bundle on a noncommutative deformation is by definition a locally free sheaf of Aσ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' These vector bundles and their moduli are our objects of study here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' When we work with a fixed Poisson structure, we use the abbreviated notations A, { · , · } and ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We also use the cut to order n represented as A(n) = O[[ℏ]]/ℏn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The existence of star products on Poisson manifolds was proven in the seminal papers of Kontse- vich [Ko1, Ko2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a complex algebraic variety X with structure sheaf OX, if both H1(X, OX) and H2(X, OX) vanish, then there is a bijection {Poisson deformations of OX}/∼ ↔ {associative deformations of OX}/∼ where ∼ denotes gauge equivalence [Y, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' These cohomological hypothesis are verified in the cases of W1 and W2 (but not for W3, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We now recall the basic properties of Poisson structures on Wk or k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All Poisson structures on Wk may be described by giving their generators over global functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is a consequence of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' [BGKS, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1] Let X be a smooth complex threefold and σ a Poisson structure on X, then fσ is integrable for all f ∈ O(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Local Calabi–Yau threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For k ≥ 1, we set Wk = Tot(OP1(−k) ⊕ OP1(k − 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The canonical charts for the complex manifold structure of Wk is obtained by gluing the open sets U = C3 {z,u1,u2} and V = C3 {ξ,v1,v2} by the relation (ξ, v1, v2) = (z−1, zku1, z−k+2u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All Poisson structures on W1 can be obtained using the following generators [BGKS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2] σ1 = ∂z ∧ ∂u1, σ2 = ∂z ∧ ∂u2, σ3 = u1∂u1 ∧ ∂u2 − z∂z ∧ ∂u2, σ4 = u2∂u1 ∧ ∂u2 + z∂z ∧ ∂u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The W1-Poisson structures σ1, σ2, σ3, σ4 are pairwise isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All Poisson structures on W2 can be obtained using the following generators [BGKS, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3] σ1 = ∂z ∧ ∂u1, σ2 = ∂z ∧ ∂u2, σ3 = z∂z ∧ ∂u2, σ4 = u1∂u1 ∧ ∂u2, σ5 = 2zu1∂u1 ∧ ∂u2 − z2∂z ∧ ∂u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The W2-Poisson structures σ2 and σ5 on are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moreover, the Poisson structures σ1, σ2, σ3, σ4 on W2 are pairwise inequivalent, giving 4 distinct Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Vector bundles on noncommutative deformations To discuss moduli of vector bundles on noncommutative deformations of Wk, for k = 1 or 2 we will consider those bundles that are formally algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We say that p = � pnℏn ∈ O[[ℏ]] is formally algebraic if pn is a polynomial for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We say that a vector bundle over (Wk, σ) is formally algebraic if it is isomorphic to a vector bundle given by formally algebraic transition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In addition, if there exists N such that pn = 0 for all n > N, we then say that p is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let A be a deformation quantization of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Then an A-module S is acyclic if and only if S = S/ℏS is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consider the short exact sequence 0 −→ S ℏ −→ S −→ S −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It gives, for j > 0 surjections Hj(X, S) ℏ −→ Hj(X, S) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This immediately implies that Hj(X, S) = 0 for j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The converse is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let Wk be a noncommutative deformation of Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Denote by A(j) the line bundle over Wk with transition function z−j, hence the pull back of O(j) on P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For k = 1, 2 any line bundle on Wk is isomorphic to A(j) for some j ∈ Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=', Pic(Wk) = Z when k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let f = f0 + �∞ n=1 �fn ℏn ∈ A∗(U ∩ V ) be the transition function for the line bundle L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Then there exist functions a0 ∈ O∗(U) and α0 ∈ O∗(V ) such that α0f0a0 = z−j and viewing a0 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' α0 as elements in A∗(U) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A∗(V ) one has α0 ⋆ f ⋆ a0 = z−j + �∞ n=1 fnℏn for some fn ∈ O(U ∩ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We may thus assume that the transition function of L is z−j + �∞ n=1 fnℏn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To give an isomorphism L ≃ A(j) it suffices to define functions an ∈ O(U) and αn ∈ O(V ) satisfying �1 + �∞ n=1 αnℏn� ⋆ �z−j + �∞ n=1 fnℏn� ⋆ �1 + �∞ n=1 anℏn� = z−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5) Collecting terms by powers of ℏ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5) is equivalent to the system of equations Sn + z−jan + z−jαn = 0 n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' where Sn is a finite sum involving fi, Bi for i ≤ n, but only ai, αi for i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The first terms are S1 = f1 S2 = f2 + α1f1 + a1f1 + B1 �α1, z−j� + B1 �z−j, a1 � + α1z−ja1 S3 = f3 + B2 �α1, z−j� + B2 �z−j, a1 � + B1 �α2, z−j� + B1 �z−j, a2 � + B1 �α1, f1 � + B1 �α1, z−ja1 � + B1 �z−j, a1 � + α2f1 + α2z−ja1 + α1f2 + α1f1a1 + α1z−ja2 + f2a1 + f1a2 Since by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 we have H1(Wk, O) = 0 when k = 1, 2, we can solve these equations recur- sively, by defining an to cancel out all terms of zjSn having positive powers of z and setting αn = zjSn − an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Note that this is essentially the same proof as [BG, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7], and it does not work for k ≥ 3, in fact Pic(W3) is much larger, see Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We now consider vector bundles of higher rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For k = 1, 2, vector bundles over Wk(σ) are filtrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is a generalisation of Ballico–Gasparim–K¨oppe [BGK1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2] to the noncom- mutative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let E be a sheaf of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2 gives that the classical limit E0 = E/ℏE is acyclic as a sheaf of A-modules (and equivalently as a sheaf of O-modules) if and only if E is acyclic as a sheaf of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Filtrability for a bundle E over Wk, for k = 1, 2 was proved in [K] and is obtained from the vanishing of cohomology groups Hi(Wk, E ⊗ SymnN ∗) for i = 1, 2, where N ∗ is the conormal QUANTIZATION OF CALABI–YAU THREEFOLDS 5 bundle of ℓ ⊂ Wk and n > 0 are integers, the proof proceeds by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the noncommutative case, let S denote the kernel of the projection A(n) → A(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' By construction we have that S/ℏS = SymnN ∗ and the required vanishing of cohomologies is guaranteed by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ The analogous proof does not work for W3, see [K, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It is unknown whether bundles on Wk are filtrable when k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' There are also some particular features happening only when k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Every holomorphic vector bundle on W1 is algebraic [K, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10], and W1 is formally rigid [GKRS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In contrast, if k > 1, then Wk has as infinite-dimensional family of deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, a deformation family for W2 can be given by (ξ, v1, v2) = � z−1, z2u1 + z � j>0 tjuj 2, u2 � [GKRS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 13] and this family contains infinitely many distinct manifolds [BGS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Furthermore, for k > q > 0, Wk can be deformed to Wq [BGS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For each Poisson manifold (Wk, σ), we want to study moduli spaces of vector bundles over (Wk, ⋆) where ⋆ is the corresponding star product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' [K, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1] showed that a rank 2 bundle E on Wk with first Chern class c1(E) = 0 is deter- mined by a canonical transition matrix �zj p 0 z−j � where, using ǫ = 0, 1 we have: p = 2j−2 � s=ǫ 2j−2−s � i=1−ǫ j−1 � l=i+s−j+1 pliszlui 1us 2 for k = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='8) and p = ∞ � s=ǫ j−1 � i=1−ǫ j−1 � l=2i−j+1 pliszlui 1us 2 for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='9) Accordingly, for a noncommutative deformation (Wk, σ) we define the notion of canonical tran- sition matrix as: T = �zj p 0 z−j � with p = ∞ � n=0 pnℏn ∈ Ext1(A(j), A(−j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10) Where we have that each pn can be given the same canonical form of the classical case, which can be seen using: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let A be a deformation quantization of OWk with k = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' There is an injective map of C-vector spaces Ext1 A(A(j), A(−j)) ∞ � n=0 Ext1 O(O(j), O(−j))ℏn ≃ Ext1 O(O(j), O(−j))[[ℏ]] p = p0 + ∞ � n=1 pnℏn (p0, p1ℏ, p2ℏ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' ) where pi ∈ Ext1(O(j), O(−j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Ext1 A(A(j), A(−j)) is the quotient of Ext1 O(O(j), O(−j))[[ℏ]] by the relations qn ≃ qn + � pipn−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ QUANTIZATION OF CALABI–YAU THREEFOLDS 6 We wish to describe the structure of moduli spaces of vector bundles on Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Using the results of this section, we may proceed analogously to the classical (commutative) setup, to extract moduli spaces out of extension groups of line bundles, by considering extension classes up to bundle isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moduli of bundles on noncommutative deformations We recall the notion of isomorphism of vector bundles on a noncommutative deformation of Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let E and E′ be vector bundles over (Wk, σ) defined by transition matrices T and T ′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' An isomorphism between E and E′ is given by a pair of matrices AU and AV with entries in Aσ(U) and Aσ(V ), respectively, which are invertible with respect to ⋆ and such that T ′ = AV ⋆ T ⋆ AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Denoting by Ext1 Alg(A(j), A(−j)) the subset of formally algebraic extension classes, we denote by Mj(Wk) the quotient Mj(Wk) := Ext1 Alg(A(j), A(−j))/∼ consisting of those classes of formally algebraic vector bundles (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1), whose classical limit is a stable vector bundle of charge j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here ∼ denotes bundle isomorphism as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 and following [BGK2] stability means that the classical limit does not split on the 0-th formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We denote by Mℏn j (Wk, σ) the moduli of bundles obtained by imposing the cut-off ℏn+1 = 0, that is, the superscript ℏn means quantised to level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Note that Mj(Wk, σ) := Mℏ0 j (Wk, σ) = Mj(Wk) recovers the classical moduli space obtained when ℏ = 0, while Mℏ j(Wk, σ) denotes the moduli on the first order quantization, which will be the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Accordingly: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We call Mj(Wk, σ) the classical moduli space and Mℏ j(Wk, σ) the quantum moduli space of bundles on Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' [BGS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7] The classical moduli spaces of vector bundles of rank 2 and splitting type j on Wk has dimension 4j − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The splitting type of a vector bundle E on (Wk, σ) is the one of its classical limit [BG, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, when the classical limit is an SL(2, C) bundle, the splitting type of E is the smallest integer j such that E can be written as an extension of A(j) by A(−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We fix a splitting type j and look at rank 2 bundles on the first formal neighbourhood ℓ(1) of ℓ ≃ P1 ⊂ W1 together with their extensions up to first order in ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We now calculate isomorphism classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let p + p′ℏ and q + q′ℏ be two extension classes in Ext1 A(A(j), A(−j)) which are of splitting type j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' in canonical U-coordinates p, p′, q, q′ are multiples of u1, u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' According to Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 bundles defined by p + p′ℏ and q + q′ℏ are isomorphic, if there exist invertible matrices �a + a′ℏ b + b′ℏ c + c′ℏ d + d′ℏ � and �α + α′ℏ β + β′ℏ γ + γ′ℏ δ + δ′ℏ � whose entries are holomorphic on U and V , respectively, such that �α + α′ℏ β + β′ℏ γ + γ′ℏ δ + δ′ℏ � ⋆ �zj q + q′ℏ 0 z−j � = �zj p + p′ℏ 0 z−j � ⋆ �a + a′ℏ b + b′ℏ c + c′ℏ d + d′ℏ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6) QUANTIZATION OF CALABI–YAU THREEFOLDS 7 We wish to determine the constraints such an isomorphism imposes on the coefficients of q and q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is more conveniently rewritten by multiplying by the right-inverse of � zj q+q′ℏ 0 z−j � , which (modulo ℏ2) is �z−j −q − q′ℏ + 2z−j{zj, q}ℏ 0 zj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We have that the zero section ℓ ≃ P1 is cut out inside Wk by u1 = u2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, the n-th formal neighbourhood of ℓ is by definition ℓ(n) = OW1 In+1 where I =< u1, u2 >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, on ℓ(1) we have that u2 1 = u2 2 = u1u2 = 0 and therefore we may write a = a0 + a1 1u1 + a2 1u2, α = α0 + α1 1u1 + α2 1u2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=', where ai 1, αi 1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' are holomorphic functions of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Following the details of the proof of [G, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3] we assume in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6) that a0 = α0, d0 = δ0 are constant and b = β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since we already know that on the classical limit the only equivalence on ℓ(1) is projectivization [G, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2], we assume p = q, keeping in mind a projectivization to be done in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We may also assume that the determinants of the changes of coordinates on the classical limit are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Accordingly, we rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6) as: �α + α′ℏ β′ℏ γ + γ′ℏ δ + δ′ℏ � = �zj p + p′ℏ 0 z−j � ⋆ �a + a′ℏ b′ℏ c + c′ℏ d + d′ℏ � ⋆ �z−j −p − q′ℏ + 2{zj, p}z−jℏ 0 zj � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7) where a0 = d0 = α0 = δ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since we already know the moduli in the classical limit, we only need to study terms containing ℏ, which after multiplying are: (1, 1) = a′ + {zja, z−j} + {zj, a}z−j + {pc, z−j} + {p, c}z−j + (pc′ + p′c)z−j (1, 2) = {p, d}zj − {a, p}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + z2jb′ − (pa′ + q′a)zj + (pd′ + p′d)zj − (pc′ + p′c + q′c)p (2, 1) = z−2jc′ (2, 2) = d′ + {z−jd, zj} + {z−j, d}zj − {z−jc, p} − {z−j, c}p − (pc′ + q′c)z−j + 2{zj, p}z−2jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All four terms must be adjusted using the free variables to only contain expressions which are holomorphic on V to satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For example, in the (2, 1) term this condition is satisfied precisely when c′ is a section of O(2j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Computing Poisson brackets, we see that the (1, 1) and (2, 2) terms can always be made holomorphic on V by appropriate choices of c and d′, leaving the coefficients of a′ free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We will need to use these free coefficients for the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It remains to analyse the (1, 2) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Because we are working on the first formal neighbourhood of ℓ, terms in u2 1, u1u2, u2 2 or higher vanish (recall that we assume that p, p′, q′ are multiples of u1 or u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since z2jb′ is there to cancel out any possible terms having power of z greater or equal to 2j, we remove it from the expression, keeping in mind that we only need to cancel out the coefficients of the monomials ziu1 and ziu2 with i ≤ 2j − 1 in the expression: (1, 2) = {p, d+a}zj −{zj, a}p+{zj, p}a+{pd, zj}+2z−j{zj, p}pc+p(d′−a′)zj +(p′−q′)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (�) To determine the quantum moduli spaces, we must verify what restrictions are imposed on q′ so that p′ and q′ define isomorphic bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since this requires computing brackets, the analysis must be carried out separately for each noncommutative deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W1 The Calabi–Yau threefold we consider in this section is the crepant resolution of the conifold singularity xy − zw = 0, that is, W1 := Tot(OP1(−1) ⊕ OP1(−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We will carry out calculations using the canonical coordinates W1 = U ∪ V where U ≃ C3 ≃ V with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates on U ∩ V ≃ C∗ × C × C given by �ξ = z−1 , v1 = zu1 , v2 = zu2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently, global functions on W1 are generated over C by the monomials 1, u1, zu1, u2, zu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For each specific noncommutative deformation (W1, Aσ), we wish to compare the quantum and classical moduli spaces of vector bundles, see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is part of the general quest to understand how deformations of a variety affect moduli of bundles on it, and it is worth noting that no commutative deformation of W1 is known to exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a rank 2 bundle E on a noncommutative deformation W1 with a canonical matrix � zj p 0 z−j � as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10) where p = �∞ n=0 pnℏn, expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='8) gives us the general form of the coefficients pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, on the first formal neighbourhood, we have: p = j−1 � l=−j+2 pl10zlu1 + j−1 � l=−j+2 pl01zlu2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1) where p = 0 if j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Each noncommutative deformation comes from some Poisson structure which determines the first order terms of the corresponding star product, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The most basic Poisson structures σ on W1 are those which generate all others over global functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We call these generators the basic Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is a basic Poisson structure on W1, then the quantum moduli space Mℏ j(Wk, σ) and its classical limit are isomorphic: Mℏ j(W1, σ) ≃ Mj(W1) ≃ P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We perform the computations using the bracket σ1 = ∂z ∧ ∂u1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' the choice of such a generator is irrelevant, since all the 4 generators give pairwise isomorphic Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To obtain an isomorphism, we need to cancel out all coefficients of the terms z2u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and z2u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2 appearing in expression �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Calculating σ1 brackets, we have {zj, f} = jzj−1 ∂f ∂u1 , and following expressions for a and d coming from the classical part a = 1 + a1 1u1 + a2 1u2, d = 1 − a1 1u1 − a2 1u2, where ai 1 and di 1 are functions of z, gives ∂a ∂u1 = a1 1, ∂d ∂u1 = −a1 1, so that {p, d} − {a, p}zj = −2 �∂p ∂za1 1 − ∂a ∂z ∂p ∂u1 � zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Therefore, expression � becomes � = −2 �∂p ∂z a1 1 − ∂a ∂z ∂p ∂u1 � zj + 2j � ∂p ∂u1 (a1 1u1 + a2 1u2 + pc) � zj−1 + p(d′ − a′)zj + (p′ − q′)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now we need to cancel out separately the coefficients of each monomial ziu1 and ziu2 for 2 ≤ i ≤ 2j − 1, that is, all those terms potentially giving nonholomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To determine QUANTIZATION OF CALABI–YAU THREEFOLDS 9 the classes in the moduli space we need to verify what constraints are imposed on q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Take for instance the monomial ziu1 in (p′d − q′a)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since a′ remains free we can always choose its corresponding coefficient in order to cancel out the term in ziu1 in the entire expression of (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Indeed, notice that the expressions p(d′ − a′)zj and (p′d − q′a)zj contain monomials of the same orders, all of which may be adjusted to zero by choosing a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moreover the first three summands in � also contain the same list of monomials, hence may also be absorbed by the appropriate choices of coefficients of a, a′ and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since this process can be independently carried out for each monomial, we then conclude that the expression � can be made holomorphic on V for any choice of q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, there are no restrictions on q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Thus, we obtain an equivalence p + p′ℏ ∼ p + q′ℏ for all q′ and the projection onto the classical limit (the first coordinate) π1 : Mℏ j(W1, σ) → Mj(W1) taking (p, p′) to p is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The isomorphism type of the moduli space is given in [BGS, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2] as P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ We now calculate the quantum moduli space for the particular choice of splitting type j = 2 and for a different choice of Poisson structure on W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We use the notation p ∈ Mj(W1) to refer to a point in the classical moduli space, that is, a rank 2 bundle is labelled by its extension class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3 (j = 2 and σ = u1σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here we write p = p0zu1 + p1u1 + p2zu2 + p3u2, p′ = p′ 0zu1 + p′ 1u1 + p′ 2zu2 + p′ 3u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' for the first order part of the extension class, where we have renamed the coefficients to simplify notation ( p0 := p110, p1 := p010, p2 := p101, p3 := p001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 implies that σ = u1σ1 is also a Poisson structure on W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' With this choice, all brackets acquire an extra u1 in comparison to the bracket σ1 used in the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2, so that in the first formal neighbourhood the (1, 2)-term described in � simplifies to just: � = z2p(d′ − a′) + z2(p′ − q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here a′ = a′ 0 + a′1u1 + a′2u2, d′ = d′ 0 − d′1u1 − d′2u2, so that d′ − a′ = (d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, the total expression of � is � = (p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)((d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2)) +(p′ 0 − q′ 0)z3u1 + (p′ 1 − q′ 1)z2u1 + (p′ 2 − q′ 2)z3u2 + (p′ 3 − q′ 3)z2u2, where we canceled out all the monomials containing u2 1, u1u2, and u2 2, since we work on the first formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We rename (d′ − a′)0(z) = λ0 + λ1z + λ2z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' to simplify notation, and since all terms in (1, 2) having powers of z equal to 4 and higher can be cancelled out by the appropriate choice of the z2jb′, it suffices to analyse the expression � = (p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)(λ0 + λ1z) +(q′ 0 − p′ 0)z3u1 + (q′ 1 − p′ 1)z2u1 + (q′ 2 − p′ 2)z3u2 + (q′ 3 − p′ 3)z2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z3u1, z2u1, z3u2, z2u2 in � with appropriate choices of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently, q′ ∼ p′ if and only if the following equality holds for some choice of λ0 and λ1: \uf8eb \uf8ec \uf8ec \uf8ed q′ 0 − p′ 0 q′ 1 − p′ 1 q′ 2 − p′ 2 q′ 3 − p′ 3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 = λ0 \uf8eb \uf8ec \uf8ec \uf8ed p0 p1 p2 p3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 + λ1 \uf8eb \uf8ec \uf8ec \uf8ed p1 0 p3 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 10 When the vectors v1 = (p0, p1, p2, p3) and v2 = (0, p1, 0, p3) are linearly independent, the point q′ belongs to the plane that passes through the point p′ with v1 and v2 as direction vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Therefore, whenever v1 and v2 are linearly independent vectors, the fibre over p = (p0, p1, p2, p3) is a copy of C4 foliated by 2-planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The leaf containing a point p′ forms the equivalence class of p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Thus, the moduli space over the fibre over p is parametrised by the 2-plane through the origin in the direction perpendicular to v1, v2 over the point p, except when p1 = p3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In contrast, the fibre over a point p = (p0, 0, p2, 0) is a copy of C4 foliated by lines in the direction of v1 = (p0, 0, p2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In this case, the moduli space over p is parametrised by a copy of C3 perpendicular to v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W1, σ) → M2(W1) ≃ P3 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf, whose stalks have dimension 2 over the Zariski open set (p1, p3) ̸= (0, 0), and dimension 3 over the P1 cut out by p1 = p3 = 0 in P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The same techniques readily generalise to give a description of the quantum moduli spaces for other choices of noncommutative deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is an extremal Poisson structure on W1, then the quantum moduli space Mℏ j(W1, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 2 over the classical moduli space Mj(W1) with singular stalks up to rank 4j − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We give the details of the case j = 3, for an extremal Poisson structure, that is, the case when all brackets vanish on the first formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The general case is clear from these calculations, just notationally more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' When j = 3 and σ = u1σ1, expression � becomes: � = p(d′ − a′)z3 + (p′d − q′a)z3, and we get a system of equations: � = � p0z5u1 + p1z4u1 + p2z3u1 + p3z2u1 + p4z5u2 + p5z4u2 + p6z3u2 + p7z2u2 � (λ0 + λ1z + λ2z2 + λ3z3 + λ4z4) + +(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1 + (p′ − q′)3z2u1 +(p′ − q′)4z5u2 + (p′ − q′)5z4u2 + (p′ − q′)6z3u2 + (p′ − q′)7z2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z5u1, z4u1, z3u1, z2u1, z5u2, z4u2, z3u2, z2u2 in � with appropriate choices of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' q′ ∼ p′ if and only if the following equality holds for some choice of λ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' λ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' λ3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='+ λ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='p3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='p7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 11 Consider now the family U of vector spaces over M2(W1) ≃ P7 whose fibre at p is given by Up = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p0 p1 p2 p3 p1 p2 p3 0 p2 p3 0 0 p3 0 0 0 p4 p5 p6 p7 p5 p6 p7 0 p6 p7 0 0 p7 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, the quantum moduli space is obtained from this family after dividing by the equivalence relation ∼ over each point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence Mℏ 2(W1, σ) = U/ ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W1, σ) → M2(W1) ≃ P7 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf or rank 4, with stalk at p having dimension equal to the corank of Up, in this case 4 ≤ dim Mℏ 2(W1, σ)p = 8 − rk Up ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the general case we have 2j − 2 ≤ dim Mℏ j(W1, σ)p = corank Up =≤ 4j − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W2 The Calabi–Yau threefold we consider in this section is a crepant resolution of the singularity xy − w2 = 0 in C4, that is W2 := Tot(OP1(−2) ⊕ OP1) = Z2 × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Similarly to what we did for W1, we will carry out calculations using the canonical coordinates W2 = U ∪V where U ≃ C3 ≃ V with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates on U ∩ V ≃ C∗ × C × C given by �ξ = z−1 , v1 = z2u1 , v2 = u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently, global holomorphic functions on W2 are generated by 1, u1, zu1, z2u1, u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For each specific noncommutative deformation (W2, Aσ), we wish to compare the quantum and classical moduli spaces of vector bundles, see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a rank 2 bundle E on a noncommutative deformation W2 with a canonical matrix � zj p 0 z−j � as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10) where p = �∞ n=0 pnℏn, expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='8) gives us the general form of the coefficients pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, on the first formal neighbourhood, we have: p = j−1 � l=−j+3 pl10zlu1 + j−1 � l=−j+1 pl01zlu2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1) where in case j = 1 we have only p001u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To describe the quantum moduli for Poisson structures on W2, we consider the expression �: � = {p, d + a}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + p(d′ − a′)zj + (p′d − q′a)zj, where we need to cancel out the coefficients of z3u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and zu2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 12 Each noncommutative deformation comes from some Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The most basic Poisson structures σ on W2 are those which generate all others over global functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We call these generators the basic Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, we compute the quantum moduli of bundles for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We observe that the 4 Poisson manifolds (W2, σi) for i = 1, 2, 3, 4, are pairwise nonisomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This can be verified by the table of their degeneracy loci: W2 Poisson structures bracket degeneracy σ1 σ2 ∅ σ3 σ4 ∪ Nevertheless, the 4 quantum moduli spaces defined by these basic Poisson structures turn out to be all isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is a basic Poisson structure on W2, then the quantum moduli space Mℏ j(Wk, σ) and its classical limit are isomorphic: Mℏ j(W2, σ) ≃ Mj(W2) ≃ P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We carry out calculations for the basic bracket σ4 = u1∂u1 ∧ ∂u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It does turn out that the result is the same for the the basic brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The calculation for σ4 is shorter, since any of the brackets having one entry equal to zj vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Because we work on the first formal neighbourhood, we also remove the expressions that are quadratic in the ui variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, the expression � that remains to be analysed simplifies to: � = {p, d + a}zj + p(d′ − a′)zj + (p′d − q′a)zj, where we must cancel out the coefficients of the monomials z3u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and zu2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' On the first formal neighbourhood, we write a = 1 + a1(z)u1 + a2(z)u2, d = 1 + d1(z)u1 + d2(z)u2, and a′ = a′ 0(z) + a′ 1(z)u1 + a′ 2(z)u2, d′ = d′ 0(z) + d′ 1(z)u1 + d′ 2(z)u2, so that the partials are ∂uia = ai(z) ∂uid = di(z) and ∂u2a = a2(z) ∂u2d = d2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The extension class given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='9) becomes p = j−1 � l=3−j pl10zlu1 + j−1 � l=1−j pl01zlu2, and computing the bracket gives {p, d + a}zj = \uf8eb \uf8ed j−1 � l=3−j pl10zl \uf8f6 \uf8f8 (d2(z) + a2(z))zju1 + \uf8eb \uf8ed j−1 � l=1−j pl01zl \uf8f6 \uf8f8 (d1(z) + a1(z))zju1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To work with a simpler notation, we present details of � when j = 2, in which case we can express the extension class as p = p0zu1 + p1zu2 + p2u2 + p3z−1u2, QUANTIZATION OF CALABI–YAU THREEFOLDS 13 having renamed the coefficients for simplicity (making p0 := p110, p1 := p101, p2 := p001, p3 := p−101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We will point out the steps for generalising to higher j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Assuming j = 2, we have {p, d + a}z2 = p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To obtain equivalence between q′ and p′, we must cancel out coefficients of z3u1, zu2, z2u2, z3u2 in the expression of �, which becomes � = p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1 +(p0z3u1 + p1z3u2 + p2z2u2 + p3zu2)(d′ 0(z) − a′ 0(z)) +(p′ 0z3u1 + p′ 1z3u2 + p′ 2z2u2 + p′ 3zu2) −(q′ 0z3u1 + q′ 1z3u2 + q′ 2z2u2 + q′ 3zu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since the highest power of z to be considered is 3, we observe that d2(z) + a2(z) may be chosen conveniently, we cancel out all terms in z3u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We may also choose d1(z) + a1(z) = 0, leaving � = (p1z3u2 + p2z2u2 + p3zu2)(d′ 0(z) − a′ 0(z)) +(p′ 1z3u2 + p′ 2z2u2 + p′ 3zu2) −(q′ 1z3u2 + q′ 2z2u2 + q′ 3zu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now we may choose d′ 0 − a′ 0 appropriately to cancel out all terms in u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that there are no conditions imposed on q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In other words, here p + p′ℏ is equivalent to p + q′ℏ for any choice of q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, the quantum and classical moduli spaces are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The generalisation to higher j works out similarly, we can first choose di + ai for i > 0 to cancel out the coefficients of u1 and then choose d′ 0 −a′ 0 to take care of the coefficients of u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, for all j using the bracket σ4 we conclude that the quantum and classical moduli spaces are isomorphic Mℏ j(W2, σ4) ≃ Mj(W2) ≃ P4j−5 where the second isomorphism is proven in [K, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now choose any Poisson structure of W2 for which all brackets in � vanish on neighbourhood 1, for example σ = u1σ4 = u2 1∂u1 ∧ ∂u2 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In such a case, the expression for � reduces to: � = p(d′ − a′)zj + (p′d − q′a)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, consider the case of j = 2, when we have: � = (p0z3u1 + p1z3u2 + p2z2u2 + p3zu2) + (d′ 0(z) − a′ 0(z)) +(p′ 0z3u1 + p′ 1z3u2 + p′ 2z2u2 + p′ 3zu2) −(q′ 0z3u1 + q′ 1z3u2 + q′ 2z2u2 + q′ 3zu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Setting d′ 0(z) − a′ 0(z) = λ0 + λ1z + λ2z2, we get a system of equations: \uf8eb \uf8ec \uf8ec \uf8ed q′ 0 − p′ 0 q′ 1 − p′ 1 q′ 2 − p′ 2 q′ 3 − p′ 3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ed λ0 0 0 0 0 λ0 λ1 λ2 0 0 λ0 λ1 0 0 0 λ0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed p0 p1 p2 p3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since we can choose λ1 and λ2 to solve the second and third equations, we see that q′ 1 and q′ 2 are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence (q′ 0, q′ 1, q′ 2, q′ 3) ∼ λ0(q′ 0, ∗, ∗, q′ 3), and our system of equations reduces to �q′ 0 − p′ 0 q′ 3 − p′ 3 � = λ0 �p0 p3 � , QUANTIZATION OF CALABI–YAU THREEFOLDS 14 which is the parametric equation of a line in the (q′ 0, q′ 3)-plane whenever (p0, p3) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The entire question of moduli now reduces to the 2-dimensional case, disregarding p1, p2 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If (p0, p3) ̸= (0, 0), then the equivalence class of q′ in the fibre over the point p is the 1-dimensional subspace L directed by the vector (p0, p3) and passing through (q′ 0, q′ 3) in the (p′ 0, p′ 3)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If p0 = p3 = 0, then we must have the equality (q′ 0, q′ 3) = (p′ 0, p′ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, its the equivalence class consists of a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Accordingly, the set of equivalence classes over p can be represented either by the line L⊥ by the origin perpendicular to L (directed by (−p3, p0) when (p0, p3) ̸= (0, 0) or else by the entire (p′ 0, p′ 3)-plane over (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W2, σ) → M2(W2) ≃ P3 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf, whose stalks have dimension 1 over the Zariski open set (p0, p3) ̸= (0, 0), and dimension 2 over the P1 cut out by p0 = p3 = 0 in P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In fact, we could express this moduli space as a sheaf given by an extension of OP3(+1) by a torsion sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is an extremal Poisson structure on W2, then the quantum moduli space Mℏ j(W2, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 3 over the classical moduli space Mj(W2) with singular stalks up to rank 4j − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, for j = 3, we write down the extremal example when the brackets vanish on the first formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The generalisation of the extremal cases to all j becomes clear from this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Where, assuming all brackets vanish on the first formal neighbourhood, we need to cancel out the coefficients of z3u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and zu2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2 in � = p(d′ − a′)zj + (p′d − q′a)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For j = 3 we have p = 2 � l=0 pl10zlu1 + 2 � l=−2 pl01zlu2, which we rewrite as p = p0z2u1 + p1zu1 + p2u1 + p3z2u2 + p4z1u2 + p5u2 + p6z–1u2 + p7z−2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Setting d′ 0(z) − a′ 0(z) = λ0 + λ1z + λ2z2 + λ3z3 + λ4z4, expression � = p(d′ − a′)z3 + (p′d − q′a)z3 becomes � = � p0z5u1 + p1z4u1 + p2z3u1 + p3z5u2 + p4z4u2 + p5z3u2 + p6z2u2 + p7zu2 � (λ0 + λ1z + λ2z2 + λ3z3 + λ4z4) +(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1 +(p′ − q′)3z5u2 + (p′ − q′)4z4u2 + (p′ − q′)5z3u2 + (p′ − q′)6z2u2 + (p′ − q′)7zu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To start with we notice that λ3 and λ4 can always be chosen to solve the equations involving q′ 3 and q′ 4 so that these 2 coordinates can take any value, that is, there are isomorphisms (q′ 0, q′ 1, q′ 2, q′ 3, q′ 4, q′ 5, q′ 6, q′ 7) ∼ (q′ 0, q′ 1, q′ 2, ∗, ∗, q′ 5, q′ 6, q′ 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 15 Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' we may remove q′ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' q′ 4 and rewrite the reduced system as: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed q′ 0 − p′ 0 q′ 1 − p′ 1 q′ 2 − p′ 2 q′ 5 − p′ 5 q′ 6 − p′ 6 q′ 7 − p′ 7 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = λ0 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p0 p1 p2 p5 p6 p7 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + λ1 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p1 p2 0 p6 p7 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + λ2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p2 0 0 p7 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here q′ ∼ p′ if and only if the equality holds for some choice of λ0, λ1, λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consider now the family U of vector spaces over M2(W2) ≃ P7 whose fibre at p is given by Up = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p0 p1 p2 p1 p2 0 p2 0 0 p5 p6 p7 p6 p7 0 p7 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, the quantum moduli space is obtained from this family after dividing by the equivalence relation ∼ over each point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence Mℏ 2(W2, σ) = U/ ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W2, σ) → M2(W2) ≃ P7 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf, with stalk at p having dimension equal to the corank of Up, in this case 3 ≤ dim Mℏ 2(W2, σ)p = corank Up = 6 − rk Up ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the general case we then have 2j − 3 ≤ dim Mℏ j(W2, σ)p = corank Up = 2j − rk Up ≤ 4j − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Computations of H1 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' H1(W1, O) = H1(W2, O) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A 1-cocycle τ ∈ O(U ∩ V ) may be written in the form τU = ∞ � l=−∞ ∞ � i=0 ∞ � s=0 τliszlui 1us 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since terms containing only positive powers of z are holomorphic on the U-chart τU ∼ −1 � l=−∞ ∞ � i=0 ∞ � s=0 τliszlui 1us 2, where ∼ denotes cohomological equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Changing to V coordinates we have τV = −1 � l=−∞ ∞ � i=0 ∞ � s=0 τlisξ−l+ki+(−k+2)svi 1vs 2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2) where, for k = 1, 2 exponents of ξ are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Thus, τV is holomorphic on V , and τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' H1(W3, O) is infinite dimensional over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' As in the proof of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 we arrive at the expression (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2) for the 1-cocycle τ on the V -chart, which in the case k = 3, gives τV ∼ −1 � l=−∞ ∞ � i=0 ∞ � s=0 τlisξ−l+3i−svi 1vs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The terms that are not holomorphic on V are all of those satisfying −l + 3i − s < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that all terms having s > 3i − l, namely all of −1 � l=−∞ ∞ � i=0 ∞ � s=3i−l+1 τliszlui 1us 2 are nontrivial in first cohomology, so that dim H1(W3, O) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Ballico is a member of GNSAGA of INdAM (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' ballico@science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='it, Gasparim - Depto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Matem´aticas, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Cat´olica del Norte, Chile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' etgasparim@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='com, Rubilar - Depto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Matem´aticas, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Sant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Concepci´on, Chile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' francisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='rubilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='arriagada@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='com, Suzuki - Depto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Matem´atica, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' de S˜ao Paulo, Brazil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' obrunosuzuki@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} diff --git a/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/2301.05073v1.pdf.txt b/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/2301.05073v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3f570eb27ecf9dcc9144c6d718ffdebcaf09e80 --- /dev/null +++ b/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/2301.05073v1.pdf.txt @@ -0,0 +1,2636 @@ +Gradient TRIX +CHRISTOPH LENZEN, CISPA Helmholtz Center for Information Security, Germany +SHREYAS SRINIVAS, CISPA Helmholtz Center for Information Security, Germany and Saarbrucken +Graduate School for Computer Science, Saarland University, Germany +Gradient clock synchronization (GCS) algorithms minimize the worst-case clock offset between the nodes in a +distributed network of diameter 𝐷 and size 𝑛. They achieve optimal offsets of Θ(log 𝐷) locally, i.e. between +adjacent nodes [18], and Θ(𝐷) globally [2]. As demonstrated in [3], this is a highly promising approach +for improved clocking schemes for large-scale synchronous Systems-on-Chip (SoC). Unfortunately, in large +systems, faults hinder their practical use. State of the art fault-tolerant GCS [4] has a drawback that is fatal in +this setting: It relies on node and edge replication. For 𝑓 = 1, this translates to at least 16-fold edge replication +and high degree nodes, far from the optimum of 2𝑓 + 1 = 3 for tolerating up to 𝑓 faulty neighbors. +In this work, we present a self-stabilizing GCS algorithm for a grid-like directed graph with optimal node in- +and out-degrees of 3 that tolerates 1 faulty in-neighbor. If nodes fail with independent probability 𝑝 ∈ 𝑜(𝑛−1/2), +it achieves asymptotically optimal local skew of Θ(log 𝐷) with probability 1 − 𝑜(1); this holds under general +worst-case assumptions on link delay and clock speed variations, provided they change slowly relative to the +speed of the system. The failure probability is the largest possible ensuring that with probabity 1 − 𝑜(1) for +each node at most one in-neighbor fails. As modern hardware is clocked at gigahertz speeds and the algorithm +can simultaneously sustain a constant number of arbitrary changes due to faults in each clock cycle, this +results in sufficient robustness to dramatically increase the size of reliable synchronously clocked SoCs. +CCS Concepts: • Hardware → Very large scale integration design; Very large scale integration design; +• Computing methodologies → Distributed algorithms; +Additional Key Words and Phrases: Clock Synchronisation, Fault Tolerance, VLSI, Self-Stabilisation +1 +INTRODUCTION +In their seminal work from 2004 [7], Fan and Lynch introduced the task of Gradient Clock Synchro- +nization (GCS). In a network of nodes that synchronize their clocks, it requires to minimize the +worst-case clock offset between neighbors. Two key insights motivate minimizing this local skew: +• In many applications the skew between adjacent nodes is the appropriate measure of quality. +• The global skew, the maximum clock offset between any pair of nodes in the network, grows +linearly with the diameter 𝐷 of the network [2]. +Defying the intuition of many, Fan and Lynch proved a lower bound of Ω(log 𝐷/log log 𝐷) on the +local skew. Follow-up work then established that this bound was very close to the mark: the best +local skew that can be achieved is Θ(log 𝐷) [18]. This exponential gap between global and local +skew strongly suggests better scalability of systems employing this approach to synchronization. +Yet, more than a decade after these results have been published, we know of no efforts to apply +these techniques in products. This is not for want of demand! To drive this point home, consider +the case of clocking synchronous hardware. Conceptually speaking, state of the art hardware +that operates synchronously distributes a clock signal from a single source using a tree network, +see e.g. [9, 21]. However, for any tree spanning a square grid, there will be adjacent grid points +whose distance in the tree is proportional to the side length of the grid [8]. Hence, the worst-case +local skew on a computer chip clocked by a clock tree must grow linearly with the side length +of the chip [2]. Indeed, these theoretical results are reflected in the reality of hardware suppliers. +Modern systems gave up on maintaining globally synchronous operation, instead communicating +Authors’ addresses: Christoph Lenzen, lenzen@cispa.de, CISPA Helmholtz Center for Information Security, Saarbrücken, +Germany; Shreyas Srinivas, shreyas.srinivas@cispa.de, CISPA Helmholtz Center for Information Security, Saarbrücken, +Germany and Saarbrucken Graduate School for Computer Science, Saarland University, Saarbrücken, Germany. +arXiv:2301.05073v1 [cs.DC] 12 Jan 2023 + +2 +Christoph Lenzen and Shreyas Srinivas +asynchronously between multiple clock islands [1]. This comes at a steep cost, both in terms of +communication latency [12] and ease of design. +So, which obstacle prevents application? At least in the above setting, neither large hidden +constants nor an overly complex algorithm get in the way. On the contrary, recent work demon- +strates that implementation effort is easily managable and pays off already for moderately-sized +systems [3]. Instead, the main obstacle are faults. To see that this is the key issue, recall that +today’s hardware comprises an enourmous number of individual components. Recent off-the-shelf +hardware has transistor counts beyond the 10 billion mark [22], requiring either incredibly low fault +rates or some degree of fault-tolerance. In a system composed of multiple clock islands that interact +asynchronously, these islands are canonical choices for fault-containment regions. Thus, one can +get away with using a clocking scheme in each island that cannot sustain faults, interpreting a +fault of the clocking subsystem as a fault of the respective island. In contrast, when the clocking +subsystems of the islands interact with each other via a clock synchronization algorithm, we must +ensure that a clock fault in a single island does not bring down the entire system! +Fault-Tolerant Clocking. When clocking hardware, high connectivity networks are not scalable. This +limits the number of concurrent faults that can be sustained, as tolerating up to 𝑓 faults requires +a node connectivity of 2𝑓 + 1. In [4], this bound is matched asymptotically by augmenting an +arbitrary network such that the GCS algorithm from [18] is simulated in a fault-tolerant way. Here, +augmentation means to replace each node in the original network by a clique of size 3𝑓 + 1 and +each edge by a biclique. The clique then synchronizes internally using the classic Lynch-Welch +algorithm [10], and the resulting local outputs are interpreted as (an approximation of) a joint +cluster clock on which the (non-tolerant) GCS algorithm from [18] is simulated. +Unfortunately, this approach is impractical due to the large overhead in terms of edges. Leaving +asymptotics aside – the edge overhead compared to the original graph is Θ(𝑓 2) rather than 𝑂(𝑓 ) – +even for the important special case of 𝑓 = 1 node degrees will be at least 15. This is a far cry from +the simplicity of current distribution techniques, and factor 5 beyond the minimum node degree +of 2𝑓 + 1 = 3. What might look like a “moderate constant” to a theoretician will not only cause a +headache to the engineer trying to route all of these edges with few layers and precise timing, it +will also substantially increase communication delay uncertainty. This, in turn, directly translates +into an increased skew, placing the break-even point with prior art beyond relevant limits. +In summary, it is essential to get as close as possible to the minimum required connectivity. This +train of thought led to the study of fault-tolerant clock distribution in low-degree networks [6, 20]. +Both of these works have in common that they assume that the clock signal is generated at a central +location. This enables these approaches to achieve self-stabilization and tolerance to isolated faults +with very simple pulse forwarding schemes. The basic idea is to propagate the signal from layer to +layer, having each node wait for two nodes signaling a clock pulse before locally generating and +forwarding their own pulse. Moreover, it is assumed that in absence of faults delays are changing +only slowly over time. Thus, matching the input frequency to the expected delay between grid +layers results in clock pulses that are well-synchronized between adjacent layers. +The above works differ in the grid structure they use (Figure 1) and the skew bounds they provide: +• Denoting by 𝑑 − 𝑢 and 𝑑 the minimum and maximum end-to-end communication delay, in a +grid of width 𝐷 [6] bounds the local skew by 𝑑 + 𝑂(𝑢2𝐷/𝑑). Since in practice 𝑑 ≫ 𝑢, this is a +non-trivial bound. Unfortunately, the fact that 𝑑 ≫ 𝑢 also means that this bound is too large +for applications. Even worse, for each fault this bound increases by 𝑑. +• In [20], each fault adds at most 𝑢 to the local skew. Observe that the used grid also has the +minimum required connectivity, as each node has only 3 incoming and outgoing edges each. + +Gradient TRIX +3 +0 +𝑢 +2𝑢 +𝑑 +𝑑 +𝑑 +𝑑−𝑢 +𝑑−𝑢 +𝑑−𝑢 +𝑑 +𝑑 +𝑑 +Fig. 1. TRIX [20] (top) and HEX [6] (bottom) grids. TRIX uses the naive pulse forwarding scheme of waiting +for the second copy of each pulse before forwarding it. We see how the TRIX grid can accumulate a skew of +Θ(𝑢𝐷). In the HEX grid, each node waits for two copies of a pulse from in-neighbours. However, 2 of the 4 +in-neighbors are on the same layer, causing a skew of 𝑑 if a neighbor on the preceding layer crashes. +Alas, these advantages come at the expense of poor scaling of worst-case skews with the +number of layers: on layer ℓ, adjacent nodes may pulse up to 𝑢ℓ time apart. +Note that in order to tolerate failure of an arbitrary component, also the clock source has to +be replicated and the replicas to be synchronized in a fault-tolerant and self-stabilizing manner. +However, here one can employ techniques for fully connected networks [11, 19]; using them in a +single location for 𝑓 = 1 does not constitute a scalability issue. +In light of the above, in this work we ask the question +“Can a small local skew be achieved in a fault-tolerant way at minimal connectivity?” +Our Contribution +We provide a positive answer to the above question for the special case of 𝑓 = 1. This is achieved by +using the same grid as in [20], but with a different rule for forwarding pulses. Our novel algorithm +is designed as a discrete and fault-tolerant counterpart to the GCS algorithm from [18]. +Making this work requires substantial conceptual innovation and technical novelty. On the +conceptual level, our algorithm simulates a discretized variant of the (non-fault-tolerant!) GCS +algorithm from [18] on an arbitrary base graph of minimum degree 2. In more detail, each copy of +the graph, referred to as layer, represents a “time step” of the GCS algorithm. For each node, there +is an edge from its copy on a given layer to the copies of itself and its neighbors on the next layer. +The forwarded pulses along these edges serve two very different functions: +• The pulse messages sent to copies of neigbhors correspond to the GCS algorithm’s messages +for estimating clock offsets to neighbors. +• The pulse messages sent between copies of the same node convey its local time from one of +its copies to the next. +Note that this turns a permanently faulty node in the grid into a simulated node being faulty in a +single time step only. This is of vital importance, because it enables us to rely on the self-stabilization +properties of the GCS algorithm from [18]. These are implicitly shown in [13]; we prove them +explicitly in the different setting of this work. +However, by itself this does not guarantee bounded skew between correct nodes, since we +also need to contain the effect of such a “transient” fault on the state of the simulated algorithm. + +4 +Christoph Lenzen and Shreyas Srinivas +Otherwise, a fault would increase skews arbitrarily, effectively corrupting downstream nodes: at +any given node, the smallest or largest time at which a pulse from neighbors on the preceding +layer is received could be determined by a faulty node. We can overcome this issue if there is at +most one faulty in-neighbor. The key observation to controlling the impact of a faulty node on the +pulse time lies in that it can indeed affect only one of three times: the smallest or largest time at +which a pulse from copies of neighbors on the previous layer is received, or the time at which the +pulse from the copy of the node itself is received. In particular, the median of these three times lies +within the interval spanned by the correct in-neighbors’ pulse times. By imposing the additional +constraint to always tie the time at which a pulse is generated closely to this median, we can limit +the local impact of a fault on skews. +In summary, we seek to simultaneously simulate a time-discrete variant of the GCS algorithm +from [18], while also guaranteeing that pulse forwarding times are, up to a sufficiently small +deviation, identical to median reception times plus a fixed offset. Unfortunately, no existing GCS +algorithm that achieves a small local skew [14, 15, 17, 18] can be used for this purpose as-is, since +their decision rules are in conflict with the above “stick to the median” requirement. +As our main technical contribution, we resolve this conflict, simultaneously adapting the resulting +algorithm to the discrete setting. To do so, we determine suitably weakened discrete variants of the +slow and fast conditions introduced in [15]. In essence, we allow that a simulated node whose pulse +time is ahead all of its neighbors’ pulse times to delay its next pulse by the difference to the fastest +neighbor; an analogous rule applies to nodes pulsing later than all of their neighbors. From the +perspective of the GCS algorithm in [18], this constitutes a potentially arbitrarily large clock “jump,” +which we leverage to implement the stick-to-the-median requirement despite the arbitrary changes +in timing faulty nodes may apply to their pulse messages. To prevent uncontrolled oscillatory +behavior arising from adjacent nodes “jumping” in opposite directions, we introduce an additional +condition, which we refer to as jump condition. Essentially, it slightly reduces how large jumps +are to avoid that uncertainty in message delays and local clock speeds cause nodes to “overswing,” +potentially resulting in arbitrarily large skews, cf. Figure 5. +Turning so many knobs at once meant that it was not clear that such a scheme would work. +Indeed, bounding the skew of this novel algorithm turned out to be highly challenging, as jumps +that delay pulses rather than speeding them up invalidate the fundamental assumption that clocks +progress at rate at least 1 present in all prior work [14, 15, 17, 18]. As a result, the main technical +hurdle and contribution turned out to be proving a bound on the local skew Lℓ between neighbors +in the same layer ℓ for the fault-free case. +Theorem 2. If there are no faults, then Lℓ ≤ 4𝜅(2 + log 𝐷) for all ℓ ∈ N. +Here, choosing the input clock frequency to be 1/(2𝑑) results in 𝜅 ∈ Θ(𝑢 + (𝜗 − 1)𝑑), where +it is assumed that local clocks run at rates between 1 and 𝜗 > 1. All of our results require that +𝑑 ≫ 𝑢 + (𝜗 − 1)𝑑, or equivalently, that the local skew remains small compared to 𝑑. Note that if +this condition does not hold, we are outside the parameter range of interest: then skews become +large compared to the length of a clock cycle under ideal conditions and clock frequency has to be +reduced substantially. +To address faults, we bound by how much faults can affect timing. Due to the aforementioned +stick to the median rule, we can bound the local impact of a fault on timing in terms of the local +skew. However, applying this argument repeatedly, skews would grow exponentially in the number +of faults. While tolerating a constant number of faults is certainly better than tolerating none, this +is unsatisfactory, since the requirement of one faulty in-neighbor holds with probability 1 − 𝑜(1) +for a fairly high independent probability of 𝑝 ∈ 𝑜(1/√𝑛). Given that the topology we are most + +Gradient TRIX +5 +interested in is roughly a square grid, i.e., there are roughly √𝑛 layers, the naive approach outlined +above does not result in a non-trivial bound on the skew unless 𝑝 is very close to 1/𝑛. +We provide an improved analysis exploiting that our base graph is almost a line. Hence, the 𝑑-hop +neighborhood grows linearly with 𝑑 and hence the number of nodes in layers ℓ′ ∈ [ℓ − 𝑛1/12, ℓ] +that affect the pulse time of a node in layer ℓ is in Θ(𝑛1/6). Thus, if nodes fail with probability +𝑝 ∈ 𝑜(1/√𝑛), the probability that there are more than 2 faulty nodes within distance 𝑛1/12 that +affect a given node is 𝑜(1/𝑛). Intuitively, this buys enough time for the self-stabilization properties +of the simulated algorithm to reduce its local skew again before it spirals out of control. +Theorem 5. With probability 1 − 𝑜(1), Lℓ ∈ 𝑂(𝜅 log 𝐷) for all ℓ ∈ N. +The final step is to extend this bound on the local skew within a layer to one that includes +adjacent nodes in different layers. As we propagate pulses layer by layer, we cannot hope to match +pulse times of the 𝑘-th pulse between different layers. Instead, we match the input period to the +nominal time a pulse spends on each layer. This works neatly so long as there are no changes in +message delay, clock speed, and behavior of faulty nodes between consecutive pulses. +Theorem 7. If faulty nodes do not change the timing of their output pulses, then L ∈ 𝑂(𝜅 log 𝐷) +with probability 1 − 𝑜(1). +To a large extent, this strong assumption is justified in our specific context. Clock speeds of +modern systems are in the gigahertz range, and the amount of change in timing that occurs within a +single clock cycle is much smaller than over the lifetime of a system [23]. Similarly, the by far most +common timing faults are stuck-at faults, i.e., the signal observed by downstream nodes remains +constant logical 0 or 1, and broken connections. From the point of view of the receiving node, this +is equivalent to an early or late pulse, respectively, without any change between pulses. +Of course, timing will still change slowly, the above benign faults will occur at some point, before +which the nodes worked correctly, and some faults may be more severe. Using once more that +faulty nodes’ impact on timing is bounded by the local skew, the bound from Theorem 7 extends to +a constant number of arbitrary faults in each pulse alongside small changes in delays and hardware +clock speeds. +Corollary 7. With probability 1−𝑜(1), L ∈ 𝑂(𝜅 log 𝐷) even when in each pulse (i) a constant number +of faulty nodes change their output behavior and timing, (ii) link delays vary by up to 𝑛−1/2𝑢 log 𝐷, +and (iii) hardware clock speeds vary by up to 𝑛−1/2(𝜗 − 1) log 𝐷. +Finally, if all else fails, we can fall back on the ability of the pulse progation algorithm to +recover from arbitrary transient faults. In constrast to the simulated GCS algorithm, achieving +self-stabilization of the pulse propagation scheme itself is straightforward due to the directionality +of the propagation. +Theorem 6. The pulse propagation algorithm can be implemented in a self-stabilizing way. It +stabilizes within 𝑂(√𝑛) pulses. +In light of these results, we view this work as a major step towards simultaneously achieving +high performance and strong robustness in the practical setting of clock distribution in hardware. +In alignment with the theoretical question motivating this work, we achieve an asymptotically +optimal local skew at the minimum possible node degree under the assumption of node failures +with probability 𝑜(𝑛−1/2). +Organization of this Article. In Section 2, we discuss the system model, introduce the graph on +which we run our synchronization algorithm, and motivate our modeling choices, including its non- +standard aspects. We then present a simplified version of the algorithm that better highlights the + +6 +Christoph Lenzen and Shreyas Srinivas +conceptual approach in Section 3; the full algorithm and its equivalence without faulty predecessors +is shown in Appendix B. We follow with the formal derivation of the skew bounds in Section 4. +2 +MODELING +The model we use is non-standard, as it is tailored to the specific setting outlined in the introduction. +Accordingly, we will emphasize and discuss model choices where this seems prudent. +Setting. Recall that our goal is to provide a synchronized clock signal to a large System-on-Chip. +Physically, this means that we need to provide the clock signal to a rectangular area; for simplicity, +we will assume it to be square. We want to supply a uniform grid of nodes in the square area with +this signal, which then will serve as roots of relatively small local clock trees supplying the low-level +components with the clock signal. If these trees contribute a maximum clock skew of Δ and the +skew between adjacent grid points is at most L, the triangle inequality guarantees a worst-case +skew of L + 2Δ between adjacent components of the System-on-Chip. The local clock trees can be +designed using standard methodology. Therefore, in the following we will focus exclusively on the +grid of their roots. +A key assumption we make is that communication delay between correct adjacent nodes changes +only slowly with time. This enables us to generate synchronized pulses at all grid nodes by matching +the input frequency with the (inverse) propagation time between consecutive layers. This is justified +for two reasons: +• The dominant sources of uncertainty in propagation delay are inaccuracies in component +fabrication, aging, and temperature and frequency variations that are slow relative to the +time it takes to propagate an input clock pulse across even a large System-on-Chip [23] +• Changing delays of all links between a pair of adjacent layers by up to 𝛿 increases skew +bounds by at most 𝛿, cf. Lemma 22. +In order to generate sufficiently synchronized pulses at the nodes of layer 0, a straightforward +solution is to use a redundant path, i.e., a path of 3-cliques in which adjacent cliques are fully +bipartitely connected, to propagate pulses from the clock reference along an edge of the chip. As +we show in Corollary 6, this results in input pulses of small enough local skew. For each clique, +one of the nodes will be the layer-0 node providing its output pulse to close-by nodes of layer 1. +In a perfect grid, all layers would consist of a path. Unfortunately, this results in the issue that the +endpoints of the path, lacking one neighbor, would have only two adjacent nodes in the preceding +and subsequent layer. A naive solution is to insert a additional edges between the boundary nodes, +turning the layer into a cycle and the entire graph into a cylinder (with some special treatment +of layer 0). However, realizing such a solution on the square would result in far too long edges +between boundary nodes or require to, essentially, replicate each layer, effectively doubling the +number of nodes and edges in the graph. +Instead, we choose to replicate the boundary nodes only, which then provides the “missing” input +to the next layer. Note that this increases the degree of the nodes next to the boundary nodes by +one. We cope with this by a general analysis allowing for the layers to be copies of an arbitrary base +graph of minimum degree 2. In Figure 2 and Figure 3, we show the base graph and the connectivity +of nodes between adjacent layers of our synchronization network in our assumed setting. +Network Graph. We are given a simple connected base graph 𝐻 = (𝑉, 𝐸) of minimum degree 2 and +diameter 𝐷 ∈ N>0. For 𝑣,𝑤 ∈ 𝑉 , denote by 𝑑(𝑣,𝑤) ≤ 𝐷 the distance from 𝑣 to 𝑤 in 𝐻. To derive the +graph 𝐺 = (𝑉𝐺, 𝐸𝐺) we use for synchronization, for each ℓ ∈ N we create a copy 𝑉ℓ of 𝑉 . Denoting +by (𝑣, ℓ) the copy of 𝑣 ∈ 𝑉 in 𝑉ℓ, we define 𝐸ℓ := {((𝑣, ℓ), (𝑤, ℓ + 1)) | {𝑣,𝑤} ∈ 𝐸 ∨ 𝑣 = 𝑤}. We now +obtain 𝐺 by setting 𝑉𝐺 := � +ℓ ∈N 𝑉ℓ and 𝐸𝐺 := � +ℓ ∈N 𝐸ℓ. That is, for each layer ℓ ∈ N we have a copy + +Gradient TRIX +7 +Fig. 2. Base graph 𝐻 used in this work. Rather than using a cycle, which would result in a TRIX grid, we +replicate the end nodes of a line to ensure a minimum degree of 2. Alternatively, one could use a line and +exploit that the probability that one of the 𝑂(√𝑛) boundary nodes fails is 𝑜(1). +Fig. 3. Layer structure of 𝐺 resulting from our choice of 𝐻. Most nodes have in- and out-degree 3, some 4. +of 𝑣 ∈ 𝑉 , which has outgoing edges to the copies of itself and all its neighbors on layer ℓ + 1.1 Sine +𝑉𝐺 is a DAG, we refer to out-neighbors as successors and in-neighbors as predecessors. +Fault Model. An unknown subset 𝐹 ⊂ 𝑉𝐺 is Byzantine faulty, meaning that these nodes may violate +the protocol arbitrarily. Edge faults are mapped to node faults, i.e., if edge ((𝑣, ℓ), (𝑤, ℓ +1)) is faulty, +we instead consider (𝑣, ℓ) faulty. We impose the constraint that no node has two faulty predecessors. +Formally, for all ℓ ∈ N and 𝑣 ∈ 𝑉 , |({(𝑣, ℓ)} ∪ � +{𝑣,𝑤}∈𝐸{(𝑤, ℓ)}) ∩ 𝐹 | ≤ 1. When analyzing the +system under random faults, we will assume that each node fails independently with probability +𝑝 ∈ 𝑜(1/√𝑛), which ensures that the above constraint is met with probability 1 − 𝑜(1). In addition, +we impose the restriction that at most a constant number of such faulty nodes change their timing +behavior between consecutive pulses. +Communication. Each node has the ability to broadcast pulse messages on its outgoing edges. If +node 𝑣ℓ ∈ 𝑉ℓ broadcasts at time 𝑡𝑣,ℓ, its successors receive its message at a (potentially different) +time from [𝑡𝑣,ℓ + 𝑑 − 𝑢,𝑡𝑣,ℓ + 𝑑]. The maximum end-to-end delay 𝑑 includes any delay caused by +computation. Typically, the delay uncertainty 𝑢 is much smaller than 𝑑. As discussed above, we +assume delays to be static, i.e., each edge 𝑒 = ((𝑣, ℓ), (𝑤, ℓ +1)) has an unknown, but fixed associated +delay 𝛿𝑒 ∈ [𝑑 − 𝑢,𝑑] applied to each pulse sent from (𝑣, ℓ) to (𝑤, ℓ + 1). +Note that faulty nodes can send pulses at arbitrary times, without being required to broadcast; +even if physical node implementations disallow point-to-point communication, edge faults could +still result in this behavior. +Local Clocks and Computations. Each node is able to approximately measure the progress of time +by means of a local time reference. We model this by node (𝑣, ℓ) having query access to a hardware +clock 𝐻𝑣,ℓ : R≥0 → R≥0 satisfying +∀𝑡 < 𝑡 ′ ∈ R≥0, 𝑡 ′ − 𝑡 ≤ 𝐻𝑣,ℓ (𝑡 ′) − 𝐻𝑣,ℓ (𝑡) ≤ 𝜗(𝑡 ′ − 𝑡). +for some 𝜗 > 1. No known phase relation is assumed between the hardware clocks. The algorithm +will use them exclusively to measure how much time passes between local events. Analogous to +delays, we assume that hardware clock speeds are static. This is justified in the same way as for +delays. +1This is an abuse of notation, since in a (roughly) square grid of 𝑛 := |𝑉𝐺 | nodes, we have Θ(√𝑛) layers. Since 𝑛, i.e., the +size of the grid, will only play a role when making probabilistic statements, we opted for this more convenient notation. + +8 +Christoph Lenzen and Shreyas Srinivas +Computations are deterministic. However, in addition to receiving a message, the hardware clock +reaching a time value previously determined by the algorithm can also trigger computations and +possibly broadcasting a pulse. +Output and Skew. The goal of the algorithm is to synchronize the pulses generated by correct nodes. +We assume that correct nodes on layer 0 generate well-synchronized pulses at times 𝑡𝑘 +𝑣,0 for 𝑘 ∈ N>0 +at a frequency we control. In Appendix A, we discuss how to realize this assumption in detail. All +other correct nodes generate pulses 𝑡𝑘 +𝑣,ℓ, 𝑘 ∈ N>0, based on the pulse messages received from their +predecessors. +Our measure of quality is the worst-case local skew the algorithm guarantees. We define the local +skew as the largest offset between the 𝑘-th pulses of adjacent nodes on the same layer or pulses 𝑘 +and 𝑘 + 1 of adjacent nodes on layers ℓ and ℓ + 1, whichever is larger. Formally, for ℓ ∈ N we define +Lℓ := sup +𝑘 ∈N +max +{𝑣,𝑤}∈𝐸 +(𝑣,ℓ),(𝑤,ℓ)∉𝐹 +{|𝑡𝑘 +𝑣,ℓ − 𝑡𝑘 +𝑤,ℓ|}, +Lℓ,ℓ+1 := sup +𝑘 ∈N +max +((𝑣,ℓ),(𝑤,ℓ+1)) ∈𝐸ℓ +(𝑣,ℓ),(𝑤,ℓ+1)∉𝐹 +{|𝑡𝑘 +𝑣,ℓ − 𝑡𝑘+1 +𝑤,ℓ+1|}, +and L := supℓ ∈N max{Lℓ, Lℓ,ℓ+1}. This deviates from the standard definition of the local skew: +• The definition is adjusted to pulse synchronization, which can be viewed as an essentially +equivalent time-discrete variant of clock synchronization [16]. +• Between consecutive layers, we synchronize consecutive pulses. After initialization, which is +complete once the first pulse propagated through the (in practice finite) grid, this is equivalent +to a layer-dependent index shift of pulse numbers. +3 +ALGORITHM +In this section, we discuss the pulse forwarding algorithm. We provide a simplified version of the +algorithm that behaves identical so long as the predecessors of the executing node are correct. The +full algorithm needs to handle the possibility that faulty nodes send multiple messages or none at +all. This complicates bookkeeping and loop control, distracting from the principles underlying the +algorithm’s operation. Accordingly, we defer the full algorithm to Appendix B, where we show the +equivalence to the simplified variant when there are no faulty predecessors. +3.1 +Simplified Pulse Forwarding Algorithm +The algorithm proceeds in iterations corresponding to pulses. In each iteration, node (𝑣, ℓ) +(1) timestamps the arrival times of the pulses of its predecessors using its hardware clock, +(2) determines a correction value C𝑣,ℓ based on these timestamps, and +(3) forwards the pulse Λ − 𝑑 − C𝑣,ℓ time after receiving the pulse from 𝑣ℓ−1, measured by its +hardware clock. +If all reception times are close to each other, then C𝑣,ℓ will be small. Recalling that messages are +in transit for roughly 𝑑 time, this translates to Λ being the nominal time for a pulse to propagate +from layer ℓ − 1 to layer ℓ. We need to choose Λ large enough such that the above sequence can be +always realized. That is, we need to consider how far apart the reception times of messages from +the previous layer can be, and ensure that Λ − 𝑑 exceeds this value plus the resulting C𝑣,ℓ. +Assuming that this precondition holds, Algorithm 1 implements the above approach. In each +loop iteration, it initializes three reception times to ∞: +• 𝐻own, which stores the arrival time of the pulse from (𝑣, ℓ − 1). From the perspective of the +simulated GCS algorithm, this reflects the state of the node 𝑣 ∈ 𝑉 simulated by (𝑣, ℓ), ℓ ∈ N. +• 𝐻min, which stores the minimum arrival time of a pulse from a neighbor 𝑤ℓ−1, 𝑤 ≠ 𝑣. This +corresponds to the first pulse received from a neighbor 𝑤 of 𝑣 in 𝐺 in this iteration. + +Gradient TRIX +9 +Algorithm 1 Simplified pseudocode for discrete GCS at node (𝑣, ℓ), ℓ > 0. As shown in Lemma 29, +this code is equivalent to Algorithm 3 in the absence of faults. The parameters Λ and 𝜅 will be +determined later, based on the analysis. +loop +𝐻own, 𝐻min, 𝐻max := ∞ +do +if received pulse from (𝑣, ℓ − 1) then +𝐻own := 𝐻𝑣,ℓ (𝑡) +if received pulse from first (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸 then +𝐻min := 𝐻𝑣,ℓ (𝑡) +if received pulse from last (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸 then +𝐻max := 𝐻𝑣,ℓ (𝑡) +until 𝐻own, 𝐻min, 𝐻max < ∞ +C𝑣,ℓ := min𝑠 ∈N{max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅/2 +if C𝑣,ℓ < 0 then +C𝑣,ℓ := min{𝐻own − 𝐻min − 𝜅/2 + 2𝜅, 0} +else if C𝑣,ℓ > 𝜗𝜅 then +C𝑣,ℓ := max{𝐻own − 𝐻max − 𝜅/2 − 𝜅,𝜗𝜅} +wait until 𝐻𝑣,ℓ (𝑡) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ +broadcast pulse +• 𝐻max, which stores the maximum arrival time of a pulse from a neighbor 𝑤ℓ−1, 𝑤 ≠ 𝑣. This +corresponds to the last pulse received from a neighbor 𝑤 of 𝑣 in 𝐺 in this iteration. +The do-until loop fills these variables with the correct values. At the heart of the algorithm lies the +computation of 𝐶𝑣,ℓ. If there were no faults, one could always compute +Δ := min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 +and then choose the closest value from the range [0,𝜗𝜅], i.e., set +C𝑣,ℓ := + + +Δ +if Δ ∈ [0,𝜗𝜅], +0 +if Δ < 0, and +𝜗𝜅 +if Δ > 𝜗𝜅. +To get intuition on this choice, observe that min𝑥 ∈R{max{𝐻own − 𝐻min − 𝑥, 𝐻own − 𝐻max + 𝑥}} is +attained when 𝐻own − 𝐻max + 𝑥 = 𝐻own − 𝐻min − 𝑥, which is equivalent to 𝑥 = (𝐻max − 𝐻min)/2, +i.e., 𝐻own − Δ = (𝐻max − 𝐻min)/2. If timing was perfectly accurate, the reception times of the pulse +messages could serve as exact proxies for the actual pulse forwarding times of the nodes on layer +ℓ − 1. In iteration 𝑘, this would mean to generate the pulse at (𝑣, ℓ) faster if (𝑣, ℓ − 1) generated +its pulse later than the average of min{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1} and max{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1}. Thus, any (𝑣, ℓ) for +which 𝑡𝑘 +𝑣,ℓ−1 − min{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1} > max{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1} − 𝑡𝑘 +𝑣,ℓ−1 would choose 𝐶𝑣,ℓ > 0, attempting +to reduce max{𝑣,𝑤}∈𝐸{|𝑡𝑘 +𝑣,ℓ − 𝑡𝑘 +𝑤,ℓ|} compared to max{𝑣,𝑤}∈𝐸{|𝑡𝑘 +𝑣,ℓ−1 − 𝑡𝑘 +𝑤,ℓ−1|}. This can be viewed as +trying to reduce the local skew by a greedy strategy. +Unfortunately, this naive strategy fails to account for inaccuracies due to message delay uncer- +tainty and drifting hardware clocks. Nonetheless, we follow this strategy up to deviations of 𝑂(𝜅). +The additional terms serve the following purposes: + +10 +Christoph Lenzen and Shreyas Srinivas +• Considering only discrete choices for 𝑥 ∈ 4𝜅N rather than arbitrary 𝑥 ∈ R is the key +ingredient that makes the algorithmic approach succeed, cf. [15]. Essentially, this is necessary +because there is no way to determine 𝑡𝑣ℓ−1,𝑘 − 𝑡𝑤ℓ−1,𝑘 precisely. Discretizing observed skews +in units of 𝜅 ∈ Θ(𝑢 + (𝜗 − 1)(Λ − 𝑑)) enables a delicate strategy that alternates between +overestimating skews to locally generate the next pulse earlier for the sake of “catching up” +with others and underestimating skews to “wait” for others catch up. +• Substracting 𝜅/2 accounts for errors in measuring skews, which are caused by uncertainty in +message delay and hardware clock speed. +• To limit the damage that a faulty predecessor of (𝑣, ℓ) can do, we ensure that (𝑣, ℓ) generates +its pulse without too large of a deviation from the median of 𝑡𝑣,ℓ−1, min{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1}, and +max{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1} (plus the nominal offset of Λ). This is achieved by permitting corrections +𝐶𝑣,ℓ < 0 if (𝑣, ℓ − 1) clearly generated its pulse earlier than min{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1} and 𝐶𝑣,ℓ > 𝜗𝜅 +if it clearly generated its pulse later than max{𝑣,𝑤}∈𝐸{𝑡𝑘 +𝑤,ℓ−1}, respectively. +To further motivate the last point, recall that there can be at most one fault among the predecessors +of (𝑣, ℓ). A single faulty predecessor can only affect only one of the three values 𝐻own, 𝐻min, and +𝐻max: control 𝐻own arbitrarily, 𝐻min to be smaller than the minimum reception time from a correct +node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, or 𝐻max to exceed the maximum reception time from correct nodes +(𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸. Hence, ensuring that pulses are generated with only a small offset relative to +median {𝐻own, 𝐻min, 𝐻max} + Λ − 𝑑 indeed limits the damage that a fault can do. +Achieving all of the desired properties is non-trivial, leading to the fairly involved choice of C𝑣,ℓ. +It can be viewed as simultaneously implementing relaxed fast and slow conditions (as introduced +in [15]), an additional jump condition required to make the GCS algorithm work under these relaxed +fast and slow conditions, and the requirement to stick close to the median of predecessors’ pulse +times. In Section 4.2, we specify the (relaxed) slow and fast condition, as well as the jump condition, +and show that the algorithm implements them. Lemmas 19 and 20 show that the algorithm also +enforces deviates little from the time interval spanned by correct predecessors (offset by Λ). +There is some freedom in the choice of parameters. For simplicity, we fix a good choice of 𝜅 and +note that 𝑑 must satisfy a lower bound 𝐵 ∈ 𝑂(supℓ ∈N{Lℓ} +𝜅). Observe that this constraint simply +means that the skew bounds are useful, as a skew that is of similar size as the maximum end-to-end +delay requires to slow the system down substantially. Finally, Λ must be at least 𝑑 +𝑂(supℓ ∈N{Lℓ}), +which due to the previous constraint holds e.g. for the choice Λ = 2𝑑. Formally, for a sufficiently +large constant 𝐶,2 +𝜅 := 2 +� +𝑢 + +� +1 − 1 +𝜗 +� +(Λ − 𝑑) +� +, +(1) +Λ ≥ 𝐶𝜗(sup +ℓ ∈N +{Lℓ} + 𝑢) + 𝑑, and +(2) +𝑑 ≥ 𝐶(𝜗(sup +ℓ ∈N +{Lℓ} + 𝑢) + 𝜅). +(3) +Complete Algorithm. The complete algorithm cannot wait for messages from all predecessors to +determine when to send its pulse, as a faulty node not sending its pulse then would deadlock all its +descendants. As discussed above, the hardware clock time of the next pulse time does not deviate +much from median {𝐻own, 𝐻min, 𝐻max} + Λ −𝑑, but does depend on max{𝐻min, 𝐻own, 𝐻max} in some +cases. However, we will prove that Lℓ−1 is small enough such that all pulse messages from correct +nodes will be received in time. Hence, it is sufficient to wait until median {𝐻own, 𝐻min, 𝐻max}+𝜗Lℓ−1 +(or later) according to 𝐻𝑣,ℓ. Provided that Λ − 𝑑 is large enough, this implies that any message for +2We do not attempt to optimize constants in this work. + +Gradient TRIX +11 +computing C𝑣,ℓ missing is due to a fault; in fact, at the point in time when this becomes clear, C𝑣,ℓ +is already determined, regardless of how late the message would arrive. +The complete algorithm differs from Algorithm 1 by covering the case that a signal does not +arrive in time. Intuitively, one can treat the respective message arrival time (𝐻own or 𝐻max, 𝐻min is +not possible) as ∞, while allowing such an ∞ to cancel out in substraction: +• If 𝐻own = ∞, then 𝐶𝑣,ℓ ∈ 𝐻own − 𝐻max − 𝑂(𝜅), and (𝑣, ℓ) will generate its pulse at local time +𝐻own + Λ − 𝑑 − 𝐶𝑣,ℓ ∈ 𝐻max + Λ − 𝑑 + 𝑂(𝜅). +• If 𝐻max = ∞ and 𝐻own ≥ 𝐻min, then 𝐶𝑣,ℓ ∈ 𝐻own − 𝐻min ± Θ(𝜅) and (𝑣, ℓ) will generate its +pulse at local time 𝐻own + Λ − 𝑑 − 𝐶𝑣,ℓ ∈ 𝐻min + Λ − 𝑑 ± 𝑂(𝜅). +• If 𝐻max = ∞ and 𝐻own < 𝐻min, then 𝐶𝑣,ℓ ∈ [0, 2𝜅] and (𝑣, ℓ) will generate its pulse at local +time 𝐻own + Λ − 𝑑 − 𝐶𝑣,ℓ ∈ 𝐻own + Λ − 𝑑 − 𝑂(𝜅). +Note that in all cases, the pulse is generated with an offset of Λ−𝑑 −Θ(𝜅) from the median reception +time. The complete algorithm follows the above intuition, leveraging the fact that there is no need +to wait indefinitely to determine that the third signal is late, and is given in Appendix B. +Last, but not least, it is of interest to make the pulse forwarding algorithm self-stabilizing [5]. +Due to the design choice of propagating the clock signal from a single source along a DAG, this will +immediately translate to the overall scheme being self-stabilizing, so long as the clock generation +is self-stabilizing, too. This is straightforward, because one can assume that the signals from the +previous layer are already well-synchronized. Thus, all that nodes need to do is to detect when all +but possibly one (faulty) pulse signal arrive in close temporal proximity to determine when to clear +their memory and start a new iteration of the main loop. In Appendix B.1, we sketch how this can +be achieved using standard techniques. +4 +ANALYSIS +We now analyze the pulse progagation scheme under the assumption that layer 0 generates well- +synchronized pulses. We discuss a suitable method for achieving this in Appendix A. Our analysis +proceeds along the following lines: +(1) We show that, if the local skew is small enough compared to Λ, i.e., Equation (2) holds, all +correct nodes execute their iterations as intended. That is, each correct node on layer ℓ > 0 +receives the 𝑘-th pulses of its correct predecessors in its 𝑘-th loop iteration. This is deferred +to Appendix B. We then proceed under the assumption that this holds true, which will be +justified retroactively once we establish that the local skew is bounded. +(2) Since delays and hardware clock speeds are (approximated as being) static, any (substantial) +change in relative timing of consecutive pulses is due to faulty nodes. Thus, the task of +bounding the local skew reduces to bounding the intra-layer skew Lℓ for a single pulse, since +such a bound must take into account the full variability introduced by faulty nodes. This +reasoning is deferred to Appendix C. +(3) Based on potential functions, we analyze Lℓ in the absence of faults. The results entail not +only bounded skew, but also that the potentials recover if they become unexpectedly large. +(4) We show that faulty nodes have limited impact on the potentials. From this and the above +recovery property, we conclude that skews behave favorably also when there are faults. +As stated above, the first two steps of our line of reasoning are deferred to the appendix. The main +challenge is to bound Lℓ for a single pulse. Due to the first step, we know that the 𝑘-th pulse at +correct nodes depends only on the 𝑘-th pulses of their predecessors (Lemma 28). Therefore, in the +following fix 𝑘 and denote the 𝑘-th pulse time of correct (𝑣, ℓ) ∈ 𝑉𝐺 by 𝑡𝑣,ℓ. +Recall that for 𝑣,𝑤 ∈ 𝑉 , we denote by 𝑑(𝑣,𝑤) their distance in the base graph 𝐻. Our analysis is +built around the following potential functions. + +12 +Christoph Lenzen and Shreyas Srinivas +Definition 1 (Potential Functions). Let 𝑣,𝑤 ∈ 𝑉 and 𝑠, ℓ ∈ N. We define +𝜓𝑠 +𝑣,𝑤(ℓ) := 𝑡𝑣,ℓ − 𝑡𝑤,ℓ − 4𝑠𝜅𝑑(𝑣,𝑤), +Ψ𝑠 (ℓ) := max +𝑣,𝑤∈𝑉{𝜓𝑠 +𝑣,𝑤(ℓ)}, +𝜉𝑠 +𝑣,𝑤(ℓ) := 𝑡𝑣,ℓ − 𝑡𝑤,ℓ − (4𝑠 − 2)𝜅𝑑(𝑣,𝑤), and +Ξ𝑠 (ℓ) := max +𝑣,𝑤∈𝑉{Ξ𝑠 +𝑣,𝑤(ℓ)}. +Bounding Ψ𝑠 (ℓ) readily translates to bounding Lℓ. +Observation 1. If for 𝑠, ℓ ∈ N and some Ψ𝑠 ∈ R≥0 it holds that Ψ𝑠 (ℓ) ≤ Ψ𝑠, then Lℓ ≤ Ψ𝑠 + 4𝑠𝜅. +Proof. Fix 𝑘 ∈ N and suppose that {𝑣,𝑤} ∈ 𝐸 maximizes |𝑡𝑣,ℓ − 𝑡𝑤,ℓ|. W.l.o.g., assume that +𝑡𝑣,ℓ ≥ 𝑡𝑤,ℓ. Since {𝑣,𝑤} ∈ 𝐸, we have that 𝑑(𝑣,𝑤) = 1. Hence, +|𝑡𝑣,ℓ − 𝑡𝑤,ℓ| = 𝑡𝑣,ℓ − 𝑡𝑤,ℓ = 𝜓𝑠 +𝑣,𝑤(ℓ) + 4𝑠𝜅 ≤ Ψ𝑠 (ℓ) + 4𝑠𝜅 ≤ Ψ𝑠 + 4𝑠𝜅. +Since 𝑘 ∈ N is arbitrary, it follows that Lℓ ≤ Ψ𝑠 + 4𝑠𝜅. +□ +In summary, the goal of our analysis will be to bound Ψ𝑠 (ℓ) by a small value for some 𝑠 satisfying +4𝑠𝜅 ∈ 𝑂(𝑢 log 𝐷). +We first study the behavior of the algorithm if there are no faults. Accordingly, this will be tacitly +assumed in all statements of this section, with the expection of Section 4.4. Note that by Lemma 29, +this means that we may also tacitly assume that Algorithm 1 is run by all nodes in layers ℓ ∈ N>0. +In Section 4.4, we will then bound the impact of faulty layers on the potential. +4.1 +Basic Statements +We first show three basic lemmas. The first relates the local reception times of pulses to the actual +sending times, bounding the error by 𝜅. +Lemma 1. For (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0, set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and 𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}. +Then +𝑡𝑣,ℓ−1 − 𝑡max − 𝜅 ≤ 𝐻own − 𝐻max − 𝜅 +2 ≤ 𝑡𝑣,ℓ−1 − 𝑡max +𝑡𝑣,ℓ−1 − 𝑡min − 𝜅 ≤ 𝐻own − 𝐻min − 𝜅 +2 ≤ 𝑡𝑣,ℓ−1 − 𝑡min. +Proof. We prove the first inequality; the second is shown analogously. Let 𝑡 ′ +𝑣,ℓ−1 and 𝑡 ′ +max denote +the times when the pulse messages sent at time 𝑡𝑣,ℓ−1 and 𝑡max are received at 𝑣ℓ, respectively. From +the bounds on message delays, it follows that +𝑡𝑣,ℓ−1 + 𝑑 − 𝑢 ≤ 𝑡 ′ +𝑣,ℓ−1 ≤ 𝑡𝑣,ℓ−1 + 𝑑 and +𝑡max + 𝑑 − 𝑢 ≤ 𝑡 ′ +max ≤ 𝑡max + 𝑑. +Thus, +𝑡𝑣,ℓ−1 − 𝑡max − 𝑢 ≤ 𝑡 ′ +𝑣,ℓ−1 − 𝑡 ′ +max ≤ 𝑡𝑣,ℓ−1 − 𝑡max + 𝑢. +Using the bounds on hardware clock rates, we get that +|𝑡 ′ +𝑣,ℓ−1 − 𝑡 ′ +max − (𝐻own − 𝐻max)| ≤ (𝜗 − 1)|𝑡 ′ +𝑣,ℓ−1 − 𝑡 ′ +max| ≤ (𝜗 − 1)(|𝑡𝑣,ℓ−1 − 𝑡max| + 𝑢). + +Gradient TRIX +13 +Applying Equation (2), we infer that +|𝑡𝑣,ℓ−1 − 𝑡max − (𝐻own − 𝐻max)| ≤ |𝑡 ′ +𝑣,ℓ−1 − 𝑡 ′ +max − (𝐻own − 𝐻max)| + 𝑢 +≤ (𝜗 − 1)|𝑡𝑣,ℓ−1 − 𝑡max| + 𝜗𝑢 +≤ (𝜗 − 1)Lℓ−1 + 𝜗𝑢 +≤ (𝜗 − 1) +�Λ − 𝑑 +𝜗 +− 𝑢 +� ++ 𝜗𝑢 += +� +1 − 1 +𝜗 +� +(Λ − 𝑑) + 𝑢. +Finally, using Equation (1), we conclude that +𝑡𝑣,ℓ−1 − 𝑡max − 𝜅 ≤ 𝑡𝑣,ℓ−1 − 𝑡max − 2 +�� +1 − 1 +𝜗 +� +(Λ − 𝑑) + 𝑢 +� +≤ 𝐻own − 𝐻max − 𝜅 +2 +≤ 𝑡𝑣,ℓ−1 − 𝑡max. +□ +The second lemma shows that corrections are not too large. +Lemma 2. For all 𝑣 ∈ 𝑉 and ℓ ∈ N>0, C𝑣,ℓ ≤ Λ − 𝑑. +Proof. Abbreviate +Δ = min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 . +We distinguish three cases. +• Δ < 0. Then Algorithm 1 sets +C𝑣,ℓ ≤ min +� +𝐻own − 𝐻min − 𝜅 +2 + 2𝜅, 0 +� +≤ 0. +As Λ ≥ 𝑑 by Equation (2), the claim of the lemma holds in this case. +• 0 ≤ Δ ≤ 𝜗𝜅. Then, using the notation of Lemma 1, +C𝑣,ℓ = Δ < min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 +≤ min +𝑠 ∈N +� +max{𝑡𝑣,ℓ−1 − 𝑡max + 4𝑠𝜅,𝑡𝑣,ℓ−1 − 𝑡min − 4𝑠𝜅} +� +≤ min +𝑠 ∈N {max{Lℓ−1 + 4𝑠𝜅, Lℓ−1 − 4𝑠𝜅}} += Lℓ−1, +which is smaller than Λ − 𝑑 by Equation (2). +• Δ > 𝜗𝜅. Note that then +𝜗𝜅 < Δ ≤ max{𝐻own − 𝐻max, 𝐻own − 𝐻min} − 𝜅 +2 = 𝐻own − 𝐻max − 𝜅 +2, + +14 +Christoph Lenzen and Shreyas Srinivas +as 𝐻max ≥ 𝐻min. Therefore, applying Lemma 2, +C𝑣,ℓ = max +� +𝐻own − 𝐻max − 𝜅 +2 − 𝜅,𝜗𝜅 +� +≤ 𝐻own − 𝐻max − 𝜅 +2 +≤ 𝑡𝑣,ℓ−1 − 𝑡max +≤ Lℓ−1 +< Λ − 𝑑. +□ +The third lemma bounds the time difference between the pulses of (𝑣, ℓ − 1) and (𝑣, ℓ). +Lemma 3. For all 𝑣 ∈ 𝑉 and ℓ ∈ N>0 it holds that +𝑑 − 𝑢 + Λ − 𝑑 − C𝑣,ℓ +𝜗 +≤ 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 ≤ Λ − C𝑣,ℓ. +Proof. Let 𝑡 ′ +𝑣,ℓ−1 denote the time at which (𝑣, ℓ) receives the pulse sent by (𝑣, ℓ − 1) at time 𝑡𝑣,ℓ−1. +Inspecting the code of Algorithm 1, we see that +𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ = 𝐻𝑣,ℓ (𝑡 ′ +𝑣,ℓ−1) + Λ − 𝑑 − C𝑣,ℓ. +Since C𝑣,ℓ ≤ Λ −𝑑 by Lemma 2, it follows that 𝐻𝑣,ℓ (𝑡 ′ +𝑣,ℓ−1) ≥ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) and hence 𝑡𝑣,ℓ ≥ 𝑡 ′ +𝑣,ℓ−1. Using +the bounds on message delays and hardware clock speeds, we get that +𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 = 𝑡𝑣,ℓ − 𝑡 ′ +𝑣,ℓ−1 + 𝑡 ′ +𝑣,ℓ−1 − 𝑡𝑣,ℓ−1 +≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) − 𝐻𝑣,ℓ (𝑡 ′ +𝑣,ℓ−1) + 𝑑 += Λ − C𝑣,ℓ +and +𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 = 𝑡𝑣,ℓ − 𝑡 ′ +𝑣,ℓ−1 + 𝑡 ′ +𝑣,ℓ−1 − 𝑡𝑣,ℓ−1 +≥ +𝐻𝑣,ℓ (𝑡𝑣,ℓ) − 𝐻𝑣,ℓ (𝑡 ′ +𝑣,ℓ−1) +𝜗 ++ 𝑑 − 𝑢 += Λ − 𝑑 − C𝑣,ℓ +𝜗 ++ 𝑑 − 𝑢, +which can be rearranged into the claimed inequalities. +□ +4.2 +The Slow, Fast, and Jump Conditions +The key to bounding the local skew without faults is to find the right balance between two conflicting +goals: choosing C𝑣,ℓ large enough to “catch up” to predecessors 𝑤ℓ−1 ≠ 𝑣ℓ−1 that generated their +pulse earlier than 𝑣ℓ−1, but small enough to “wait” for predecessors 𝑤ℓ−1 ≠ 𝑣ℓ−1 that generated their +pulse later than 𝑣ℓ−1. The following condition captures what we need regarding the latter. +Definition 2 (Slow Condition). For all 𝑠 ∈ N, correct layers ℓ − 1 ∈ N, and 𝑣ℓ ∈ 𝑉ℓ \ 𝐹, we require +the slow condition SC(𝑠) := SC-1(𝑠) ∨ SC-2(𝑠) ∨ SC-3 to hold, where +SC-1(𝑠) : C𝑣,ℓ +𝜗 +≤ 𝑡𝑣,ℓ−1 − max +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} + 4𝑠𝜅 +SC-2(𝑠) : C𝑣,ℓ +𝜗 +≤ 𝑡𝑣,ℓ−1 − +min +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − 4𝑠𝜅 +SC-3: C𝑣,ℓ ≤ 0. + +Gradient TRIX +15 +This can be viewed as a variant of the slow condition from [15], adjusted to our setting by +quantifying by how much 𝑣ℓ may safely shift the timing of its pulse. The main conceptual difference +to [15] is that we relax the slow condition by adding SC-3. In what follows, we drop 𝑠 from the +notation when it is clear from context. +Lemma 4. For all 𝑠 ∈ N and (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0, SC(𝑠) holds at (𝑣, ℓ). +Proof. Using Lemma 29, we prove the claim for Algorithm 1. Set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} +and 𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}. If C𝑣,ℓ ≤ 0, SC-3 is trivially satisfied. Hence, assume that C𝑣,ℓ > 0. +Abbreviate +Δ = min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 += max{𝐻own − 𝐻max + 4𝑠min𝜅, 𝐻own − 𝐻min − 4𝑠min𝜅} − 𝜅 +2, +where 𝑠min ∈ N is an index for which the minimum is attained. +If Δ ≤ 𝜗𝜅, then C𝑣,ℓ = Δ. Otherwise, +C𝑣,ℓ = max +� +𝐻own − 𝐻max − 𝜅 +2 − 𝜅,𝜗𝜅 +� +≤ max{Δ,𝜗𝜅} = Δ. +Either way, we get that C𝑣,ℓ/𝜗 < C𝑣,ℓ ≤ Δ. +We distinguish two cases. +• 𝐻own − 𝐻max + 4𝑠min𝜅 ≥ 𝐻own − 𝐻min − 4𝑠min𝜅. Then for 𝑠 ∈ N, 𝑠 ≥ 𝑠min, by Lemma 1 we have +that +Δ ≤ 𝐻own − 𝐻max + 4𝑠min𝜅 − 𝜅 +2 ≤ 𝑡𝑣,ℓ−1 − 𝑡max + 4𝑠𝜅, +i.e., SC-1 holds. Now consider 𝑠 ∈ N, 𝑠 < 𝑠min. Since 𝐻own −𝐻max −𝜗𝑢 + 4𝑠𝜅 < 𝐻own −𝐻max − +𝜗𝑢 + 4𝑠min𝜅 ≤ Δ, but the minimum is attained at index 𝑠min, we must have that +Δ ≤ 𝐻own − 𝐻min − 4𝑠𝜅 − 𝜅 +2 ≤ 𝑡𝑣,ℓ−1 − 𝑡min − 4𝑠𝜅, +where the second step again applies Lemma 1. Thus, SC-2 holds. +• 𝐻own − 𝐻max + 4𝑠min𝜅 < 𝐻own − 𝐻min − 4𝑠min𝜅. In this case, we analogously infer that SC-1 +holds for 𝑠 > 𝑠min and SC-2 holds for 𝑠 ≤ 𝑠min. +□ +The fast condition is the counterpart to Definition 2 addressing the need to “catch up” to neighbors +that are ahead. +Definition 3 (Fast Condition). For all 𝑠 ∈ N>0, correct layers ℓ − 1 ∈ N>0, and 𝑣ℓ ∈ 𝑉ℓ \ 𝐹, we +require the fast condition FC(𝑠) := FC-1(𝑠) ∨ FC-2(𝑠) ∨ FC-3 to hold, where +FC-1(𝑠) : C𝑣,ℓ ≥ 𝑡𝑣,ℓ−1 − max +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} + (4𝑠 − 2)𝜅 + 𝜅 +FC-2(𝑠) : C𝑣,ℓ ≥ 𝑡𝑣,ℓ−1 − +min +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − (4𝑠 − 2)𝜅 + 𝜅 +FC-3: C𝑣,ℓ ≥ 𝜅. +This can be viewed as a variant of the fast condition from [15], adjusted to our setting by +quantifying by how much 𝑣ℓ may safely shift the timing of its pulse. The main conceptual difference +to [15] is that we relax the fast condition by adding FC-3. +In addition, note that there is an additive term of 𝜅 that does not change sign. Its purpose is to +account for the fact that our simulation of the GCS algorithm from [18] operates in discrete time +steps corresponding to the layers. The continuous versions of the GCS algorithm in [14, 15, 18] +can choose this term arbitrarily small. In contrast, we need it to exceed the maximum error in +time measurement accumulated in a step. We remark that, in principle, one could choose this term + +16 +Christoph Lenzen and Shreyas Srinivas +𝑣 +𝑡𝑣 − 4𝑠𝜅𝑑(𝑣,𝑤) +𝑤 +𝑡𝑤 − (4𝑠 − 2)𝜅𝑑(𝑣,𝑤) +Fig. 4. Slow condition (left) and fast condition (right). SC(𝑠) is tailored to ensuring that max𝑤∈𝑉 {𝜓𝑠𝑣,𝑤(ℓ)} +(the length of the green arrow) cannot grow quickly. Nodes 𝑤 with C𝑤,ℓ ≤ 0 (SC-3 holds) cannot apply a +correction pushing them below the red line. If C𝑤,ℓ > 0, then both SC-1 and SC-2 will ensure that there is a +neighbor 𝑥 of 𝑤 such that the offset of 𝑡𝑤,ℓ−1 − C𝑤,ℓ/𝜗 to the black line does not exceed the one of 𝑡𝑥,ℓ−1. +In other words, SC ensures that the blue arrows indicating C𝑤,ℓ/𝜗 do not reach below the red line. This +means that any increase of max𝑤∈𝑉 {𝜓𝑠𝑣,𝑤(ℓ)} is caused by delay and clock speed variation, which in turn is +bounded by 𝜅/2 per layer. Similarly, FC(𝑠) is tailored to ensuring that max𝑣∈𝑉 {𝜉𝑠𝑣,𝑤(ℓ)} (the length of the +green arrow), if positive, decreases by at least 𝜅/2. To ensure this, C𝑤,ℓ (indicated by blue arrows) must be +large enough to reach below the red line. This is achieved by FC(𝑠) having an additional “slack” term of 𝜅, +which overcomes the “loss” of 𝜅/2 due to uncertainty. +different from 𝜅. However, since both need to meet the same lower bound of 𝑢 + (1 − 1/𝜗)(Λ − 𝑑), +there is no asymptotic gain in introducing a separate parameter. +Lemma 5. For all 𝑠 ∈ N and (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0, FC(𝑠) holds at (𝑣, ℓ). +Proof. Using Lemma 29, we prove the claim for Algorithm 1. Set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and +𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}. If C𝑣,ℓ ≥ 𝜗𝜅, trivially FC-3 is satisfied. Hence, assume that C𝑣,ℓ < 𝜗𝜅. +Abbreviate +Δ = min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 += max{𝐻own − 𝐻max + 4𝑠min𝜅, 𝐻own − 𝐻min − 4𝑠min𝜅} − 𝜅 +2, +where 𝑠min ∈ N is an index for which the minimum is attained. +If Δ ≥ 0, then C𝑣,ℓ = Δ. Otherwise, +C𝑣,ℓ = min +� +𝐻own − 𝐻min − 𝜅 +2 + 2𝜅, 0 +� +≥ Δ. +Either way, we get that C𝑣,ℓ ≥ Δ. +For 𝑠 ∈ N, 𝑠 ≤ 𝑠min, by Lemma 1 and Equation (1) it holds that +Δ ≥ 𝐻own − 𝐻max + 4𝑠𝜅 − 𝜅 +2 ≥ 𝑡𝑣,ℓ−1 − 𝑡max + (4𝑠 − 2)𝜅 + 𝜅, +proving that FC-1 holds. For 𝑠 ∈ N, 𝑠 > 𝑠min, by Lemma 1 and Equation (1) we get that +Δ ≥ 𝐻own − 𝐻min − 4(𝑠 − 1)𝜅 − 𝜅 +2 ≥ 𝑡𝑣,ℓ−1 − 𝑡min − (4𝑠 − 2)𝜅 + 𝜅, +showing that FC-2 holds. +□ +Our relaxation of the slow and fast conditions adds a substantial complication. From the per- +spective of the time-continuous variant of the algorithm in [15], we now allow for arbitrarily large +clock “jumps,” rather than bounded clock rates. In our discrete version, the rate bound from [15] +corresponds to C𝑣,ℓ ∈ [0,𝜗𝜅]. Without this additional constraint, the slow and fast conditions are +insufficient to bound skews. + +Gradient TRIX +17 +𝑣ℓ+2 +𝑣ℓ+1 +𝑣ℓ +𝑣ℓ+2 +𝑣ℓ+1 +𝑣ℓ +Fig. 5. On the left, it is shown how skews increase without JC. While SC(0) disallows that (𝑣, ℓ) speeds up +its pulse by more than the equivalent of (𝑣, ℓ − 1) matching the earliest pulse of any (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, +FC permits that a node (𝑣, ℓ) with slow (𝑣, ℓ − 1) to “overshoot,” i.e., C𝑣,ℓ (shown as blue arrow) gets large. +This results in an amplifying oscillatory behavior. On the right, the same scenario is shown with JC in effect. +JC forces the corrections to stop 𝜅 before the earliest or latest neighbor, respectively, resulting in a dampened +oscillation. +This is illustrated in Figure 5, showing an execution that satisfies SC and FC, but suffers from +skews that grow without bound. The key issue is that adjacent nodes could “jump” in opposite +directions, resulting in an oscillatory behavior in which measurement errors accumulate indefinitely. +To avoid this kind of behavior, we add an additional condition that “dampens” such oscillations, yet +limits by how much a faulty predecessor can cause an increase in skew. +Definition 4 (Jump Condition). For all correct layers ℓ − 1 ∈ N>0 and 𝑣ℓ ∈ 𝑉ℓ \ 𝐹, we require the +jump condition JC := JC-1 ∨ JC-2 ∨ JC-3 to hold, where +JC-1: 𝜅 < C𝑣,ℓ +𝜗 +≤ 𝑡𝑣,ℓ−1 − max +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − 𝜅 +JC-2: 0 > C𝑣,ℓ ≥ 𝑡𝑣,ℓ−1 − +min +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} + 𝜅 +JC-3: 0 ≤ C𝑣,ℓ +𝜗 +≤ 𝜅. +Lemma 6. Suppose that layer ℓ − 1 ∈ N and 𝑣ℓ ∈ 𝑉ℓ are correct. Then JC holds at 𝑣ℓ. +Proof. Using Lemma 29, we prove the claim for Algorithm 1. Set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and +𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}. We distinguish three cases. +• 0 ≤ C𝑣,ℓ ≤ 𝜗𝜅. Then JC-3 is satisfied trivially. + +18 +Christoph Lenzen and Shreyas Srinivas +• C𝑣,ℓ < 0. By Lemma 1 and Equation (1), then +C𝑣,ℓ = 𝐻own − 𝐻min − 𝜅 +2 + 2𝜅 ≥ 𝑡𝑣,ℓ−1 − 𝑡min + 𝜅, +i.e., JC-2 holds. +• C𝑣,ℓ > 𝜗𝜅. By Lemma 1, then +C𝑣,ℓ = 𝐻own − 𝐻max − 𝜅 +2 − 𝜅 ≤ 𝑡𝑣,ℓ−1 − 𝑡max − 𝜅, +i.e., JC-3 holds. +□ +4.3 +Bounding Ψ𝑠 in the Absence of Faults +With the conditions established, we are ready to study how Ψ𝑠 (ℓ) evolves in the fault-free setting. +The main technical challenge in bounding Ψ𝑠 lies in performing the induction step from 𝑠 − 1 ∈ N +to 𝑠. We will argue that for Ψ𝑠 ( ¯ℓ ) to be large for some ¯ℓ, Ξ𝑠 (ℓ ) must have been large for some +ℓ < ¯ℓ, with an additive term growing with ¯ℓ − ℓ. +Theorem 1. For 𝑠 ∈ N>0 and layers ℓ ≤ ¯ℓ, it holds that +Ψ𝑠 ( ¯ℓ ) ≤ max +� +0, Ξ𝑠 (ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 +� ++ ( ¯ℓ − ℓ ) · 𝜅 +2 . +Proof strategy. Intuitively, we intend to argue that if Ψ𝑠 ( ¯ℓ ) is large, so must be Ξ𝑠 (ℓ ). Tracing +back the cause for this, we show that in every step, we have that Ξ𝑠 (ℓ − 1) is larger than Ξ𝑠 (ℓ) by at +least 𝜅/2. Since Ξ𝑠 ( ¯ℓ ) ≥ Ψ𝑠 ( ¯ℓ ), as 𝜓𝑠 +𝑣,𝑤(ℓ) ≥ 𝜉𝑠 +𝑣,𝑤(ℓ) for all 𝑣, 𝑤, 𝑠, and ℓ, this yields the claim. To +formalize that Ξ𝑠 (ℓ) must have been decreasing steadily, we seek to show that the minimal layer ℓ +for which there are nodes 𝑣ℓ,𝑤ℓ ∈ 𝑉 satisfying that 𝜉𝑠 +𝑣ℓ,𝑤ℓ (ℓ) is large enough is ℓ. To this end, we +identify nodes 𝑤 and 𝑣 – either 𝑤ℓ and 𝑣ℓ themselves or neighbors of them – which cause the large +skew on layer ℓ by a having a large skew on layer ℓ − 1. This is done based on SC(𝑠) and FC(𝑠), +with JC kicking in for the special case that 𝑤 = 𝑣ℓ and 𝑣 = 𝑤ℓ. +A key obstacle is that if 𝑤 is a neighbor of 𝑤ℓ, this results in a larger difference in skew than if 𝑣 +is a neighbor of 𝑣ℓ, namely 4𝑠𝜅 versus (4𝑠 − 2)𝜅. Thus, when 𝑤 is closer to 𝑣ℓ than 𝑤ℓ, we “lose” 2𝜅 +relative to the skew bound on layer ℓ. For 𝑑(𝑣 ¯ℓ,𝑤 ¯ℓ) many steps, we can compensate for this based +on the initial skew between 𝑣 ¯ℓ and 𝑤 ¯ℓ, but not more. To address this, essentially we need to show +that for any additional steps “towards” 𝑣ℓ there will be a corresponding step “away” from 𝑣ℓ, on +which we “gain” additional 2𝜅 relative to the skew bound on the layer ℓ. +If corrections were always positive, this would be straightforward: Steps towards 𝑣ℓ would also +be steps towards 𝑣 ¯ℓ, and upon 𝑤ℓ = 𝑣 ¯ℓ we would reach a contradiction to the skew bounds shown. +Unfortunately, negative corrections foreclose this simple argument. To address this, we introduce a +third “prover” node 𝑝ℓ, where 𝑝 ¯ℓ = 𝑣 ¯ℓ, which never increases its distance to 𝑤ℓ; if 𝑝ℓ performs a +negative correction, then 𝑝 is a neighbor of 𝑝ℓ that is closer to 𝑤ℓ. We then can infer that 𝑝 ≠ 𝑤 +from the skew bounds. +A major complication this approach faces is the special case 𝑝 = 𝑤ℓ and 𝑤 = 𝑝ℓ. Again, JC kicks +in to show that we have sufficiently large skew between 𝑝 and 𝑤. However, now 𝑝 lies “behind” 𝑤 +from the perspective of 𝑣. A later reversal of this situation by repeating the case that 𝑝 = 𝑤ℓ and +𝑤 = 𝑝ℓ results in 𝑤 being farther away from 𝑣ℓ, yet 𝑑(𝑝,𝑤) = 𝑑(𝑝ℓ,𝑤ℓ). The proof covers this case +by adding an additional (4𝑠 − 2)𝜅 to the skew bound if the above situation occured an odd number +of times. +Finally, we seek to avoid the case that 𝑣 = 𝑝ℓ and 𝑝 = 𝑣ℓ for analogous reasons. Fortunately, +here we can exploit that the skew bound between 𝑣ℓ and 𝑤ℓ is stronger than the one between 𝑝ℓ +and 𝑤ℓ, meaning that we can simply choose 𝑝 = 𝑣 instead in this situation. In the proof, we do so + +Gradient TRIX +19 +whenever 𝑣 lies on the path connecting 𝑝ℓ and 𝑤ℓ that we maintain to keep track of hop counts in +the construction. +□ +Proof of Theorem 1. Assume towards a contradiction that the statement of Theorem 1 is false +for minimal ¯ℓ, i.e., there are 𝑣 ¯ℓ and 𝑤 ¯ℓ such that +𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ > ( ¯ℓ − ℓ ) · 𝜅 +2 +(4) +and +𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ > Ξ𝑠 (ℓ ) − ( ¯ℓ − ℓ ) · 𝜅 +2 − 𝜅 +(5) +and there is no smaller ¯ℓ′ for which this applies for some pair of nodes. +Let ℓ ∈ [ℓ, ¯ℓ] be minimal such that are 𝑣ℓ, 𝑝ℓ,𝑤ℓ ∈ 𝑉 , a path 𝑄ℓ in 𝐻 from 𝑝ℓ to 𝑣ℓ, and a path 𝑃ℓ +in 𝐻 from 𝑝ℓ to 𝑤ℓ with the following properties: +(P1) 𝑤ℓ ≠ 𝑝ℓ. +(P2) 𝑤ℓ ≠ 𝑣ℓ. +(P3) +𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅|𝑃ℓ| ≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − ℓ) · 𝜅 +2 > 0. +(P4) Denote by |𝑃ℓ| and |𝑄ℓ| the length of 𝑃ℓ and 𝑄ℓ, respectively. With the shorthand +Δℓ := +� +|𝑃ℓ| + |𝑄ℓ| − 1 +if 𝑃ℓ and 𝑄ℓ have the same first edge +|𝑃ℓ| + |𝑄ℓ| +else, +it holds that +𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅Δℓ ≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ) · 𝜅 +2 + 2𝜅|𝑃ℓ|. +(P5) If 𝑣ℓ ∈ 𝑃ℓ, then 𝑝ℓ = 𝑣ℓ. +To see that such an index must indeed exist, let +• 𝑝 ¯ℓ := 𝑣 ¯ℓ, +• 𝑃 ¯ℓ be a shortest path in 𝐻 from 𝑝 ¯ℓ to 𝑤 ¯ℓ, and +• 𝑄 ¯ℓ := (𝑝 ¯ℓ) = (𝑣 ¯ℓ), i.e., the 0-length path from 𝑝 ¯ℓ to 𝑣 ¯ℓ. +This choice satisfies +• (P1) and (P2), because Ψ𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) ≠ 0 implies that 𝑣 ¯ℓ ≠ 𝑤 ¯ℓ; +• (P4), because +𝑡𝑣 ¯ℓ,¯ℓ − 𝑡𝑤 ¯ℓ,¯ℓ − (4𝑠 − 2)𝜅Δℓ = 𝑡𝑣 ¯ℓ,¯ℓ − 𝑡𝑤 ¯ℓ,¯ℓ − (4𝑠 − 2)𝜅|𝑃 ¯ℓ| = 𝜓𝑣 ¯ℓ,𝑤 ¯ℓ + 2𝜅|𝑃 ¯ℓ|; and +• (P3) and (P5), because 𝑝 ¯ℓ = 𝑣 ¯ℓ (i.e., 𝑡𝑝 ¯ℓ,¯ℓ = 𝑡𝑣 ¯ℓ,¯ℓ and Δ¯ℓ = |𝑃 ¯ℓ|) and (P4) holds. +Corollary 2 proves that in fact ℓ = ℓ. Note that +𝑑(𝑣ℓ,𝑤ℓ) ≤ +� +|𝑃ℓ| + |𝑄ℓ| − 2 +if 𝑃ℓ and 𝑄ℓ share the first edge +|𝑃ℓ| + |𝑄ℓ| +else +≤ Δℓ + +20 +Christoph Lenzen and Shreyas Srinivas +and that |𝑃ℓ| ≥ 1 due to (P1). Therefore, (P4) yields that +Ξ𝑠 (ℓ ) ≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅𝑑(𝑣ℓ,𝑤ℓ) +≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅Δℓ +≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ) · 𝜅 +2 + 2𝜅|𝑃ℓ| +≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ) · 𝜅 +2 + 2𝜅, +contradicting Equation (5) and completing the proof. +□ +The remainder of Section 4.3 is dedicated to proving Corollary 2, which is the missing step in +the proof of Theorem 1. To this end, until the end of Section 4.3 we consider the setting of the +proof of Theorem 1 and assume for contradiction that ℓ > ℓ. We take note of some straightforward +implications. +Observation 2. For any fixed index ℓ, we have the following implications: +• (P3) ⇒ (P1) +• (P4) ⇒ (P2) +• (𝑣ℓ = 𝑝ℓ∧ (P4)) ⇒ (P3). +Moreover, +𝜓𝑠 +𝑣ℓ,𝑤ℓ (ℓ) − ( ¯ℓ − ℓ) · 𝜅 +2 > 0. +Proof. We prove each implication separately. +• From (P3), 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ > 4𝑠𝜅|𝑃ℓ| ≥ 0. This implies 𝑡𝑝ℓ,ℓ > 𝑡𝑤ℓ,ℓ and hence 𝑤ℓ ≠ 𝑝ℓ, i.e., (P1). +• Note that Δℓ ≥ 0, |𝑃ℓ| ≥ 0, and 4𝑠 − 2 > 0. Hence, (P4) and Equation (4) imply that +𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ ≥ 𝜓𝑠 +𝑣ℓ,𝑤ℓ ( ¯ℓ ) > 0. +It follows that 𝑤ℓ ≠ 𝑣ℓ, i.e., (P2). +• If 𝑣ℓ = 𝑝ℓ, then 𝑡𝑣ℓ,ℓ = 𝑡𝑝ℓ,ℓ, |𝑄ℓ| = 0, and Δℓ = |𝑃ℓ|. Thus, (P4) implies that +𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅|𝑃ℓ| ≥ 𝜓𝑠 +𝑣ℓ,𝑤ℓ (¯𝑙) + ( ¯ℓ − ℓ) · 𝜅 +2 + 2𝜅|𝑃ℓ| ≥ 𝜓𝑠 +𝑣ℓ,𝑤ℓ (¯𝑙) − ( ¯ℓ − ℓ) · 𝜅 +2 + 2𝜅|𝑃ℓ|, +which can be rearranged to yield (P3). +□ +A Step in the Construction. We now identify nodes that are suitable for taking the role of 𝑣ℓ, 𝑝ℓ, and +𝑤ℓ on layer ℓ − 1. These are either the nodes themselves or neighbors of them in 𝐻, where FC(𝑠), +SC(𝑠), and JC serve to relate respective pulse times. +Lemma 7. There is a node 𝑣 ∈ 𝑉 such that +𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅Δ𝑣 − 𝜅, +where +Δ𝑣 = + + +0 +and 𝑣 = 𝑣ℓ, +−1 +and {𝑣, 𝑣ℓ} is the last edge of 𝑄ℓ or the first edge of 𝑃ℓ, or +1 +and {𝑣ℓ, 𝑣} ∈ 𝐸. +Proof. By Lemma 5, 𝑣ℓ obeys the fast condition. Thus one of three things is true for 𝑣ℓ. +• FC-1(𝑠) holds. In this case, let 𝑣 = arg max{𝑥,𝑣ℓ }∈𝐸{𝑡𝑥,ℓ−1} and bound +𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ +max +{𝑥,𝑣ℓ }∈𝐸 +� +𝑡𝑥,ℓ−1 +� +− (4𝑠 − 2)𝜅 − 𝜅 = 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅 − 𝜅, +i.e., the claim of the lemma holds with Δ𝑣 = 1. + +Gradient TRIX +21 +• FC-2(𝑠) holds. In this case, let {𝑣, 𝑣ℓ} be the last edge of 𝑄ℓ if |𝑄ℓ| ≠ 0 or the first edge of 𝑃ℓ +otherwise; the latter is feasible, because then 𝑣ℓ = 𝑝ℓ, and |𝑃ℓ| ≠ 0 due to (P1). We get that +𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ +min +{𝑥,𝑣ℓ }∈𝐸 +� +𝑡𝑥,ℓ−1 +� ++ (4𝑠 − 2)𝜅 − 𝜅 ≤ 𝑡𝑣,ℓ−1 + (4𝑠 − 2)𝜅 − 𝜅. +Thus, the claim of the lemma holds with Δ𝑣 = −1. +• FC-3 holds. In this case, +𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣ℓ,ℓ−1 − 𝜅, +i.e., the claim of the lemma holds with Δ𝑣 = 0. +□ +Lemma 8. There is a node 𝑤 ∈ 𝑉 such that +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +≥ 𝑡𝑤,ℓ−1 + 4𝑠𝜅Δ𝑤, +where +Δ𝑤 = + + +0 +and 𝑤 = 𝑤ℓ, +−1 +and {𝑤,𝑤ℓ} is the last edge of 𝑃ℓ, or +1 +and {𝑤ℓ,𝑤} ∈ 𝐸. +Proof. By Lemma 4, 𝑤ℓ satisfies SC. We make a case distinction based on which one of SC-1, +SC-2, and SC-3 applies. +• SC-1(𝑠) holds. Let {𝑤,𝑤ℓ} be the last edge of 𝑃ℓ; by (P1), |𝑃ℓ| ≠ 0, i.e., this edge exists. Then +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,𝑙 +𝜗 +≥ +max +{𝑥,𝑤ℓ }∈𝐸{𝑡𝑥,ℓ−1} − 4𝑠𝜅 ≥ 𝑡𝑤,ℓ−1 − 4𝑠𝜅, +i.e., the claim of the lemma holds with Δ𝑤 = −1. +• SC-2(𝑠) holds. In this case, let 𝑤 = arg min{𝑥,𝑣ℓ }∈𝐸{𝑡𝑥,ℓ−1} and bound +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ, ℓ +𝜗 +≥ +min +{𝑥,𝑤ℓ }∈𝐸{𝑡𝑥,ℓ−1} + 4𝑠𝜅 = 𝑡𝑤,ℓ−1 + 4𝑠𝜅. +Thus, the lemma holds with Δ𝑤 = 1. +• SC-3 holds. Then +𝑡𝑤ℓ,ℓ−1 − C𝑤,ℓ ≥ 𝑡𝑤ℓ,ℓ−1, +i.e., the claim of the lemma holds with Δ𝑤 = 0. +□ +Lemma 9. There is a node 𝑝 ∈ 𝑉 such that +𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ ≤ +� +𝑡𝑝,ℓ−1 +and 𝑝 = 𝑝ℓ, or +𝑡𝑝,ℓ−1 − 𝜅 +and {𝑝ℓ, 𝑝} is the first edge of 𝑃ℓ. +Proof. If C𝑝ℓ,ℓ ≥ 0, the claim holds with 𝑝 = 𝑝ℓ. Hence, suppose that C𝑝ℓ,ℓ < 0. Let {𝑝ℓ, 𝑝} be +the first edge of 𝑃ℓ; such an edge exists, as by (P1) we have that 𝑝ℓ ≠ 𝑤ℓ and hence |𝑃ℓ| ≠ 0. By +Lemma 6, 𝑝ℓ satisfies JC. As C𝑝ℓ,ℓ < 0, JC-2 must apply. We conclude that +C𝑝ℓ,ℓ ≥ 𝑡𝑝ℓ,ℓ−1 − +min +{𝑥,𝑝ℓ }∈𝐸 +� +𝑡𝑥,ℓ−1 +� ++ 𝜅 ≥ 𝑡𝑝ℓ,ℓ−1 − 𝑡𝑝,ℓ−1 + 𝜅. +Rearranging terms, the desired inequality follows. +□ + +22 +Christoph Lenzen and Shreyas Srinivas +In the following, let (𝑣, 𝑝,𝑤) be the triple of nodes guaranteed by Lemmas 7 to 9. Denote by ◦ +concatenation of paths, by prefix(𝑅,𝑥) the prefix of path 𝑅 ending at node 𝑥 ∈ 𝑅, and by suffix(𝑅,𝑥) +the suffix of path 𝑅 starting at node 𝑥 ∈ 𝑅. Let +𝑝′ = +� +𝑣 +if 𝑣 lies on suffix(𝑃ℓ, 𝑝), +𝑝 +else, +𝑃 := +� +prefix(𝑃ℓ,𝑤) +if 𝑤 lies on 𝑃ℓ, +𝑃ℓ ◦ (𝑤ℓ,𝑤) +else, +𝑃 ′ := +� +suffix(𝑃, 𝑝′) +if 𝑝′ lies on 𝑃, +(𝑝′,𝑤) +else, +𝑄 := +� +prefix(𝑄ℓ, 𝑣) +if 𝑣 lies on 𝑄ℓ, +𝑄ℓ ◦ {𝑣ℓ, 𝑣} +else, +𝑄 ′ := +� +suffix(𝑄, 𝑝′) +if 𝑝′ lies on 𝑄, +(𝑝′, 𝑝ℓ) ◦ 𝑄 +else. +For notational convenience, in analogy to Δℓ we also define +Δ := +� +|𝑃 ′| + |𝑄 ′| − 1 +if 𝑃 ′ and 𝑄 ′ have the same first edge +|𝑃 ′| + |𝑄 ′| +else. +We will show that this construction satisfies properties (P1) to (P5) for layer ℓ − 1 with 𝑣ℓ−1 = 𝑣, +𝑝ℓ−1 = 𝑝′, 𝑤ℓ−1 = 𝑤, 𝑃ℓ−1 = 𝑃 ′, and 𝑄ℓ−1 = 𝑄 ′; this will constitute the desired contradiction. +However, we first point out that indeed 𝑃 ′ and 𝑄 ′ are paths in 𝐻 from 𝑝′ to 𝑤 and 𝑣, respectively. +To this end, we first cover the special case that 𝑝′ does not lie on 𝑃. +Observation 3. If 𝑝′ does not lie on 𝑃, then 𝑝′ = 𝑤ℓ and either 𝑤 = 𝑝ℓ or 𝑝′ = 𝑣. +Proof. By Lemma 9, 𝑝 lies on the first edge of 𝑃ℓ. Hence, if 𝑝′ = 𝑝, 𝑝′ lies on 𝑃 unless prefix(𝑃ℓ,𝑤) +does not contain this edge. By Lemma 8, this can only happen if the first edge of 𝑃ℓ is also the last +edge, i.e., 𝑃ℓ = (𝑝ℓ,𝑤ℓ) = (𝑤, 𝑝′). +It remains to consider the case that 𝑝′ ≠ 𝑝, i.e., 𝑝′ = 𝑣. Again, we use that all edges but the last of +𝑃ℓ are also contained in 𝑃 by Lemma 8. Thus, 𝑝′ = 𝑣 = 𝑤ℓ. +□ +Observation 4. 𝑃 ′ is a path in 𝐻 from 𝑝′ to 𝑤 and 𝑄 ′ is a path in 𝐻 from 𝑝′ to 𝑣. +Proof. To show that 𝑃 ′ is a path from 𝑝′ to 𝑤, note that by Lemma 8, 𝑃 is a path in 𝐻, which +by definition ends at 𝑤. Thus, if 𝑃 ′ = suffix(𝑃, 𝑝′), 𝑃 ′ is a path from 𝑝′ to 𝑤 in 𝐻. Otherwise, by +Observation 3, 𝑝′ = 𝑤ℓ, and {𝑝′,𝑤} = {𝑤ℓ,𝑤} ∈ 𝐸 by Lemma 8. +To show that 𝑄 ′ is a path from 𝑝′ to 𝑣, note that by Lemma 7, 𝑄 is a path in 𝐻, which by definition +ends at 𝑣. If 𝑝′ = 𝑝, by Lemma 9 𝑄 ′ is also a path in 𝐻, which by definition begins at 𝑝′ and has +the same endpoint as 𝑄, which is 𝑣. On the other hand, if 𝑝′ = 𝑣, suffix(𝑄, 𝑝′) = suffix(𝑄, 𝑣) = (𝑣), +which is the 0-length path from 𝑝′ = 𝑣 to itself. +□ +Proving the Properties. To prove Corollary 2, we establish that the tuple (𝑣, 𝑝′,𝑤, 𝑃 ′,𝑄′) satisfies +properties (P1) to (P5) for layer ℓ − 1, contradicting the minimality of ℓ. By Observation 4, indeed 𝑃 ′ +and 𝑄 ′ are paths from 𝑣 to 𝑤 and 𝑝′, respectively. In the following, we will repeatedly use this fact +and the property that {𝑥ℓ,𝑥} ∈ 𝐸 for 𝑥 ∈ {𝑣,𝑤, 𝑝} whenever 𝑥 ≠ 𝑥ℓ, without explicitly invoking +Observation 4 and Lemmas 7 to 9. +We first rule out the special case that 𝑣 = 𝑤ℓ and 𝑤 = 𝑣ℓ. + +Gradient TRIX +23 +Lemma 10. The case that 𝑣 = 𝑤ℓ and 𝑤 = 𝑣ℓ is not possible. +Proof. Assume towards a contradiction that 𝑣 = 𝑤ℓ and 𝑤 = 𝑣ℓ. We use (P4), Lemma 3, and +Lemma 8 to bound +−C𝑤,ℓ ≥ 𝑡𝑤,ℓ − 𝑡𝑤,ℓ−1 − Λ += 𝑡𝑣ℓ,ℓ − (𝑡𝑤,ℓ−1 − 4𝑠𝜅) − Λ − 4𝑠𝜅 +≥ 𝑡𝑣ℓ,ℓ − +� +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +� +− Λ − 4𝑠𝜅 += 𝑡𝑣ℓ,ℓ − +� +𝑡𝑤ℓ,ℓ−1 + 𝑑 − 𝑢 + Λ − 𝑑 − C𝑤ℓ,ℓ +𝜗 +� +− 𝜅 +2 − 4𝑠𝜅 +≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 − 𝜅 +2 +≥ 𝜓𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 +2 +> 0. +Thus, by JC, it holds that +𝑡𝑤,ℓ−1 ≤ 𝑡𝑤ℓ,ℓ−1 + C𝑤,ℓ − 𝜅. +Note that by (P1), |𝑃ℓ| ≠ 0 and hence |𝑃ℓ|, Δℓ ≥ 1. Thus, by (P4) and Equation (4) +𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 ≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ)𝜅 +2 ≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) > 0. +We distinguish two cases. +• C𝑤ℓ,ℓ ≤ 𝜗𝜅. Then by Lemma 3 +4𝑠𝜅 < 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ += 𝑡𝑤,ℓ − 𝑡𝑤ℓ,ℓ +≤ 𝑡𝑤,ℓ−1 − C𝑤,ℓ − +� +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +� ++ 𝑢 + +� +1 − 1 +𝜗 +� +(Λ − 𝑑) +≤ 𝑢 + +� +1 − 1 +𝜗 +� +(Λ − 𝑑) +< 𝜅, +which is a contradiction, because 𝑠 ≥ 1. +• C𝑤ℓ,ℓ > 𝜗𝜅. By JC, it follows that +𝑡𝑤ℓ,ℓ−1 ≥ 𝑡𝑤,ℓ−1 + C𝑤ℓ,ℓ +𝜗 ++ 𝜅, +yielding by Lemma 3 that +𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 = 𝑡𝑤ℓ,ℓ−1 − 𝑡𝑤,ℓ−1 +≥ 𝑡𝑤,ℓ−1 + C𝑤ℓ,ℓ +𝜗 ++ 𝜅 − (𝑡𝑤ℓ,ℓ−1 + C𝑤,ℓ − 𝜅) += 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ − +� +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +� ++ 2𝜅 +≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ + 2𝜅 − 𝜅 +2 +> 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ + 𝜅 +2 . + +24 +Christoph Lenzen and Shreyas Srinivas +Recall that by (P1), |𝑃ℓ| ≠ 0 and hence |𝑃ℓ|, Δℓ ≥ 1. Moreover, 𝑑(𝑣,𝑤) = 𝑑(𝑤ℓ,𝑤) ≤ 1, since +by Lemma 8 𝑤 is either 𝑤ℓ or a neighbor of 𝑤ℓ. Therefore, (P4) implies that +𝜓𝑠 +𝑣,𝑤(ℓ − 1) = 𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 − 4𝑠𝜅𝑑(𝑣,𝑤) +> 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅|𝑃ℓ| + 𝜅 +2 +≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − (ℓ − 1))𝜅 +2 . +Thus, 𝑣 and 𝑤 satisfy Equation (4) and Equation (5) with index ¯ℓ replaced by index ℓ − 1 < ¯ℓ, +contradicting the minimality of ¯ℓ. +□ +Next, we prove a helper lemma relating 𝑡𝑤ℓ,ℓ and 𝑡𝑤,ℓ−1 by a stronger bound than Lemma 8 for +the special case that 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ. This follows similar reasoning as the previous lemma. +However, it does not yield an immediate contradiction, as we need to rely on the weaker bound +provided by (P3). +Lemma 11. If 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ, then +𝑡𝑤ℓ,ℓ − 𝑡𝑤,ℓ−1 > 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +. +Proof. We use (P3) and Lemma 3 to bound +−C𝑤,ℓ ≥ 𝑡𝑤,ℓ − 𝑡𝑤,ℓ−1 − Λ += 𝑡𝑝ℓ,ℓ − (𝑡𝑤,ℓ−1 − 4𝑠𝜅) − Λ − 4𝑠𝜅 +≥ 𝑡𝑝ℓ,ℓ − +� +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +� +− Λ − 4𝑠𝜅 += 𝑡𝑝ℓ,ℓ − +� +𝑡𝑤ℓ,ℓ−1 + 𝑑 − 𝑢 + Λ − 𝑑 − C𝑤ℓ,ℓ +𝜗 +� +− 𝜅 +2 − 4𝑠𝜅 +≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 − 𝜅 +2 +≥ 𝜓𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 +2 +> 0. +Thus, by JC, it holds that +𝑡𝑤,ℓ−1 ≤ 𝑡𝑤ℓ,ℓ−1 − 𝜅. +We distinguish two cases. +• C𝑤ℓ,ℓ ≤ 𝜗𝜅. Then +𝑡𝑤ℓ,ℓ − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑤ℓ,ℓ − 𝑡𝑤ℓ,ℓ−1 + 𝜅 +≥ 𝑑 − 𝑢 + Λ − 𝑑 − C𝑤ℓ,ℓ +𝜗 ++ 𝜅 +≥ 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +. +• C𝑤ℓ,ℓ > 𝜗𝜅. By JC, it follows that +𝑡𝑤ℓ,ℓ−1 ≥ 𝑡𝑤,ℓ−1 + C𝑤ℓ,ℓ +𝜗 ++ 𝜅, + +Gradient TRIX +25 +yielding that +𝑡𝑤ℓ,ℓ − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑤ℓ,ℓ − 𝑡𝑤ℓ,ℓ−1 + C𝑤ℓ,ℓ +𝜗 ++ 𝜅 +> 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +. +□ +Using Lemma 11, we establish (P4) for the special case of 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ. Note that this +entails that 𝑤 is closer to 𝑝ℓ, yet 𝑃 is not shorter than 𝑃ℓ. This is accounted for by the case distinction +in the definition of Δℓ, which covers the difference. +Lemma 12. If 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ, then (P4) holds for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1. +Proof. Denote by Δ𝑣 ∈ {−1, 0, 1} the value such that +𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅Δ𝑣 − 𝜅 +according to Lemma 7. By Lemmas 3 and 11, +𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 +> 𝑡𝑣,ℓ−1 − 𝑡𝑤ℓ,ℓ + 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +≥ 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − 𝑡𝑤ℓ,ℓ + 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +≥ 𝑡𝑣ℓ,ℓ − Λ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − 𝑡𝑤ℓ,ℓ + 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +=𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 +2 +≥ (4𝑠 − 2)𝜅(Δℓ + Δ𝑣) +𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − (ℓ − 1))𝜅 +2 + 𝜅𝑠|𝑃ℓ|. +We claim that Δ ≤ Δℓ + Δ𝑣. Note that plugging this into the above inequality yields +𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ (4𝑠 − 2)𝜅Δ +𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − (ℓ − 1))𝜅 +2 + 𝜅𝑠|𝑃 ′|, +i.e., (P4) for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1, as desired. Therefore, proving the above claim will +complete the proof. +To show the claim, we first note that 𝑃 ′ = (𝑝′,𝑤) = (𝑤ℓ, 𝑝ℓ). Since 𝑤 = 𝑝ℓ, by Lemma 8 we also +have that 𝑃ℓ = (𝑝ℓ,𝑤ℓ) = (𝑤, 𝑝′). In particular, |𝑃ℓ| = |𝑃 ′|. We distinguish two cases. +• 𝑃ℓ and 𝑄ℓ share the first edge. It follows that |𝑄ℓ| ≥ 2, as otherwise 𝑣ℓ = 𝑤ℓ, contradicting +(P2). If 𝑣 = 𝑝′, then +|𝑄 ′| = |𝑄| = |(𝑣)| = 0 ≤ |𝑄ℓ| − 2 ≤ |𝑄ℓ| + Δ𝑣 − 1. +Otherwise, the first edge of𝑄 is the first edge of𝑄ℓ and thus 𝑃ℓ. This edge is {𝑝ℓ,𝑤ℓ} = {𝑝ℓ, 𝑝′}. +Hence, |𝑄 ′| = | suffix(𝑄, 𝑝′)| ≤ |𝑄| − 1 = |𝑄ℓ| + Δ𝑣 − 1. Either way, we get that +Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 − 1 = Δℓ + Δ𝑣. +• 𝑃ℓ and 𝑄ℓ do not share the first edge, but 𝑃 ′ and 𝑄 ′ do. Then +Δ = |𝑃 ′| + |𝑄 ′| − 1 ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 − 1 = Δℓ + Δ𝑣. +• 𝑃ℓ and 𝑄ℓ do not share the first edge and neither do 𝑃 ′ and 𝑄 ′. As the first (and only) edge of +𝑃 ′ is {𝑝′,𝑤} = {𝑝′, 𝑝ℓ}, this entails that 𝑄 ′ = suffix(𝑄, 𝑝′). We distinguish two subcases. +– | suffix(𝑄, 𝑝′)| ≤ |𝑄| − 1. Then +Δ = |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄| − 1 ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑣. + +26 +Christoph Lenzen and Shreyas Srinivas +– | suffix(𝑄, 𝑝′)| = |𝑄| and 𝑣ℓ ≠ 𝑤. Then 𝑝′ is the last node on 𝑄, i.e., 𝑣 = 𝑝′. As by Observa- +tion 4 𝑄 ′ is a path from 𝑝′ to 𝑣, it follows that |𝑄 ′| = 0 < |𝑄ℓ|. We conclude that +Δ = |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑣. +– | suffix(𝑄, 𝑝′)| = |𝑄| and 𝑣ℓ = 𝑤. As 𝑤 = 𝑝ℓ and 𝑝′ = 𝑣 = 𝑤ℓ as in the previous subcase, +this contradicts Lemma 10. +□ +Before proceeding to the case that 𝑣 ≠ 𝑤ℓ or 𝑤 ≠ 𝑣ℓ, we prove another helper statement ruling +out the specific case that 𝑣 ≠ 𝑝′ = 𝑤. +Lemma 13. It is not possible that 𝑣 ≠ 𝑝′ = 𝑤. +Proof. Assume towards a contradiction that 𝑣 ≠ 𝑝′ = 𝑤. Thus, 𝑝′ = 𝑝. Lemmas 8 and 9 yield +that +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +≥ 𝑡𝑤,ℓ−1 − 4𝑠𝜅 and +𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ ≤ 𝑡𝑝′,ℓ−1. +Using (P3) and Lemma 3, it follows that +0 = 𝑡𝑝′,ℓ−1 − 𝑡𝑤,ℓ−1 +≥ 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ − +� +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +� +− 4𝑠𝜅 +≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 − 𝜅 +2 +≥ 𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 +2 +> 0, +arriving at the desired contradiction. +□ +We now establish (P4) for the case that 𝑣 ≠ 𝑤ℓ or 𝑤 ≠ 𝑣ℓ. +Lemma 14. If 𝑝′ ≠ 𝑤ℓ or 𝑤 ≠ 𝑝ℓ, then (P4) holds for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1. +Proof. Denote by Δ𝑤, Δ𝑣 ∈ {−1, 0, 1} the values such that +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ ≥ 𝑡𝑤,ℓ−1 + 4𝑠𝜅Δ𝑤 +𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅Δ𝑣 − 𝜅 +according to Lemmas 7 and 8. Using (P4) and Lemma 3, we bound +𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 +≥ 𝑡𝑣ℓ,ℓ−1 − C𝑣,ℓ +𝜗 ++ (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − (𝑡𝑤ℓ,ℓ−1 − C𝑤,ℓ − 4𝑠𝜅Δ𝑤) +≥ 𝑡𝑣ℓ,ℓ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − 𝑡𝑤ℓ,ℓ + 4𝑠𝜅Δ𝑤 − 𝜅 +2 +≥ (4𝑠 − 2)𝜅(Δℓ + Δ𝑣 + Δ𝑤) + ( ¯ℓ − (ℓ − 1))𝜅 +2 + 𝜅𝑠 (|𝑃ℓ| + Δ𝑤) +≥ (4𝑠 − 2)𝜅(Δℓ + Δ𝑣 + Δ𝑤) + ( ¯ℓ − (ℓ − 1))𝜅 +2 + 𝜅𝑠|𝑃 ′|, +where the last step exploits that |𝑃 ′| = | suffix(𝑃, 𝑝′)| ≤ |𝑃| ≤ |𝑃ℓ| + Δ𝑤. We claim that Δ ≤ +Δℓ + Δ𝑣 + Δ𝑤. Proving this claim will complete the proof, as by the above inequality then +𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ (4𝑠 − 2)𝜅Δ + ( ¯ℓ − (ℓ − 1))𝜅 +2 + 𝜅𝑠|𝑃 ′|, + +Gradient TRIX +27 +i.e., (P4) for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1. +By Observation 3 and the prerequisites of the lemma, 𝑃 ′ = suffix(𝑃, 𝑝′) or 𝑝′ = 𝑣 = 𝑤ℓ. To cover +the possibility that 𝑃 ′ = suffix(𝑃, 𝑝′), we distinguish several cases: +• 𝑝′ = 𝑝ℓ. Then 𝑃 ′ = 𝑃 and 𝑄 ′ = 𝑄, as 𝑝′ is the first node of both 𝑃 and 𝑄. Hence, +|𝑃 ′| + |𝑄 ′| = |𝑃| + |𝑄| ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑤 + Δ𝑣. +We distinguish three subcases. +– 𝑃ℓ and 𝑄ℓ do not share their first edge. Then +Δ ≤ |𝑃 ′| + |𝑄 ′| = |𝑃ℓ| + |𝑄ℓ| + Δ𝑤 + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣. +– 𝑃ℓ, 𝑄ℓ, and 𝑄 ′ share the same first edge. By Lemma 13, 𝑤 ≠ 𝑝′. Therefore, 𝑃 ′ = 𝑃 ≠ (𝑝′), +which means that 𝑃ℓ and 𝑃 ′ have the same first edge, too. Thus, 𝑄 ′ and 𝑃 ′ have the same +first edge as well, and +Δ = |𝑃 ′| + |𝑄 ′| − 1 = |𝑃ℓ| + |𝑄ℓ| − 1 + Δ𝑤 + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣. +– 𝑃ℓ and 𝑄ℓ have the same first edge, but 𝑄 ′ does not. Since 𝑄ℓ ≠ (𝑝ℓ), we have that 𝑣ℓ ≠ 𝑝ℓ. +By (P5), this implies that 𝑣ℓ ∉ 𝑃ℓ. In particular, 𝑣ℓ cannot be part of the first edge of 𝑄ℓ and +|𝑄ℓ| ≥ 2. As 𝑝′ = 𝑝ℓ, 𝑄 and 𝑄 ′ both start with 𝑝′. Therefore, 𝑄 ′ is a prefix of 𝑄ℓ. However, +𝑄ℓ has the same first edge as 𝑃ℓ, while 𝑄 ′ does not. Thus, |𝑄 ′| = 0 ≤ |𝑄ℓ| + Δ𝑣 − 1. We +conclude that +Δ = |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄ℓ| − 1 + Δ𝑤 + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣. +• 𝑣 = 𝑝′ ≠ 𝑝ℓ. Then 𝑄 ′ = (𝑝′). Moreover, by the prerequisites of the lemma, 𝑃 ′ = suffix(𝑃, 𝑝′). +Since 𝑝′ ≠ 𝑝ℓ, we have that | suffix(𝑃, 𝑝′)| ≤ |𝑃| − 1. By construction, |𝑄 ′| ≤ |𝑄| + 1. Overall, +Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃| − 1 + |𝑄| + 1 = |𝑃ℓ| + Δ𝑤 + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣. +• 𝑣 ≠ 𝑝′ ≠ 𝑝ℓ. Thus, 𝑝′ = 𝑝 and by Lemma 9 {𝑝ℓ, 𝑝′} is the first edge of 𝑃ℓ. Hence, |𝑃 ′| = +| suffix(𝑃, 𝑝′)| ≤ |𝑃| − 1 ≤ |𝑃ℓ| + Δ𝑤 − 1. We distinguish two subcases. +– 𝑃ℓ and 𝑄ℓ do not share their first edge. Then +Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + Δ𝑤 + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣. +– 𝑃ℓ and 𝑄ℓ share their first edge. As 𝑣 ≠ 𝑝′, 𝑄 has the same first edge as 𝑄ℓ, i.e., {𝑝ℓ, 𝑝′}. +Hence, |𝑄 ′| = | suffix(𝑄, 𝑝′| = |𝑄| − 1 ≤ |𝑄ℓ| + Δ𝑣 − 1. We conclude that +Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + Δ𝑤 + |𝑄ℓ| + Δ𝑣 − 2 < Δℓ + Δ𝑤 + Δ𝑣. +It remains to consider the case that 𝑃 ′ ≠ suffix(𝑃, 𝑝′) and 𝑝′ = 𝑣 = 𝑤ℓ. Then |𝑄 ′| = (𝑣) and +|𝑃 ′| = |(𝑝′,𝑤)| = 1, implying that Δ = 1. By (P2), 𝑣ℓ ≠ 𝑤ℓ = 𝑣. If Δ𝑣 = 1, then +Δ = 1 ≤ Δℓ ≤ Δℓ + Δ𝑤 + Δ𝑣. +By Lemma 7, the remaining case is that Δ𝑣 = −1 and {𝑣, 𝑣ℓ} is the last edge of 𝑄ℓ or the first edge +of 𝑃ℓ. By Lemma 10, it is impossible that 𝑣 = 𝑤ℓ, so this edge must be the last one of 𝑄ℓ and distinct +from the first one of 𝑃ℓ. Moreover, by the prerequisites of the lemma, 𝑝ℓ ≠ 𝑤, so it must hold that +|𝑃ℓ| ≥ 2. Overall, either +• |𝑄ℓ| ≥ 2 and +Δ = 1 ≤ |𝑃ℓ| + |𝑄ℓ| − 3 ≤ Δℓ − 2 = Δℓ + Δ𝑤 + Δ𝑣, or +• |𝑄ℓ| = 1 and 𝑄ℓ and 𝑃ℓ do not share the first edge, yielding +Δ = 1 ≤ |𝑃ℓ| + |𝑄ℓ| − 2 = Δℓ − 2 = Δℓ + Δ𝑤 + Δ𝑣. +□ +Corollary 1. (P4) and (P2) hold for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1. + +28 +Christoph Lenzen and Shreyas Srinivas +Proof. Follows from Lemma 12, Lemma 14, and Observation 2. +□ +It remains to prove (P3). +Lemma 15. (P3) holds for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1. +Proof. If 𝑣 = 𝑝′, the statement readily follows from Corollary 1 and Observation 2. Therefore, +assume that 𝑣 ≠ 𝑝′ and hence 𝑝′ = 𝑝 in the following. Denote by Δ𝑤 ∈ {−1, 0, 1} the value such +that +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ ≥ 𝑡𝑤,ℓ−1 + 4𝑠𝜅Δ𝑤 +𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ ≤ 𝑡𝑝′,ℓ−1 +according to Lemmas 8 and 9. +Using (P3) and Lemma 3, it follows that +𝑡𝑝′,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ − +� +𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ +𝜗 +� ++ Δ𝑤4𝑠𝜅 +≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ + 4𝑠𝜅Δ𝑤 − 𝜅 +2 +≥ 4𝑠𝜅(|𝑃ℓ| + Δ𝑤) +𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 +2 . +If 𝑃 ′ = suffix(𝑃, 𝑝′), then |𝑃 ′| ≤ |𝑃| ≤ |𝑃ℓ| + Δ𝑤 and (P3) for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1 readily +follows from the above inequality. +Otherwise, by the assumption that 𝑣 ≠ 𝑝′ and Observation 3, it holds that 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ, +and |𝑃 ′| = |𝑃ℓ|. Using Lemmas 3 and 11 together with (P3), we arrive at +𝑡𝑝′,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑝′,ℓ−1 − 𝑡𝑤ℓ,ℓ + +� +𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +� +≥ 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ + +� +𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +� +≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 𝜅 +2 +≥ 4𝑠𝜅|𝑃ℓ| +𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 +2 +≥ 4𝑠𝜅|𝑃 ′| +𝜓𝑠 +𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 +2, +i.e., (P3) for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1. +□ +Finally, using these results it is not hard to show that (P5) is satisfied as well. +Lemma 16. (P5) holds for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1. +Proof. Suppose that 𝑣 lies on 𝑃 ′. By Corollary 1, 𝑣 ≠ 𝑤. Thus, if 𝑃 ′ = (𝑝′,𝑤), 𝑣 = 𝑝′, i.e., (P5) +holds for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1. +Otherwise, 𝑃 ′ = suffix(𝑃, 𝑝′), implying that 𝑣 lies on suffix(𝑃, 𝑝′). As 𝑣 ≠ 𝑤, this implies that 𝑣 +lies on 𝑃ℓ. Assuming for contradiction that 𝑣 ≠ 𝑝′ = 𝑝, by Lemma 9 we have that prefix(𝑃, 𝑝′) = +prefix(𝑃ℓ, 𝑝′), which equals either (𝑝ℓ) = (𝑝′) or (𝑝ℓ, 𝑝′). Thus, the above entails that 𝑣 actually lies +on suffix(𝑃ℓ, 𝑝′) = suffix(𝑃ℓ, 𝑝). As then 𝑝′ = 𝑣, this is a contradiction and we must indeed have +that 𝑝′ = 𝑣. +□ +Corollary 2. In the proof of Theorem 1, it must hold that ℓ = ℓ. + +Gradient TRIX +29 +Proof. Assuming for contradiction that ℓ > ℓ, Corollary 1, Lemmas 15 and 16, and Observation 2 +show that layer ℓ − 1 also satisfies the properties (P1) to (P5) for some 𝑣ℓ−1, 𝑝ℓ−1,𝑤ℓ−1, and paths +𝑃ℓ−1, 𝑄ℓ−1, contradicting the minimality of ℓ. +□ +Bounding Skews. With our machinery for bounding Ψ𝑠 in place, it remains to perform the induction +on 𝑠 ∈ N>0 to wrap things up. To anchor the induction at 𝑠 = 1, we exploit that Ψ1(ℓ) ≤ Ξ1(ℓ)+2𝜅𝐷. +Lemma 17. +Ψ1(ℓ) ≤ +� +Ξ1(0) +if ℓ < 4Ξ1(0)/𝜅 +4𝜅𝐷 +else. +Proof. Recall that 𝜅 = 2(𝑢 + (1 − 1/𝜗)(Λ − 𝑑)). Note that Ξ1(ℓ) ≤ Ψ1(ℓ) + 2𝜅𝐷 for all ℓ ∈ N. By +Theorem 1, we thus have for any ℓ ≤ ¯ℓ that +Ψ1( ¯ℓ ) ≤ max +� +0, Ξ1(ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 +� ++ ( ¯ℓ − ℓ )𝜅 +2 +≤ max +� +0, Ψ1(ℓ ) + 2𝜅𝐷 − ( ¯ℓ − ℓ + 1)𝜅 +� ++ ( ¯ℓ − ℓ )𝜅 +2 . +In particular, we have that +Ψ1(ℓ) ≤ +� +max +� +4𝜅𝐷, Ξ1(0) +� +if ℓ < 8𝐷 +max +� +4𝜅𝐷, Ψ1(ℓ − 8𝐷) − 2𝜅𝐷 +� +else. +By induction on 𝑘 ∈ N, we thus have that +Ψ1(ℓ) ≤ max{4𝜅𝐷, Ξ1(0) − 2𝑘𝜅𝐷} +for all ℓ ∈ [8𝑘𝐷, 8(𝑘 + 1)𝐷). The claim of the lemma follows by noting that ℓ ≥ 4Ξ1(0)/𝜅 results in +𝑘 ≥ Ξ1(0)/(2𝜅𝐷). +□ +Note that this lemma shows that Ψ1 self-stabilizes [5] within 𝑂(Ξ1(0)/𝜅) layers. +We remark that a more careful analysis reveals a bound on Ψ1(ℓ) that converges to 2𝜅𝐷. We +confine ourselves to stating this result for the small input skew that we guarantee. +Corollary 3. If L0 ≤ 4𝜅, then Ψ1(ℓ) ≤ 2𝜅𝐷 for all ℓ ∈ N. +Proof. Note that +Ξ1(0) = max +𝑣,𝑤∈𝑉{𝑡𝑣,0 − 𝑡𝑤,0 − 2𝜅𝑑(𝑣,𝑤)} ≤ max +𝑣,𝑤∈𝑉{(L0 − 2𝜅)𝑑(𝑣,𝑤)} ≤ (L0 − 2𝜅)𝐷 ≤ 2𝜅𝐷. +By replacing 8𝐷 with 4𝐷 in the induction from the proof of Lemma 17, we get that +Ψ1(ℓ) ≤ +� +max +� +2𝜅𝐷, Ξ1(0) +� +if ℓ < 4𝐷 +max +� +2𝜅𝐷, Ψ1(ℓ − 4𝐷) +� +else, +implying a uniform bound of Ψ1(ℓ) ≤ 2𝜅𝐷 for all ℓ ∈ N. +□ +For the sake of completeness, we also infer that supℓ ∈N{Ψ0(ℓ)}, also referred to as the global +skew in the literature, is in 𝑂(𝑢 + (1 − 1/𝜗)(Λ −𝑑)). Provided that Λ ∈ 𝑂(𝑑 +𝑢/(𝜗 − 1)), this bound +is asymptotically optimal [2]. +Corollary 4. If L0 ≤ 4𝜅, then Ψ0(ℓ) ≤ 6𝜅𝐷 ∈ 𝑂(𝑢 + (1 − 1/𝜗)(Λ − 𝑑)) for all ℓ ∈ N. +Proof. Follows from Corollary 3, the fact that Ψ0(ℓ) ≤ Ψ1(ℓ) + 4𝜅𝐷, and the choice of 𝜅. +□ +In order to bound the local skew, we now turn to attention to Ψ𝑠 (ℓ) for 𝑠 > 1. + +30 +Christoph Lenzen and Shreyas Srinivas +Lemma 18. For some 𝑠 ∈ N, 𝑠 > 0, suppose that Ψ𝑠−1(ℓ) ≤ Ψ𝑠−1 for all ℓ ∈ N. Then +Ψ𝑠 (ℓ) ≤ +� +Ξ𝑠 (0) + Ψ𝑠−1 +2 +if ℓ < Ψ𝑠−1/𝜅 +Ψ𝑠−1 +2 +else. +Proof. Recall that 𝜅 = 2(𝑢 + (1 − 1/𝜗)(Λ − 𝑑)). For ℓ < Ψ𝑠−1/𝜅, by Theorem 1 with ¯ℓ = ℓ and +ℓ = 0 we have that +Ψ𝑠 (ℓ) ≤ Ξ𝑠 (0) + 𝜅ℓ +2 ≤ Ξ𝑠 (0) + Ψ𝑠−1 +2 +. +Note that Ξ𝑠 (ℓ) ≤ Ψ𝑠−1(ℓ) ≤ Ψ𝑠−1 for all ℓ ∈ N. Thus, for ℓ ≥ Ψ𝑠−1/𝜅 by Theorem 1 with ¯ℓ = ℓ +and ℓ = ℓ − ⌊Ψ𝑠−1/𝜅⌋ we have that +Ψ𝑠 (ℓ) ≤ max +� +0, Ξ𝑠 +� +ℓ − +� Ψ𝑠−1 +𝜅 +�� +− +�� Ψ𝑠−1 +𝜅 +� ++ 1 +� +𝜅 +� ++ +� Ψ𝑠−1 +𝜅 +� 𝜅 +2 ≤ Ψ𝑠−1 +2 +. +□ +Using this lemma, we can bound the local skew by 𝑂(𝜅(1 + log 𝐷)) = 𝑂((𝑢 + (1 − 1/𝜗)(Λ − +𝑑))(1 + log 𝐷)). +Theorem 2. If there are no faults, then Lℓ ≤ 4𝜅(2 + log 𝐷) for all ℓ ∈ N. +Proof. By Lemma 27, L0 ≤ 4𝜅. By Corollary 3, Ψ1(ℓ) ≤ 2𝜅𝐷 for all ℓ ∈ N. By the assumption +that L0 ≤ 4𝜅, for all 𝑠 > 1 we have that +Ξ𝑠 (0) = max +𝑣,𝑤∈𝑉{𝑡𝑣,0 − 𝑡𝑤,0 − (4𝑠 − 2)𝜅𝑑(𝑣,𝑤)} ≤ max +𝑣,𝑤∈𝑉{(L0 − 6𝜅)𝑑(𝑣,𝑤)} = 0. +Hence, inductive use of Lemma 18 yields that Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷. In particular, Ψ⌊log 𝐷⌋ ≤ 8𝜅. The +claim now follows by Observation 1. +□ +Moreover, in addition we obtain the following self-stabilization property. +Theorem 3. If for 𝑠,𝑠′ ∈ N, 𝑠 ≤ 𝑠′, we have that Ψ𝑠 (ℓ) ≤ Ψ𝑠 for all ℓ ≥ ℓ ∈ N, then for ℓ ≥ ℓ +Lℓ ≤ +� +4𝑠𝜅 + Ψ𝑠 +if ℓ ≤ ℓ < ℓ + 2Ψ𝑠/𝜅 and +4𝑠′𝜅 + +Ψ𝑠 +2𝑠′−𝑠 +if ℓ ≥ ℓ + 2Ψ𝑠/𝜅. +Proof. Inductive use3 of Lemma 18 yields for 𝑠′ ≥ 𝑠 and ℓ ≥ ℓ + �𝑠′ +𝜎=𝑠+1 Ψ𝑠/(2𝜎−𝑠𝜅) that +Ψ𝑠′ ≤ +Ψ𝑠 +2𝑠′−𝑠 . +Since the sum forms a geometric series, this in particular applies to all ℓ ≥ ℓ + 2Ψ𝑠/𝜅. The claim +now follows by applying Observation 1. +□ +4.4 +Bounding Skews in the Presence of Faults +To analyze how skews evolve with faults, we relate the setting with faults to the bounds we have +for a fault-free system. The key property the algorithm guarantees is that, up to an additive 2𝜅, the +pulse time is within the interval spanned by the correct predecessors’ pulse times plus Λ. We first +show this for the case that for some node (𝑣, ℓ), (𝑣, ℓ − 1) is faulty. +3As is, the lemma applies only if ℓ = 0. However, the algorithm and hence all statements are invariant under shifting indices +by ℓ. + +Gradient TRIX +31 +Lemma 19. Suppose that the only faulty predecessor of (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, is (𝑣, ℓ − 1). Denote +𝑡min := +min +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and +𝑡max := max +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}. +Then +𝑡min + Λ − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡max + Λ + 2𝜅. +Proof. By the assumption of the lemma, for all {𝑣,𝑤} ∈ 𝐸, (𝑤, ℓ − 1) ∉ 𝐹. We have that +𝐻own − 𝐻max = min +𝑠 ∈N {𝐻own − 𝐻max + 4𝑠𝜅} +≤ min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} +≤ max{𝐻own − 𝐻max, 𝐻own − 𝐻min} += 𝐻own − 𝐻min. +Hence, abbreviating +Δ = min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2, +it holds that +𝐻own − 𝐻max − 𝜅 +2 ≤ Δ ≤ 𝐻own − 𝐻min − 𝜅 +2 . +Taking into account the adjustments in case Δ ∉ [0,𝜗𝜅] and using that 𝐻min ≤ 𝐻max we get that +𝐻own − 𝐻max − 3𝜅 +2 ≤ C𝑣,ℓ ≤ 𝐻own − 𝐻min + 3𝜅 +2 . +Therefore, the local time 𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ at which (𝑣, ℓ) generates its pulse satisfies +𝐻min + Λ − 𝑑 − 3𝜅 +2 ≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) ≤ 𝐻max + Λ − 𝑑 + 3𝜅 +2 . +If 𝐻min > 𝐻𝑣,ℓ (𝑡𝑣,ℓ), we have that +𝑡min − 𝑡𝑣,ℓ ≤ 𝐻min − 𝐻𝑣,ℓ (𝑡𝑣,ℓ). +Applying the lower bound of 𝑑 − 𝑢 on message delay and Equation (1), we get that +𝑡𝑣,ℓ ≥ 𝑡min + 𝑑 − 𝑢 + Λ − 𝑑 − 3𝜅 +2 > 𝑡min + Λ − 2𝜅. +If 𝐻min ≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ), the bounds on message delays and hardware clock drift together with Equa- +tion (1) yield that +𝑡𝑣,ℓ ≥ 𝑡min + 𝑑 − 𝑢 + Λ − 𝑑 − 3𝜅/2 +𝜗 +> 𝑡min + Λ − 3𝜅 +2 − 𝑢 − +� +1 − 1 +𝜗 +� +(Λ − 𝑑) += 𝑡min + Λ − 2𝜅. +Concerning the upper bound on 𝑡𝑣,ℓ, note that because 𝑡𝑣,ℓ is increasing in 𝐻𝑣,ℓ (𝑡𝑣,ℓ), to bound +𝑡𝑣,ℓ from above we may assume that +𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻max + Λ − 𝑑 + 3𝜅 +2 > 𝐻max, +where the last step uses Equation (2). In this case, +𝑡𝑣,ℓ − 𝑡max ≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) − 𝐻max + 𝑑 ≤ Λ + 3𝜅 +2 < Λ + 2𝜅. +□ + +32 +Christoph Lenzen and Shreyas Srinivas +Similar reasoning covers the case that for some (𝑣, ℓ) ∈ 𝑉ℓ and {𝑣,𝑤} ∈ 𝐸, (𝑤, ℓ − 1) is faulty. +Lemma 20. Suppose that for (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, (𝑣, ℓ − 1) is not faulty, and at most one predecessor is +faulty. Denoting +𝑡min := +min +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑤,ℓ−1} and +𝑡max := +max +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑤,ℓ−1}, +then +𝑡min + Λ − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡max + Λ. +Proof. By Lemma 3, C𝑣,ℓ ≥ 0 implies that +𝑡𝑣,ℓ − 𝑡min ≤ 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 ≤ Λ, +while C𝑣,ℓ ≤ 𝜗𝜅 yields that +𝑡𝑣,ℓ − 𝑡max ≥ 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 ≥ 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +− 𝜅 ≥ Λ − 2𝜅. +It remains to show the upper bound on 𝑡𝑣,ℓ if C𝑣,ℓ < 0 and the lower bound if C𝑣,ℓ > 𝜗𝜅. +Consider first the case that C𝑣,ℓ < 0. Accordingly, +C𝑣,ℓ = 𝐻own − 𝐻min − 𝜅 +2 + 2𝜅 > 𝐻own − 𝐻min. +It follows that +𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ ≤ 𝐻min + Λ − 𝑑. +Noting that the reception time of the first message from a predecessor is bounded from above by +the reception time of the message from a correct predecessor, we conclude that +𝑡𝑣,ℓ ≤ 𝑡min + Λ. +Now consider the case that C𝑣,ℓ > 𝜗𝜅. Consequently, +C𝑣,ℓ = 𝐻own − 𝐻max − 𝜅 +2 − 𝜅 > 𝐻own − 𝐻max − 𝜗𝑢. +It follows that the local time 𝐻 at which (𝑣, ℓ) generates its pulse satisfies that +𝐻 = 𝐻own + Λ − 𝑑 − C𝑣,ℓ ≥ 𝐻max + Λ − 𝑑 + 𝜗𝑢. +Noting that the reception time of the latest message from a predecessor is bounded from below by +the reception time of the latest message from a correct predecessor, by Equation (1) we conclude +that +𝑡𝑣,ℓ ≥ 𝑡max + 𝑑 + Λ − 𝑑 +𝜗 +> 𝑡𝑣,ℓ−1 + Λ − 𝜅. +□ +Corollary 5. Denote +𝑡min := +min +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑤,ℓ−1} and +𝑡max := +max +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑤,ℓ−1}. +Then +𝑡min + Λ − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡max + Λ + 2𝜅. + +Gradient TRIX +33 +Proof. Immediate from Lemmas 19 and 20 and the assumption that no node has more than one +faulty predecessor. +□ +Using this result, we can bound the impact of a fault in layer ℓ − 1 on successors via the skew +bounds of close-by nodes on layer ℓ − 1; we exploit that all bounds we show would in fact also +apply to the faulty node if it was correct. +Lemma 21. Suppose for a node (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, that one of its predecessors is faulty. Moreover, +assume that in an execution that differs only in that the faulty predecessor of (𝑣, ℓ) is correct, it holds +that max{𝑣,𝑤}∈𝐸{|𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1|} ≤ 𝐵. Then in the execution with the predecessor being faulty, the +pulse time of (𝑣, ℓ) differs by at most 2𝐵 + 4𝜅. +Proof. Denote by standard variables values in the execution without the predecessor being +faulty and by primed variables values in the one where it is. In particular, for node (𝑣, ℓ) ∈ 𝑉ℓ \ 𝐹 +𝑡min := +min +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1{𝑡𝑤,ℓ−1}, +𝑡max := +max +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 +{𝑡𝑤,ℓ−1}, +𝑡 ′ +min := +min +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑤,ℓ−1}, and +𝑡 ′ +max := +max +((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑤,ℓ−1}. +denote the earliest and latest pulsing times of (correct) predecessors without and with faults on +layer ℓ − 1, respectively. +Observe that +𝑡𝑣,ℓ−1 − 𝐵 ≤ 𝑡min ≤ 𝑡min′ ≤ 𝑡max′ ≤ 𝑡max ≤ 𝑡𝑣,ℓ−1 + 𝐵. +Hence, Corollary 5 (applied to both executions) shows that +𝑡𝑣,ℓ−1 − 𝐵 − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡𝑣,ℓ−1 + 𝐵 + 2𝜅 and +𝑡𝑣,ℓ−1 − 𝐵 − 2𝜅 ≤ 𝑡 ′ +𝑣,ℓ ≤ 𝑡𝑣,ℓ−1 + 𝐵 + 2𝜅. +□ +Finally, we observe that such a “time shift” propagates without further increase, so long as there +are no faults. However, a subtlety here is that this is only true for our bounds on timing: a change +in timing might leave more time for drift of the local clock to accumulate; since our worst-case +bounds include the maximum time error that can possibly be accumulated from drift (so long as +local skews do not become exceedingly large), this is already accounted for in the bound provided +by Lemma 3. Hence, we obtain the following generalized variant of Lemma 3. +Lemma 22. Suppose that for 𝑣 ∈ 𝑉 and ℓ ∈ N>0 the predecessors of (𝑣, ℓ) are correct. If we shift the +pulse times of these predecessors by at most 𝛿 ∈ R, where Equation (2) still holds for the shifted times, +then +𝑑 − 𝑢 + Λ − 𝑑 − C𝑣,ℓ +𝜗 +− 𝛿 ≤ 𝑡 ′ +𝑣,ℓ − 𝑡𝑣,ℓ−1 ≤ Λ − C𝑣,ℓ + 𝛿, +where 𝑡 ′ +𝑣,ℓ denotes the pulse time of (𝑣, ℓ) in the execution with the shifts applied. +Proof. Pulse times are increasing as functions of pulse times of predecessors. Therefore, in +order to maximize or minimize 𝑡 ′ +𝑣,ℓ, we need to maximize or minimize the predecessors’ pulse times, +respectively. Shifting all predecessors’ pulse times uniformly by 𝛿 also shifts 𝑡 ′ +𝑣,ℓ by 𝛿 relative to +𝑡𝑣,ℓ. The statement now follows analogously to the proof of Lemma 3, carrying the uniform shift +through all inequalities. +□ + +34 +Christoph Lenzen and Shreyas Srinivas +With these tools in place, we can conclude that skews do not grow arbitrarily in the face of faults. +Theorem 4. If there are at most 𝑓 faulty nodes in the system and none in layer 0, then Lℓ ∈ +𝑂(5𝑓 𝜅 log 𝐷). +Proof. We prove by induction on the number 𝑖 ≤ 𝑓 of layers ℓ > 0 with faults that the skew is +bounded by 𝐵𝑖 := 4𝜅(2 + log 𝐷)5𝑖 �𝑖 +𝑗=0 5−𝑗 ∈ 𝑂(5𝑓 𝜅 log 𝐷). By Corollary 6, L0 ≤ 𝜅/2 < 4𝜅. Thus, +if there are no faults in layers ℓ > 0, by Theorem 2 we have that Lℓ ≤ 𝐵0 := 4𝜅(2 + log 𝐷) for all +ℓ ∈ N. +Assume that we completed step 𝑖 ∈ N and that ℓ𝑖+1 is the next layer where faults need to be +added. Then we have that for all ℓ ≤ ℓ𝑖+1 that Lℓ′ ≤ 𝐵𝑖 = 4𝜅(2 + log 𝐷)5𝑓 �𝑖 +𝑗=0 5−𝑗 both before and +after adding the faults on layer 𝑖 + 1. By Lemma 21, it follows that pulsing times on layer ℓ𝑖+1 + 1 do +not change by more than 2𝐵𝑖 + 4𝜅 due to the addition of faults. By Lemma 22, this extends to all +bounds4 we compute on pulse times in layers ℓ > ℓ𝑖+1. Since 𝐷 ≥ 1 and thus log 𝐷 ≥ 0, we get that +the local skew in step 𝑖 + 1 is bounded by +5𝐵𝑖 + 4𝜅 = 4𝜅(2 + log 𝐷)5𝑖+1 +𝑖∑︁ +𝑗=0 +5−𝑗 + 4𝜅 ≤ 4𝜅(2 + log 𝐷)5𝑖+1 +𝑖+1 +∑︁ +𝑗=0 +5−𝑗 = 𝐵𝑖+1. +□ +Bounding Skews with Uniform Fault Distribution +The bound in Theorem 4, which is exponential in 𝑓 , seems to suggest that the system can only +support a very small number of faults or the local skew explodes. However, we have not yet +taken into account that the starting point of our entire approach is the assumption that faults are +sufficiently sparse, meaning that it is highly unlikely that many of them cluster together in a way +that causes an exponential pile-up of local skew. This enables the self-stabilization properties of +the algorithm to prevent such a build-up altogether. +In the following, assume that each node fails uniformly and independently with probability +𝑜(𝑛−1/2). This is the largest probability of error we can support while guaranteeing that no node +has more than one faulty predecessor with probability 1 − 𝑜(1). A key observation is that this +entails that within a fairly large distance of 𝑛1/12, no node has more than a constant number of +faulty nodes that can influence it. We now formalize and show this claim. +Definition 5 (Distance-𝛿 Ancestors). For node (𝑣, ℓ) ∈ 𝑉ℓ and 𝛿 ∈ N, its distance-𝛿 ancestors are +all nodes (𝑤, ℓ′) ∈ 𝑉𝐺 \ {(𝑣, ℓ)} such that there is a (directed) path of length at most 𝛿 from (𝑤, ℓ′) +to (𝑣, ℓ) in 𝐺. +Definition 6 (Distance-𝛿 𝑘-faulty). Node (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0 is distance-𝛿 𝑘-faulty if 𝑘 ∈ N is +minimal such that there are at most 𝑘 faulty nodes among the distance-((𝑘 + 1)𝛿) ancestors of +(𝑣, ℓ). +Observation 5. Suppose that 𝛿 ≤ 𝑛1/12. If nodes fail independently with probability 𝑝 ∈ 𝑜(1/√𝑛), +then with probability 1 − 𝑜(1) all nodes are distance-𝛿 𝑘-faulty for 𝑘 ≤ 2. +Proof. In order to be distance-𝛿 𝑘-faulty for 𝑘 > 2, a node must have at least 3 faults among +its distance-(3𝛿) ancestors. The number of these ancestors is bounded by (3𝛿)2 ∈ 𝑂(𝑛1/6). Since +𝑝 ∈ 𝑜(1/√𝑛), the probability for this to happen is bounded by 𝑂(𝑝3�𝑛1/6 +3 +�) = 𝑂(𝑝3√𝑛) ⊂ 𝑜(1/𝑛). +The claim follows by applying a union bound over all 𝑛 nodes. +□ +4Due to drifting hardware clocks, this does not apply to the pulse times themselves. However, we rely on Lemma 3 to prove +our bounds in the absence of faults, and this is covered by Lemma 22. + +Gradient TRIX +35 +We can exploit this to control how much skews grow as the result of faults much better. +Lemma 23. Suppose that Ψ𝑠 (ℓ) ≤ 𝐵𝑠,ℓ and Lℓ ≤ 𝐵 for all layers ℓ ≥ ℓ and 𝑠 ∈ N, where ℓ, ℓ ∈ N, if +there are no faults in these layers. If no node in a layer ℓ ≥ ℓ has more than 2 faulty nodes among its +distance-(ℓ − ℓ ) ancestors, then Ψ𝑠 (ℓ) ≤ 𝐵𝑠,ℓ + 12𝐵 + 24𝜅 for all ℓ ≥ ℓ. +Proof. We examine by how much adding faults on layers ℓ ≥ ¯ℓ might affect pulsing times. For +ℓ ≥ ¯ℓ and (𝑣, ℓ) ∈ 𝑉ℓ, denote by 𝑓𝑣,ℓ ∈ {0, 1, 2} the number of faulty distance-(ℓ − ¯ℓ) ancestors +of (𝑣, ℓ). For 𝑓𝑣,ℓ = 0, there is no change in 𝑡𝑣,ℓ. For 𝑓𝑣,ℓ > 0, consider two cases. If (𝑣, ℓ) has no +faulty predecessor, then by Lemma 22, 𝑡𝑣,ℓ is changed at most by the maximum shift that any +of its predecessors undergoes. On the other hand, if (𝑣, ℓ) does have a faulty predecessor, then +𝑓𝑣,ℓ > 𝑓𝑤,ℓ−1 for all correct predecessors of (𝑣, ℓ). Thus, by Lemma 21 we can bound shifts by 𝐵𝑓𝑣,ℓ , +where 𝐵0 := 0 and 𝐵𝑓 +1 := 2(𝐵 + 𝐵𝑓 ) + 4𝜅. +By assumption, 𝑓𝑣,ℓ ≤ 2 and hence the maximum shift is bounded by 𝐵2 = 6𝐵 + 12𝜅. We conclude +that Ψ𝑠 (ℓ) ≤ 𝐵𝑠,ℓ + 2𝐵2 = 𝐵𝑠,ℓ + 12𝐵 + 24𝜅, as claimed. +□ +Together with Lemma 23, Observation 5 shows that skews do not increase by more than a +constant factor within 𝑛1/12 layers. However, we need to handle a total of Θ(√𝑛) layers. To this end, +we slice up the task into chunks of 𝑛1/12 layers and leverage the self-stabilization properties of the +algorithm. For simplicity, in the following we assume that 𝑛1/12 is integer. As we prove asymptotic +bounds, this does not affect the results. +Definition 7 (Slices). Slice 𝑖 ∈ N>0 consists of layers ℓ ∈ [(𝑖 − 1)𝑛1/12,𝑖𝑛1/12 − 1]. +Note that there are no more than 𝑛5/12 slices, because the nodes are arranged in square grid. Due +to the duplication of nodes on layer 0 and the boundary nodes on layers ℓ > 0, the number of slices +is actually 𝑛5/12 − Θ(1). +As our next step towards a probabilistic skew bound, we prove that if the local skew remains +bounded, then for levels 𝑠 that are not too large, Ψ𝑠 remains almost as small as without faults. First, +we show a loose bound that naively accumulates shifts slice by slice. +Lemma 24. Suppose that +• L0 ≤ 4𝜅, +• each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2, and +• Lℓ ≤ 𝐵 for all ℓ ∈ N. +Then for each 𝑠 ∈ N and layer ℓ in slice 𝑖 ∈ N>0, we have that +Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷 + 𝑖(12𝐵 + 24𝜅). +for all ℓ ∈ N. +Proof. Assume first that there are no faults. In this case, analogously to the proof of Theorem 2, +we get that Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷 for all ℓ ∈ N. Now we “add” faults inductively slice by slice, by +Lemma 23 each time increasing the bound on Ψ𝑠 (ℓ) by 12𝐵 + 24𝜅 for all slices 𝑗 ≥ 𝑖. +□ +For larger values of 𝑠, 22−𝑠𝜅𝐷 ≪ 𝑛1/12, meaning that this naive bound is insufficient to show +that Ψ𝑠 (ℓ) does not increase much compared to the fault-free setting. However, we can take things +much further by leveraging Theorem 3. +Lemma 25. Suppose that +• L0 ≤ 4𝜅, +• each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2, and +• Lℓ ≤ 𝐵 ∈ 𝑜(𝑛1/12𝜅/log 𝐷) for all ℓ ∈ N. + +36 +Christoph Lenzen and Shreyas Srinivas +Then for5 𝑠 ∈ N>0, 𝑠 ≤ log 𝐷 − log(𝐵/𝜅) − 2 log log 𝐷, it holds that +Ψ𝑠 (ℓ) ≤ Ψ𝑠 ∈ (1 + 𝑜(1))22−𝑠𝜅𝐷. +Proof. Note that 𝐷 ∈ Θ(𝑛1/2) and hence log log 𝐷 ∈ 𝜔(1). Accordingly, the prerequisites of the +lemma ensure that 𝑛5/12(𝐵 + 𝜅) ∈ 𝑜(𝜅𝐷/log 𝐷) and 𝐵 + 𝜅 ∈ 𝑜(Ψ𝑠−1/log 𝐷). Hence, we may fix a +suitable 𝜀 ∈ 𝑜(1) such that +𝑛5/12(12𝐵 + 24𝜅) ≤ +𝜀 +log 𝐷 · 2𝜅𝐷 and +�� Ψ𝑠−1 +𝑛5/12𝜅 +� ++ 1 +� +(12𝐵 + 24𝜅) ≤ +𝜀 +4 log 𝐷 · Ψ𝑠−1. +We claim that if 𝑛 is sufficiently large such that 𝜀 ≤ 1, we have that +Ψ𝑠 (ℓ) ≤ Ψ𝑠 := 22−𝑠𝜅𝐷 · +� +1 + +𝜀𝑠 +log 𝐷 +� +, +which we show by induction on 𝑠 ∈ N>0. +For the base case of 𝑠 = 1, note that there are no more than 𝑛5/12 slices, yielding by Lemma 24 +that +Ψ1(ℓ) ≤ 2𝜅𝐷 + 𝑛5/12(12𝐵 + 24𝜅) ≤ +� +1 + +𝜀 +log 𝐷 +� +2𝜅𝐷, +i.e., indeed Ψ1(ℓ) ≤ Ψ1. +Now assume that the claim holds for 𝑠 − 1 ∈ N>0. Then, by Lemma 24 and the induction +hypothesis, for layers ℓ in slices 𝑖 ≤ ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉, we have that +Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷 + +� Ψ𝑠−1 +𝑛1/12𝜅 +� +(12𝐵 + 24𝜅) < Ψ𝑠−1 +2 ++ +�� Ψ𝑠−1 +𝑛1/12𝜅 +� ++ 1 +� +(12𝐵 + 24𝜅). +For a layer ℓ in a slice 𝑖 > ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉, assume first that we add only faults in slices 𝑗 < +𝑖 − ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉. Hence, we can apply Lemma 18, shifting layer indices such that “layer 0” is +the first layer of slice 𝑖 − ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉. In this setting, we thus have that Ψ𝑠 (ℓ) ≤ Ψ𝑠−1 +2 . We now +apply Lemma 23 inductively to slices 𝑗 ∈ [𝑖−⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉,𝑖], adding in total (⌈Ψ𝑠−1/(𝑛1/12𝜅)⌉+ +1)(12𝐵 + 24𝜅) to the bound, i.e., +Ψ𝑠 (ℓ) ≤ Ψ𝑠−1 +2 ++ +�� Ψ𝑠−1 +𝑛1/12𝜅 +� ++ 1 +� +(12𝐵 + 24𝜅) +≤ +�1 +2 + +� +𝜀 +4 log 𝐷 +�� +Ψ𝑠−1 += +�1 +2 + +� +𝜀 +4 log 𝐷 +�� +22−(𝑠−1)𝜅𝐷 · +� +1 + 𝜀(𝑠 − 1) +log 𝐷 +� += 22−𝑠𝜅𝐷 · +� +1 + 𝜀(𝑠 − 1/2) +log 𝐷 ++ +𝜀2 +2 log2 𝐷 +� +≤ 22−𝑠𝜅𝐷 · +� +1 + +𝜀𝑠 +log 𝐷 +� +, +where the last step assumes that 𝑛 is large enough so that 𝜀 ≤ 1. +□ +5If 𝐷 = 1, we assume the upper bound on 𝑠 to be negative and the claim is vacuously true. Note that we are making an +asymptotic statement in 𝑛 and that 𝐷 grows with 𝑛, so this case is actually of no concern here. + +Gradient TRIX +37 +Our goal is to bound Ψ⌊log 𝐷⌋ by 𝑂(𝜅 log 𝐷), since by Observation 1 this implies a bound of +𝑂(𝜅 log 𝐷) on the local skew. Thus, we will use the above lemma with 𝐵 ∈ 𝑂(𝜅 log 𝐷), which +gets us within 𝑂(log log 𝐷) levels of our “target” level ⌊log 𝐷⌋. To bridge this remaining gap, we +exploit that the time required for stabilizing the remaining 𝑂(log log 𝐷) levels after a fault-induced +increase of skews takes only log𝑂 (1) 𝐷 = log𝑂 (1) 𝑛 ⊂ 𝑜(𝑛1/12) layers, since the involved potentials +are bounded by 𝑜(𝜅𝑛1/12). +Lemma 26. Suppose that +• L0 ≤ 4𝜅 and +• each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2. +Then Lℓ ∈ 𝑂(𝜅 log 𝐷). +Proof. Assume towards a contradiction that the claim is false, and let ¯ℓ ∈ N>0 be minimal such +that Lℓ is too large. Hence, for layers ℓ < ¯ℓ, we may assume that Lℓ ≤ 𝐶𝜅 log 𝐷 for a sufficiently +large constant 𝐶. +Consider 𝑠 = ⌊log 𝐷 − log(𝐵/𝜅) − 2 log log 𝐷 − log𝐶⌋ − 5. By Lemma 25, for all ℓ ∈ N, ℓ < ¯ℓ it +holds that +Ψ𝑠 (ℓ) ∈ Ψ𝑠 := (1 + 𝑜(1))22−𝑠𝜅𝐷 ⊆ +�1 +4 + 𝑜(1) +� +log3 𝐷, +which for sufficiently large 𝑛 is smaller than ⌊log3 𝐷⌋/2. In fact, this bound also applies to layer ¯ℓ, +since the pulsing times of nodes on layer ¯ℓ depend only on the behavior of nodes on layer ¯ℓ − 1 and +the delays of messages sent to nodes on layer ¯ℓ. +Now assume that 𝑛 is sufficiently large. This ensures that log3 𝐷 ≤ 𝑛1/12, implying by the +prerequisites of the lemma that each node is distance-(log3 𝐷) 𝑘-faulty for 𝑘 ≤ 2. Consider adjacent +correct nodes (𝑣, ℓ), (𝑤, ℓ) ∈ 𝑉ℓ \ 𝐹 for any ℓ ∈ N, ℓ ≤ ¯ℓ, and {𝑣,𝑤} ∈ 𝐸. We first show that +distance-(log3 𝐷) 0-faulty nodes satisfy that +𝑡𝑣,ℓ − 𝑡𝑤,ℓ ∈ (4 + 𝑜(1))𝜅(2 + log 𝐷) ⊂ 𝑂(𝜅 log 𝐷). +(6) +Since faults that are not among the ancestry of a node cannot affect its pulse time, this follows by +applying Theorem 3 with ℓ = ℓ − ⌊(log3 𝐷)⌋ ≤ ℓ − 2Ψ𝑠 and 𝑠′ := ⌊log 𝐷⌋. +To extend this to distance-(log3 𝐷) 𝑘-faulty nodes for 𝑘 ∈ {1, 2}, we show by induction on +𝑘 ∈ {0, 1, 2} that such nodes have their pulse time shifted by no more than 𝑂(𝜅 log 𝐷) relative to +an execution in which they are distance-(log3 𝐷) 0-faulty. The base case of 𝑘 = 0 is trivial. +To perform the step from 𝑘 − 1 ∈ {0, 1} to 𝑘, assume towards a contradiction that there is a +node (𝑣, ℓ) with a larger shift, on some minimal layer. Now consider a distance-(log3 𝐷) 𝑘-faulty +node (𝑣, ℓ) ∈ 𝑉ℓ \ 𝐹, ℓ ≤ ¯ℓ, whose predecessors are all correct. There must be a distance-(log3 𝐷) +ancestor of (𝑣, ℓ) that is faulty, since otherwise (𝑣, ℓ) would be distance-(log3 𝐷) 0-faulty. Let 𝑑 +be the minimal distance in which there is a faulty ancestor of (𝑣, ℓ). Then all ancestors of (𝑣, ℓ) +in distance 𝑑 are distance-(log3 𝐷) 𝑘′-faulty for 𝑘′ < 𝑘, as otherwise (𝑣, ℓ) would be 𝑘′-faulty for +some 𝑘′ > 𝑘. +Consider an ancestor of (𝑣, ℓ) in distance 𝑑 − 1. If its predecessors are all correct, by the induction +hypothesis and Lemma 22 their pulse time is shifted by 𝑂(𝜅 log 𝐷) relative to an execution in which +they are distance distance-(log3 𝐷) 0-faulty. If there is a faulty predecessor, we infer this from the +induction hypothesis, Equation (6), and Lemma 21.6 If 𝑑 > 1, we now inductively apply Lemma 22 +until having extended this bound to all ancestors of (𝑣, ℓ) within distance 𝑑 − 1 and finally (𝑣, ℓ) +itself. This is a contradiction to (𝑣, ℓ) violating the claimed bound on the shift. +6Here the constants in the 𝑂-notation change, while Lemma 22 maintains the bound used in its prerequisites. Since we +perform only two inductive steps, we do not need to keep track of how much the constants increase. + +38 +Christoph Lenzen and Shreyas Srinivas +We conclude that indeed shifts are bounded by 𝑂(𝜅 log 𝐷). From this and Equation (6), it im- +mediately follows that L ¯ℓ ∈ 𝑂(𝜅 log 𝐷). As 𝐶 is sufficiently large, for sufficiently large 𝑛 this is a +contradiction. We conclude that Lℓ ∈ 𝑂(𝜅 log 𝐷) for all ℓ ∈ N, as claimed. +□ +Putting these results together, we arrive the desired bound on the local skew. +Theorem 5. With probability 1 − 𝑜(1), Lℓ ∈ 𝑂(𝜅 log 𝐷) for all ℓ ∈ N. +Proof. By Corollary 6, with probability 1 − 𝑜(1) it holds that L0 ≤ 𝜅/2. By Observation 5, with +probability 1 − 𝑜(1) each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2. By a union bound, both events +occur concurrently with probability 1 − 𝑜(1). Hence, the claim follows by applying Lemma 26. +□ + +Gradient TRIX +39 +REFERENCES +[1] B. Bailey. 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Optimal Gradient Clock Synchronization in Dynamic Networks. +Symposium on Principles of distributed computing (PODC), 2010. +[15] F. Kuhn and R. Oshman. Gradient Clock Synchronization Using Reference Broadcasts. In Conference on Principles of +Distributed Systems (OPODIS), pages 204–218, 2009. +[16] Clock Synchronisation and Adversarial Fault Tolerance, 2021, retrieved on 04 Jan 2023. https://www.mpi-inf.mpg.de/ +fileadmin/inf/d1/teaching/summer21/csaft/reading-material-ch09.pdf. +[17] C. Lenzen, T. Locher, and R. Wattenhofer. Clock Synchronization with Bounded Global and Local Skew. In Symposium +on Foundations of Computer Science (FOCS), pages 509–518, 2008. +[18] C. Lenzen, T. Locher, and R. Wattenhofer. Tight Bounds for Clock Synchronization. Journal of the ACM, 57(2), 2010. +[19] C. Lenzen and J. Rybicki. Self-Stabilising Byzantine Clock Synchronisation Is Almost as Easy as Consensus. Journal of +the ACM, 66(5), 2019. +[20] C. Lenzen and B. Wiederhake. TRIX: Low-Skew Pulse Propagation for Fault-Tolerant Hardware, 2020. https://arxiv. +org/abs/2010.01415. +[21] R. Shelar. Routing with Constraints for Post-Grid Clock Distribution in Microprocessors. IEEE Transactions on +Computer-Aided Design of Integrated Circuits and Systems, 29(2):245–249, 2010. +[22] Transistor Count, retreived Oct 2022. https://en.wikipedia.org/wiki/Transistor_count. +[23] T. Xanthopoulos, editor. Clocking in Modern VLSI Systems. Springer US, 2009. + +40 +Christoph Lenzen and Shreyas Srinivas +A +GENERATING SYNCHRONIZED INPUTS +In this appendix we describe a method for generating well synchronised pulses at layer 0, at a rate +of roughly one pulse per Λ time units. There are several ways of approaching this task, but even +when aiming for a fault-tolerant solution, this is an easy problem. The reason is that we merely +need to maintain a small local skew on a line topology, with no alternative propagation paths to +neighboring nodes. +Since our goal is to handle an independent probability of 𝑝 ∈ 𝑜(𝑛−1/2) of node failures, in fact +we can simply exploit that at most √𝑛 nodes are required on layer 0. We provide a trivial scheme +that is suitable for our specific setting of the base graph 𝐺 being a line (with replicated endpoints). +Algorithm 2 Pulse forwarding algorithm for nodes (𝑖, 0), 𝑖 ∈ {1, . . . , 𝐷}; node (0, 0) is the clock +source. The parameter Λ is as described in Algorithm 3. +𝐻 := ∞ +loop +do +if received pulse from (𝑖 − 1, 0) then +𝐻 := 𝐻𝑖,0(𝑡) +until 𝐻𝑖,0(𝑡) = 𝐻 + Λ − 𝑑 +broadcast pulse to (𝑖 + 1, 0) and successors on layer 1. +Lemma 27. For 𝑘 ∈ N, assume that the clock source at node (0, 0) generates its 𝑘-th pulse at time +(𝑘 − 1)Λ. If all nodes on layer 0 are correct, the scheme given in the above algorithm generates pulses +with local skew L0 ≤ 𝜅/2 and 𝑡𝑘 +𝑖,0 ∈ [(𝑘 + 𝑖 − 1)Λ − 𝑖𝜅/2, (𝑘 + 𝑖 − 1)Λ]. Moreover, it stabilizes after +transient faults within time 𝐷Λ. +Proof. Consider first the case that there are no transient faults. We prove the statement by +induction on 𝑖 ∈ N, where the base case is covered by the assumptions on node 0. +For the step from 𝑖 − 1 ∈ N to 𝑖, we perform an induction over the pulse number 𝑘 ∈ N>0. +The induction hypothesis is that pulses 1, . . . ,𝑘 − 1 have been generated in accordance with the +claim of the lemma and the first 𝑘 − 1 loop iterations at node 𝑖 have been completed by the time +the 𝑘-th pulse message from node 𝑖 − 1 arrives. Note that we can use 𝑘 = 0 as base case for this +induction, for which the claim is vacuously true. For the step from 𝑘 − 1 ∈ N to 𝑘, denote by +𝑡 ′ +𝑖−1,𝑘 ∈ [𝑡𝑖−1,𝑘 + 𝑑 − 𝑢,𝑡𝑖−1,𝑘 + 𝑑] the reception time of the pulse message from node (0,𝑖 − 1) at +node (0,𝑖). By the bounds on hardware clock rates, Equation (1), and the induction hypothesis of +the induction on 𝑖, node (0,𝑖) generates its 𝑘-th pulse at time +𝑡𝑖,𝑘 ∈ +� +𝑡𝑖−1,𝑘 + 𝑑 − 𝑢 + Λ − 𝑑 +𝜗 +,𝑡𝑖−1,𝑘 + Λ +� +⊆ +� +𝑡𝑖−1,𝑘 + Λ − 𝜅 +2,𝑡𝑖−1,𝑘 + Λ +� +⊆ +� +(𝑘 + 𝑖 − 1)Λ − 𝑖𝜅 +2 , (𝑘 + 𝑖 − 1)Λ +� +, +unless it receives another pulse message from (𝑖 − 1, 0) before doing so. This, however, is not the +case, since we assume that message delays and hardware clock rates do not vary over time, entailing +that these reception times lie Λ time apart.7 +7Note that a separation of Λ − 𝑑 time would suffice. The slack of 𝑑 means that small changes in timing between pulses are +unproblematic, which we exploit in Corollary 7. + +Gradient TRIX +41 +It remains to show the claimed bound on stabilization time. To this end, observe that the only +state information that nodes maintain is 𝐻. On reception of a pulse message, this state is overwritten. +This will remove spurious state from the system. +We would like to argue that the above induction can therefore be performed as-is, meaning that +the system has stabilized by the time each node has generated its first pulse. However, there is a +subtlety: it could happen that a spurious message that is still in transit at time 0 overwrites the +state of node (1, 0) after it received the first message from (0, 0). Node (1, 0) then behaves as if the +first message of (0, 0) arrived later, at the exact same time as the spurious message. Because also +such a spurious message is delivered within at most 𝑑 time, we can re-interpret this as a longer +delay of still at most 𝑑 of the first message sent by node (0, 0). Note that this modification reduces +the difference between the reception times of the first and second pulse from node (0, 0) at node +(1, 0) by up to 𝑢, but the separation remains at least Λ − 𝑢 ≥ Λ − 𝑑, i.e., the second message is not +received before (1, 0) generates its first pulse. We can apply the same scheme to nodes 2, . . . , 𝐷, +resulting in the desired bound on the stabilization time. +□ +Corollary 6. L0 ≤ 𝜅/2 with probability 1 − 𝑜(1). It is self-stabilizing with stabilization time Λ𝐷. +We remark that for a general base graph 𝐺, ensuring a small local skew is non-trivial. However, +so long as |𝑉 | is small enough such that faults on layer 0 occur with probability 𝑜(1), one is free to +fall back on a non-fault-tolerant GCS algorithm. This achieves L0 ∈ 𝑂(𝜅 log 𝐷), which does not +increase the asymptotic local skew bound of the pulse forwarding scheme. + +42 +Christoph Lenzen and Shreyas Srinivas +B +FULL PULSE FORWARDING ALGORITHM +Algorithm 3 Discrete GCS at node (𝑣, ℓ), ℓ > 0. The parameters Λ, and 𝜅 will be determined later, +based on the analysis. +loop +𝐻min, 𝐻own, 𝐻max := ∞ +for {𝑣,𝑤} ∈ 𝐸 do +𝑟𝑤 := 0 +do +if received pulse from 𝑣ℓ−1 and 𝐻own = ∞ then +𝐻own := 𝐻𝑣,ℓ (𝑡) +if for some {𝑣,𝑤} ∈ 𝐸 received pulse from (𝑤, ℓ − 1) and 𝑟𝑤 = 0 then +if 𝑟𝑤′ = 0 for all {𝑣,𝑤 ′} ∈ 𝐸 then +𝐻min := 𝐻𝑣,ℓ (𝑡) +𝑟𝑤 := 1 +if 𝑟𝑤′ = 1 for all {𝑣,𝑤 ′} ∈ 𝐸 then +𝐻max := 𝐻𝑣,ℓ (𝑡) +until 𝐻min < ∞ and 𝐻𝑣,ℓ (𝑡) ≥ min{𝐻max + 𝜅/2 + 𝜗𝜅, 2𝐻own − 𝐻min + 2𝜅)} +if 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅 then +wait until 𝐻𝑣,ℓ (𝑡) = 𝐻max + 3𝜅/2 + Λ − 𝑑 +else +C𝑣,ℓ := min𝑠 ∈N{max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅/2 +if C𝑣,ℓ < 0 then +C𝑣,ℓ := min {𝐻own − 𝐻min + 3𝜅/2, 0} +else if C𝑣,ℓ > 𝜗𝜅 then +C𝑣,ℓ := max {𝐻own − 𝐻max − 3𝜅/2,𝜗𝜅} +wait until 𝐻𝑣,ℓ (𝑡) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ +broadcast pulse +A basic requirement for the algorithm to work correctly is that (𝑣, ℓ) receives the 𝑘-th pulses of all +correct predecessors within its 𝑘-th iteration of the main loop of Algorithm 3. +Lemma 28. For all 𝑘 ∈ N and (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, node (𝑣, ℓ) receives the 𝑘-th pulses of all correct +predecessors within its 𝑘-th iteration of the main loop of Algorithm 3. +Proof. We show by induction on ℓ ∈ N>0 and 𝑘 ∈ N>0 that (𝑣, ℓ) broadcasts the 𝑘𝑡ℎ pulse after +receiving the 𝑘-th pulse from all correct (𝑤, ℓ − 1) satisfying that ((𝑤, ℓ − 1), (𝑣, ℓ)) ∈ 𝐸, but before +receiving the (𝑘 + 1)-th pulse from such a node. Moreover, for all 𝑘 ≥ 2, 𝑡𝑘 +𝑣,ℓ − 𝑡𝑘−1 +𝑣,ℓ += Λ. +For the induction on ℓ, we use ℓ = 0 as base case, requiring only that nodes generate pulses +at frequency 1/Λ. As delays and clock speeds do not change, this holds true. For the step from +ℓ − 1 ∈ N to ℓ, we perform the induction on 𝑘. Suppose that the claim holds for all 𝑘′ < 𝑘 ∈ N>0 +and consider the 𝑘-th iteration of the outer loop at (𝑣, ℓ). +• The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅. Then a message from each +node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, has been received in the current loop iteration. By the induction +hypotheses for layer ℓ − 1 and pulse 𝑘 − 1, respectively, for correct such nodes this is the 𝑘-th +pulse message. +We need to show that the 𝑘-th message from (𝑣, ℓ − 1) is received in time; the induction +hypothesis guarantees that it is not received too early. As the minimum degree of 𝐺 is 2, at + +Gradient TRIX +43 +least one node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, is correct. If (𝑣, ℓ − 1) is correct, too, it sent its pulse +message at the latest at time 𝑡𝑤,ℓ−1 + Lℓ−1. By the bounds on message delay and clock speed, +this message is received at a local time +𝐻 ≤ 𝐻max + 𝜗(Lℓ−1 + 𝑢) ≤ 𝐻max + Λ − 𝑑 < 𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ). +• The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own −𝐻min +2𝜅. As 𝐻min < ∞, also 𝐻own < ∞. +Using that 𝐻min ≤ 𝐻max, we get that +Δ := min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 +≤ max{𝐻own − 𝐻max, 𝐻own − 𝐻min} − 𝜅 +2 +≤ 𝐻own − 𝐻min − 𝜅 +2 +and hence C𝑣,ℓ ≤ 𝐻own − 𝐻min + 3𝜅/2 ≤ 3𝜅/2. It follows that +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) ≥ max{𝐻min, 𝐻own} + Λ − 𝑑 − 3𝜅 +2 . +We distinguish two subcases. +– (𝑣, ℓ − 1) is correct. Then by the bounds on message delay and clock speed, for each correct +(𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, its 𝑘-th pulse message is received at a local time +𝐻 ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) ≤ 𝐻own + Λ − 𝑑 − 3𝜅 +2 < 𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ), +where the last step uses Equation (2). +– (𝑣, ℓ − 1) is faulty, implying that all (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, are correct. Then by the bounds +on message delay and clock speed, for each correct (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, its 𝑘-th pulse +message is received at a local time +𝐻 ≤ 𝐻min + Λ − 𝑑 − 3𝜅 +2 < 𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ), +where we use that in order to guarantee that Λ − 𝑑 ≥ 𝜗(2Lℓ−1 + 𝑢) (i.e., Equation (2)), this +must also hold in an execution that differs by (𝑣, ℓ − 1) being correct; in such an execution, +we have that +max +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − +min +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} ≤ +max +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − 𝑡𝑣,ℓ−1 + 𝑡𝑣,ℓ−1 − +min +{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} ≤ 2Lℓ−1. +It remains to show that (𝑣, ℓ) generates its pulse before receiving a (𝑘 + 1)-th pulse message from a +correct predecessor. We distinguish two cases. +• (𝑣, ℓ − 1) is not faulty. Then the earliest local time 𝐻 at which (𝑣, ℓ) has received a 𝑘-th pulse +from a correct predecessor is bounded from below by +𝐻 ≥ 𝐻own − 𝜗(Lℓ−1 + 𝑢). +As delays and clock speeds do not change, the earliest message reception time for a (𝑘 + 1)- +th pulse from a correct predecessor is Λ time later. Hence, it is sufficient to show that +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) ≤ 𝐻 + Λ. We distinguish three subcases. +– The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅 and at local time 𝐻min a +message from a correct predecessor (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, was received by (𝑣, ℓ). Thus, +𝐻own + 𝜗(Lℓ−1 + 𝑢) + 2𝜅 ≥ 2𝐻own − 𝐻min + 2𝜅 ≥ 𝐻max + 𝜅 +2 + 𝜗𝜅. + +44 +Christoph Lenzen and Shreyas Srinivas +and, by Equation (3), +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) = 𝐻max + 3𝜅 +2 + Λ − 𝑑 +≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) + 2𝜅 + Λ − 𝑑 +≤ 𝐻own − 𝜗(Lℓ−1 + 𝑢) +≤ 𝐻 + Λ. +– The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅 and at local time 𝐻max a +message from a correct predecessor (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, was received by (𝑣, ℓ). Therefore, +𝐻own + 𝜗(Lℓ−1 + 𝑢) ≥ 𝐻max +and, by Equation (3), +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) = 𝐻max + 3𝜅 +2 + Λ − 𝑑 +≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) + 3𝜅 +2 + Λ − 𝑑 +≤ 𝐻own − 𝜗(Lℓ−1 + 𝑢) +≤ 𝐻 + Λ. +– The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own−𝐻min+2𝜅 and C𝑣,ℓ ≥ 0. By Equation (3), +then +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ ≤ 𝐻own + Λ − 𝑑 ≤ 𝐻 + Λ. +– The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own − 𝐻min + 2𝜅 and C𝑣,ℓ < 0. Then +C𝑣,ℓ = 𝐻own − 𝐻min + 3𝜅 +2 +and +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ = 𝐻min − 3𝜅 +2 + Λ − 𝑑. +Since 𝐻min is bounded from above by the earliest local reception time of a message from a +correct node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, we have that +𝐻min ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢). +By Equation (3), we conclude that +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) − 3𝜅 +2 + Λ − 𝑑 < 𝐻 + Λ. +• (𝑣, ℓ − 1) is faulty. Then 𝐻 = 𝐻min. Checking all cases in a similar fashion, we see that +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) ≤ 𝐻max + 3𝜅 +2 + Λ − 𝑑. +Using that Equation (3) must also apply in an execution where (𝑣, ℓ − 1) is not faulty and +hence max{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} ≤ 2Lℓ−1, it follows that +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) ≤ 𝐻max + 3𝜅 +2 + Λ − 𝑑 +≤ 𝐻min + 2𝜗(Lℓ−1 + 𝑢) + 3𝜅 +2 + Λ − 𝑑 +≤ 𝐻min + Λ +≤ 𝐻 + Λ. +□ +We are now ready to show that Algorithm 3 is equivalent to Algorithm 1 in the absence of faults. + +Gradient TRIX +45 +Lemma 29. Suppose that for (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, and the predecessors of (𝑣, ℓ) are correct. Then running +Algorithm 1 instead of Algorithm 3 results in the same pulse times of node (𝑣, ℓ). +Proof. Assume towards a contradiction that the claim is false. Denote by 𝑡𝑘 +𝑣,ℓ and (𝑡𝑘 +𝑣,ℓ)′ the +pulse times of Algorithm 1 and Algorithm 3 in executions with identical delays, clock speeds, and +behavior of faulty nodes. W.l.o.g., let 𝑡𝑘 +𝑣,ℓ be minimal with the property that 𝑡𝑘 +𝑣,ℓ ≠ (𝑡𝑘 +𝑣,ℓ)′. +Consider the 𝑘-th loop iteration of Algorithm 3 at node (𝑣, ℓ). We distinguish cases according to +why the inner loop terminated. +• The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅. Then in Algorithm 1, we have +that +𝐻own ≥ 𝐻max + 𝜅 +2 + 𝜗𝜅, +implying that +Δ := min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 +≥ min +𝑠 ∈N {𝐻own − 𝐻max + 4𝑠𝜅} − 𝜅 +2 +≥ 𝐻own − 𝐻min − 𝜅 +2 +≥ 𝜗𝜅. +Hence, Algorithm 1 computes +C𝑣,ℓ = 𝐻own − 𝐻max − 3𝜅 +2 +and generates its 𝑘-th pulse at local time +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) = 𝐻max + Λ − 𝑑 − C𝑣,ℓ = 𝐻max + 3𝜅 +2 + Λ − 𝑑 = 𝐻𝑣,ℓ ((𝑡𝑘 +𝑣,ℓ)′), +a contradiction. +• The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own −𝐻min +2𝜅. As 𝐻min < ∞, also 𝐻own < ∞ +for Algorithm 3. We distinguish two subcases. +– In Algorithm 1, we have +Δ := min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 < 0. +Then the same holds in Algorithm 3, as there 𝐻max is either identical to that of Algorithm 1 +of ∞. Hence, both algorithms compute 𝐶𝑣,ℓ = min{𝐻own −𝐻min +3𝜅/2, 0} and subsequently +𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ = 𝐻𝑣,ℓ ((𝑡𝑘 +𝑣,ℓ)′), a contradiction. +– In Algorithm 1, we have +Δ := min +𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 +2 ≥ 0 +Let 𝑠min ∈ N be such that +Δ := max{𝐻own − 𝐻max + 4𝑠min𝜅, 𝐻own − 𝐻min − 4𝑠min𝜅} − 𝜅 +2 . +If Δ = 𝐻own − 𝐻min − 4𝑠min𝜅 − 𝜅/2, the fact that 𝐻own and 𝐻min are identical in both +algorithms, while 𝐻max is either also identical or −∞ in Algorithm 3, again leads to the + +46 +Christoph Lenzen and Shreyas Srinivas +contradiction 𝐻𝑣,ℓ (𝑡𝑘 +𝑣,ℓ) = 𝐻𝑣,ℓ ((𝑡𝑘 +𝑣,ℓ)′). Hence, suppose that Δ = 𝐻own −𝐻max + 4𝑠min𝜅 −𝜅/2 +in Algorithm 1. Therefore, +0 ≤ Δ += 𝐻own − 𝐻max + 4𝑠min𝜅 − 𝜅/2 +≤ max{𝐻own − 𝐻max + 4(𝑠min − 1)𝜅, 𝐻own − 𝐻min − 4(𝑠min − 1)𝜅} − 𝜅 +2 += 𝐻own − 𝐻min − 4(𝑠min − 1)𝜅 − 𝜅 +2 . +Thus, +2𝐻own − 𝐻min + 2𝜅 ≥ 𝐻own + 4𝑠min𝜅 − 3𝜅 +2 ≥ 𝐻max − 𝜅 < 𝐻max − 𝜅 +2 − 𝜗𝜅. +This is a contradiction, as then the inner loop in Algorithm 3 would have terminated at an +earlier time. +□ +B.1 +Self-Stabilization +Making Algorithm 3 self-stabilizing follows standard techniques. Accordingly, we confine ourselves +to a brief high-level discussion of how this is achieved. +Theorem 6. The pulse propagation algorithm can be implemented in a self-stabilizing way. It +stabilizes within 𝑂(√𝑛) pulses. +Proof sketch. The key observation is that self-stabilization can proceed layer by layer, where +Corollary 6 shows that layer 0 stabilizes fast enough. Thus, we can assume that the correct nodes of +the previous layer generate pulses at a stable frequency of Λ satisfying the skew bounds obtained +in the analysis. +This allows us to make sure that the timing of its listening loop aligns with the pulse signals +from the previous layer: From all but one predecessor, the pulse signals must be received while the +inner loop is running. Moreover, the inner loop will terminate within Λ time. Instead of restarting +the inner loop dependent on the own generated pulse, we can instead start a loop iteration when +receiving the first pulse after a quiet period of, say, Λ/10 (where too frequent pulses of a faulty +predecessor are filtered out). As such a quiet period must occur by Equation (2), this will align the +loop correctly with the 𝑘-th pulses of correct predecessors for some 𝑘 ∈ N. +Once the inner loop terminates, we look for the next quiet period, and start a new instance of +the inner loop on reception of the next pulse from a predecessor, and so on. Whenever the inner +loop terminates correctly, i.e., not due to a timeout, we also compute the time to generate the next +pulse as in Algorithm 3. However, we do not wait until the pulse is generated before willing to start +a new instance of the inner loop. This way, we ensure that we do not miss the first pulse message +of a correct predecessor for pulse 𝑘 + 1 in case the inner loop for pulse 𝑘 was misaligned. +□ +C +OBTAINING THE FINAL SKEW BOUNDS +Recall that our model assumes that message delays and clock speeds do not vary. If the behavior of +faulty nodes is static, i.e., the timing of their output pulse messages is identical in each pulse as +well, a stable input frequency of 1/Λ results in repeating the exact same message pattern with the +same timing every 1/Λ time. We can exploit this to bound Lℓ,ℓ+1 in terms of Lℓ. +Theorem 7. If faulty nodes do not change the timing of their output pulses, then L ∈ 𝑂(𝜅 log 𝐷) +with probability 1 − 𝑜(1). + +Gradient TRIX +47 +Proof. By Corollary 5, for correct (𝑣, ℓ + 1) ∈ 𝑉ℓ+1, ℓ ∈ N, +min +((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ +(𝑤,ℓ)∉𝐹 +{𝑡𝑘 +𝑤,ℓ} + Λ − 2𝜅 ≤ 𝑡𝑘 +𝑣,ℓ+1 ≤ +max +((𝑤,ℓ),(𝑣,ℓ)) ∈𝐸ℓ +(𝑤,ℓ)∉𝐹 +{𝑡𝑘 +𝑤,ℓ} + Λ + 2𝜅. +Because the behavior of fault nodes does not change between pulses, a simple induction shows +that 𝑡𝑘+1 +𝑥,ℓ′ = 𝑡𝑘 +𝑥,ℓ′ + Λ for all correct nodes (𝑥, ℓ′) ∈ 𝑉ℓ′, ℓ′ ∈ N. In particular, +min +((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑘+1 +𝑤,ℓ } − 2𝜅 ≤ 𝑡𝑘 +𝑣,ℓ+1 ≤ +max +((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ +(𝑤,ℓ)∉𝐹 +{𝑡𝑘+1 +𝑤,ℓ } + 2𝜅. +By Theorem 5, Lℓ ∈ 𝑂(𝜅 log 𝐷). Note that this bound applies uniformly over all executions. +Thus, even if (𝑣, ℓ) is faulty, using that its neighbors are within distance 2 of each other, it holds +that +min +((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ +(𝑤,ℓ−1)∉𝐹 +{𝑡𝑘+1 +𝑤,ℓ } − +max +((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ +(𝑤,ℓ)∉𝐹 +{𝑡𝑘+1 +𝑤,ℓ } ∈ 𝑂(𝜅 log 𝐷), +by virtue of comparing to an execution in which (𝑣, ℓ) is correct. As (𝑣, ℓ + 1) was an arbitrary +correct node, the claim of the theorem follows. +□ +It remains to argue that some variation can be sustained. +Corollary 7. With probability 1−𝑜(1), L ∈ 𝑂(𝜅 log 𝐷) even when in each pulse (i) a constant number +of faulty nodes change their output behavior and timing, (ii) link delays vary by up to 𝑛−1/2𝑢 log 𝐷, +and (iii) hardware clock speeds vary by up to 𝑛−1/2(𝜗 − 1) log 𝐷. +Proof. The maximum length of a directed path in 𝐻 is bounded by 2√𝑛: at most 𝐷 ≤ √𝑛 hops +in layer 0, followed by at most √𝑛 links from layer to layer. Thus, accumulating all changes in +timing due to link delay and clock speed variation along a path results in a deviation of 𝑂((𝑢 + +(𝜗 − 1)(Λ − 𝑑)) log 𝐷 = 𝑂(𝜅 log 𝐷). This is trivial for layer 0 and applies to pulse propagation +through the layers as well, because our respective analysis relies on Corollary 5 and Lemma 22. In +order to take into account a constant number of faulty nodes with arbitrary behavior, we reason +analogously to the proof of Theorem 4, i.e., rely on Corollary 5 as well. +□ + diff --git a/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/load_file.txt b/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ef675ce95556cc38179512767df33655c263274 --- /dev/null +++ b/4dE4T4oBgHgl3EQfbgxF/content/tmp_files/load_file.txt @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf,len=1447 +page_content='Gradient TRIX CHRISTOPH LENZEN, CISPA Helmholtz Center for Information Security, Germany SHREYAS SRINIVAS, CISPA Helmholtz Center for Information Security, Germany and Saarbrucken Graduate School for Computer Science, Saarland University, Germany Gradient clock synchronization (GCS) algorithms minimize the worst-case clock offset between the nodes in a distributed network of diameter 𝐷 and size 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' They achieve optimal offsets of Θ(log 𝐷) locally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' between adjacent nodes [18], and Θ(𝐷) globally [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As demonstrated in [3], this is a highly promising approach for improved clocking schemes for large-scale synchronous Systems-on-Chip (SoC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, in large systems, faults hinder their practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' State of the art fault-tolerant GCS [4] has a drawback that is fatal in this setting: It relies on node and edge replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑓 = 1, this translates to at least 16-fold edge replication and high degree nodes, far from the optimum of 2𝑓 + 1 = 3 for tolerating up to 𝑓 faulty neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this work, we present a self-stabilizing GCS algorithm for a grid-like directed graph with optimal node in- and out-degrees of 3 that tolerates 1 faulty in-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If nodes fail with independent probability 𝑝 ∈ 𝑜(𝑛−1/2), it achieves asymptotically optimal local skew of Θ(log 𝐷) with probability 1 − 𝑜(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' this holds under general worst-case assumptions on link delay and clock speed variations, provided they change slowly relative to the speed of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The failure probability is the largest possible ensuring that with probabity 1 − 𝑜(1) for each node at most one in-neighbor fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As modern hardware is clocked at gigahertz speeds and the algorithm can simultaneously sustain a constant number of arbitrary changes due to faults in each clock cycle, this results in sufficient robustness to dramatically increase the size of reliable synchronously clocked SoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' CCS Concepts: • Hardware → Very large scale integration design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Very large scale integration design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Computing methodologies → Distributed algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Additional Key Words and Phrases: Clock Synchronisation, Fault Tolerance, VLSI, Self-Stabilisation 1 INTRODUCTION In their seminal work from 2004 [7], Fan and Lynch introduced the task of Gradient Clock Synchro- nization (GCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In a network of nodes that synchronize their clocks, it requires to minimize the worst-case clock offset between neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Two key insights motivate minimizing this local skew: In many applications the skew between adjacent nodes is the appropriate measure of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The global skew, the maximum clock offset between any pair of nodes in the network, grows linearly with the diameter 𝐷 of the network [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Defying the intuition of many, Fan and Lynch proved a lower bound of Ω(log 𝐷/log log 𝐷) on the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Follow-up work then established that this bound was very close to the mark: the best local skew that can be achieved is Θ(log 𝐷) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This exponential gap between global and local skew strongly suggests better scalability of systems employing this approach to synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Yet, more than a decade after these results have been published, we know of no efforts to apply these techniques in products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is not for want of demand!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To drive this point home, consider the case of clocking synchronous hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Conceptually speaking, state of the art hardware that operates synchronously distributes a clock signal from a single source using a tree network, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [9, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, for any tree spanning a square grid, there will be adjacent grid points whose distance in the tree is proportional to the side length of the grid [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, the worst-case local skew on a computer chip clocked by a clock tree must grow linearly with the side length of the chip [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Indeed, these theoretical results are reflected in the reality of hardware suppliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Modern systems gave up on maintaining globally synchronous operation, instead communicating Authors’ addresses: Christoph Lenzen, lenzen@cispa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='de, CISPA Helmholtz Center for Information Security, Saarbrücken, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Shreyas Srinivas, shreyas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='srinivas@cispa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='de, CISPA Helmholtz Center for Information Security, Saarbrücken, Germany and Saarbrucken Graduate School for Computer Science, Saarland University, Saarbrücken, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='05073v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='DC] 12 Jan 2023 2 Christoph Lenzen and Shreyas Srinivas asynchronously between multiple clock islands [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This comes at a steep cost, both in terms of communication latency [12] and ease of design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' So, which obstacle prevents application?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' At least in the above setting, neither large hidden constants nor an overly complex algorithm get in the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the contrary, recent work demon- strates that implementation effort is easily managable and pays off already for moderately-sized systems [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead, the main obstacle are faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To see that this is the key issue, recall that today’s hardware comprises an enourmous number of individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recent off-the-shelf hardware has transistor counts beyond the 10 billion mark [22], requiring either incredibly low fault rates or some degree of fault-tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In a system composed of multiple clock islands that interact asynchronously, these islands are canonical choices for fault-containment regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, one can get away with using a clocking scheme in each island that cannot sustain faults, interpreting a fault of the clocking subsystem as a fault of the respective island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In contrast, when the clocking subsystems of the islands interact with each other via a clock synchronization algorithm, we must ensure that a clock fault in a single island does not bring down the entire system!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fault-Tolerant Clocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' When clocking hardware, high connectivity networks are not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This limits the number of concurrent faults that can be sustained, as tolerating up to 𝑓 faults requires a node connectivity of 2𝑓 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In [4], this bound is matched asymptotically by augmenting an arbitrary network such that the GCS algorithm from [18] is simulated in a fault-tolerant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Here, augmentation means to replace each node in the original network by a clique of size 3𝑓 + 1 and each edge by a biclique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The clique then synchronizes internally using the classic Lynch-Welch algorithm [10], and the resulting local outputs are interpreted as (an approximation of) a joint cluster clock on which the (non-tolerant) GCS algorithm from [18] is simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, this approach is impractical due to the large overhead in terms of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Leaving asymptotics aside – the edge overhead compared to the original graph is Θ(𝑓 2) rather than 𝑂(𝑓 ) – even for the important special case of 𝑓 = 1 node degrees will be at least 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is a far cry from the simplicity of current distribution techniques, and factor 5 beyond the minimum node degree of 2𝑓 + 1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' What might look like a “moderate constant” to a theoretician will not only cause a headache to the engineer trying to route all of these edges with few layers and precise timing, it will also substantially increase communication delay uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This, in turn, directly translates into an increased skew, placing the break-even point with prior art beyond relevant limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In summary, it is essential to get as close as possible to the minimum required connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This train of thought led to the study of fault-tolerant clock distribution in low-degree networks [6, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Both of these works have in common that they assume that the clock signal is generated at a central location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This enables these approaches to achieve self-stabilization and tolerance to isolated faults with very simple pulse forwarding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The basic idea is to propagate the signal from layer to layer, having each node wait for two nodes signaling a clock pulse before locally generating and forwarding their own pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, it is assumed that in absence of faults delays are changing only slowly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, matching the input frequency to the expected delay between grid layers results in clock pulses that are well-synchronized between adjacent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The above works differ in the grid structure they use (Figure 1) and the skew bounds they provide: Denoting by 𝑑 − 𝑢 and 𝑑 the minimum and maximum end-to-end communication delay, in a grid of width 𝐷 [6] bounds the local skew by 𝑑 + 𝑂(𝑢2𝐷/𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since in practice 𝑑 ≫ 𝑢, this is a non-trivial bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, the fact that 𝑑 ≫ 𝑢 also means that this bound is too large for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Even worse, for each fault this bound increases by 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In [20], each fault adds at most 𝑢 to the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observe that the used grid also has the minimum required connectivity, as each node has only 3 incoming and outgoing edges each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 3 0 𝑢 2𝑢 𝑑 𝑑 𝑑 𝑑−𝑢 𝑑−𝑢 𝑑−𝑢 𝑑 𝑑 𝑑 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' TRIX [20] (top) and HEX [6] (bottom) grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' TRIX uses the naive pulse forwarding scheme of waiting for the second copy of each pulse before forwarding it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We see how the TRIX grid can accumulate a skew of Θ(𝑢𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the HEX grid, each node waits for two copies of a pulse from in-neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, 2 of the 4 in-neighbors are on the same layer, causing a skew of 𝑑 if a neighbor on the preceding layer crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Alas, these advantages come at the expense of poor scaling of worst-case skews with the number of layers: on layer ℓ, adjacent nodes may pulse up to 𝑢ℓ time apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that in order to tolerate failure of an arbitrary component, also the clock source has to be replicated and the replicas to be synchronized in a fault-tolerant and self-stabilizing manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, here one can employ techniques for fully connected networks [11, 19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' using them in a single location for 𝑓 = 1 does not constitute a scalability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In light of the above, in this work we ask the question “Can a small local skew be achieved in a fault-tolerant way at minimal connectivity?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our Contribution We provide a positive answer to the above question for the special case of 𝑓 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is achieved by using the same grid as in [20], but with a different rule for forwarding pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our novel algorithm is designed as a discrete and fault-tolerant counterpart to the GCS algorithm from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Making this work requires substantial conceptual innovation and technical novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the conceptual level, our algorithm simulates a discretized variant of the (non-fault-tolerant!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=') GCS algorithm from [18] on an arbitrary base graph of minimum degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In more detail, each copy of the graph, referred to as layer, represents a “time step” of the GCS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For each node, there is an edge from its copy on a given layer to the copies of itself and its neighbors on the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The forwarded pulses along these edges serve two very different functions: The pulse messages sent to copies of neigbhors correspond to the GCS algorithm’s messages for estimating clock offsets to neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The pulse messages sent between copies of the same node convey its local time from one of its copies to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this turns a permanently faulty node in the grid into a simulated node being faulty in a single time step only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is of vital importance, because it enables us to rely on the self-stabilization properties of the GCS algorithm from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' These are implicitly shown in [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' we prove them explicitly in the different setting of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, by itself this does not guarantee bounded skew between correct nodes, since we also need to contain the effect of such a “transient” fault on the state of the simulated algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4 Christoph Lenzen and Shreyas Srinivas Otherwise, a fault would increase skews arbitrarily, effectively corrupting downstream nodes: at any given node, the smallest or largest time at which a pulse from neighbors on the preceding layer is received could be determined by a faulty node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We can overcome this issue if there is at most one faulty in-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key observation to controlling the impact of a faulty node on the pulse time lies in that it can indeed affect only one of three times: the smallest or largest time at which a pulse from copies of neighbors on the previous layer is received, or the time at which the pulse from the copy of the node itself is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, the median of these three times lies within the interval spanned by the correct in-neighbors’ pulse times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By imposing the additional constraint to always tie the time at which a pulse is generated closely to this median, we can limit the local impact of a fault on skews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In summary, we seek to simultaneously simulate a time-discrete variant of the GCS algorithm from [18], while also guaranteeing that pulse forwarding times are, up to a sufficiently small deviation, identical to median reception times plus a fixed offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, no existing GCS algorithm that achieves a small local skew [14, 15, 17, 18] can be used for this purpose as-is, since their decision rules are in conflict with the above “stick to the median” requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As our main technical contribution, we resolve this conflict, simultaneously adapting the resulting algorithm to the discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To do so, we determine suitably weakened discrete variants of the slow and fast conditions introduced in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In essence, we allow that a simulated node whose pulse time is ahead all of its neighbors’ pulse times to delay its next pulse by the difference to the fastest neighbor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' an analogous rule applies to nodes pulsing later than all of their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the perspective of the GCS algorithm in [18], this constitutes a potentially arbitrarily large clock “jump,” which we leverage to implement the stick-to-the-median requirement despite the arbitrary changes in timing faulty nodes may apply to their pulse messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To prevent uncontrolled oscillatory behavior arising from adjacent nodes “jumping” in opposite directions, we introduce an additional condition, which we refer to as jump condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Essentially, it slightly reduces how large jumps are to avoid that uncertainty in message delays and local clock speeds cause nodes to “overswing,” potentially resulting in arbitrarily large skews, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Turning so many knobs at once meant that it was not clear that such a scheme would work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Indeed, bounding the skew of this novel algorithm turned out to be highly challenging, as jumps that delay pulses rather than speeding them up invalidate the fundamental assumption that clocks progress at rate at least 1 present in all prior work [14, 15, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As a result, the main technical hurdle and contribution turned out to be proving a bound on the local skew Lℓ between neighbors in the same layer ℓ for the fault-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there are no faults, then Lℓ ≤ 4𝜅(2 + log 𝐷) for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Here, choosing the input clock frequency to be 1/(2𝑑) results in 𝜅 ∈ Θ(𝑢 + (𝜗 − 1)𝑑), where it is assumed that local clocks run at rates between 1 and 𝜗 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' All of our results require that 𝑑 ≫ 𝑢 + (𝜗 − 1)𝑑, or equivalently, that the local skew remains small compared to 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that if this condition does not hold, we are outside the parameter range of interest: then skews become large compared to the length of a clock cycle under ideal conditions and clock frequency has to be reduced substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To address faults, we bound by how much faults can affect timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the aforementioned stick to the median rule, we can bound the local impact of a fault on timing in terms of the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, applying this argument repeatedly, skews would grow exponentially in the number of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' While tolerating a constant number of faults is certainly better than tolerating none, this is unsatisfactory, since the requirement of one faulty in-neighbor holds with probability 1 − 𝑜(1) for a fairly high independent probability of 𝑝 ∈ 𝑜(1/√𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Given that the topology we are most Gradient TRIX 5 interested in is roughly a square grid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', there are roughly √𝑛 layers, the naive approach outlined above does not result in a non-trivial bound on the skew unless 𝑝 is very close to 1/𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We provide an improved analysis exploiting that our base graph is almost a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, the 𝑑-hop neighborhood grows linearly with 𝑑 and hence the number of nodes in layers ℓ′ ∈ [ℓ − 𝑛1/12, ℓ] that affect the pulse time of a node in layer ℓ is in Θ(𝑛1/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if nodes fail with probability 𝑝 ∈ 𝑜(1/√𝑛), the probability that there are more than 2 faulty nodes within distance 𝑛1/12 that affect a given node is 𝑜(1/𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Intuitively, this buys enough time for the self-stabilization properties of the simulated algorithm to reduce its local skew again before it spirals out of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1 − 𝑜(1), Lℓ ∈ 𝑂(𝜅 log 𝐷) for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The final step is to extend this bound on the local skew within a layer to one that includes adjacent nodes in different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As we propagate pulses layer by layer, we cannot hope to match pulse times of the 𝑘-th pulse between different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead, we match the input period to the nominal time a pulse spends on each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This works neatly so long as there are no changes in message delay, clock speed, and behavior of faulty nodes between consecutive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If faulty nodes do not change the timing of their output pulses, then L ∈ 𝑂(𝜅 log 𝐷) with probability 1 − 𝑜(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To a large extent, this strong assumption is justified in our specific context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Clock speeds of modern systems are in the gigahertz range, and the amount of change in timing that occurs within a single clock cycle is much smaller than over the lifetime of a system [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Similarly, the by far most common timing faults are stuck-at faults, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the signal observed by downstream nodes remains constant logical 0 or 1, and broken connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the point of view of the receiving node, this is equivalent to an early or late pulse, respectively, without any change between pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Of course, timing will still change slowly, the above benign faults will occur at some point, before which the nodes worked correctly, and some faults may be more severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using once more that faulty nodes’ impact on timing is bounded by the local skew, the bound from Theorem 7 extends to a constant number of arbitrary faults in each pulse alongside small changes in delays and hardware clock speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1−𝑜(1), L ∈ 𝑂(𝜅 log 𝐷) even when in each pulse (i) a constant number of faulty nodes change their output behavior and timing, (ii) link delays vary by up to 𝑛−1/2𝑢 log 𝐷, and (iii) hardware clock speeds vary by up to 𝑛−1/2(𝜗 − 1) log 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, if all else fails, we can fall back on the ability of the pulse progation algorithm to recover from arbitrary transient faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In constrast to the simulated GCS algorithm, achieving self-stabilization of the pulse propagation scheme itself is straightforward due to the directionality of the propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The pulse propagation algorithm can be implemented in a self-stabilizing way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It stabilizes within 𝑂(√𝑛) pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In light of these results, we view this work as a major step towards simultaneously achieving high performance and strong robustness in the practical setting of clock distribution in hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In alignment with the theoretical question motivating this work, we achieve an asymptotically optimal local skew at the minimum possible node degree under the assumption of node failures with probability 𝑜(𝑛−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Organization of this Article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Section 2, we discuss the system model, introduce the graph on which we run our synchronization algorithm, and motivate our modeling choices, including its non- standard aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We then present a simplified version of the algorithm that better highlights the 6 Christoph Lenzen and Shreyas Srinivas conceptual approach in Section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the full algorithm and its equivalence without faulty predecessors is shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We follow with the formal derivation of the skew bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 2 MODELING The model we use is non-standard, as it is tailored to the specific setting outlined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, we will emphasize and discuss model choices where this seems prudent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that our goal is to provide a synchronized clock signal to a large System-on-Chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Physically, this means that we need to provide the clock signal to a rectangular area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for simplicity, we will assume it to be square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We want to supply a uniform grid of nodes in the square area with this signal, which then will serve as roots of relatively small local clock trees supplying the low-level components with the clock signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If these trees contribute a maximum clock skew of Δ and the skew between adjacent grid points is at most L, the triangle inequality guarantees a worst-case skew of L + 2Δ between adjacent components of the System-on-Chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The local clock trees can be designed using standard methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, in the following we will focus exclusively on the grid of their roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A key assumption we make is that communication delay between correct adjacent nodes changes only slowly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This enables us to generate synchronized pulses at all grid nodes by matching the input frequency with the (inverse) propagation time between consecutive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is justified for two reasons: The dominant sources of uncertainty in propagation delay are inaccuracies in component fabrication, aging, and temperature and frequency variations that are slow relative to the time it takes to propagate an input clock pulse across even a large System-on-Chip [23] Changing delays of all links between a pair of adjacent layers by up to 𝛿 increases skew bounds by at most 𝛿, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In order to generate sufficiently synchronized pulses at the nodes of layer 0, a straightforward solution is to use a redundant path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', a path of 3-cliques in which adjacent cliques are fully bipartitely connected, to propagate pulses from the clock reference along an edge of the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As we show in Corollary 6, this results in input pulses of small enough local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For each clique, one of the nodes will be the layer-0 node providing its output pulse to close-by nodes of layer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In a perfect grid, all layers would consist of a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, this results in the issue that the endpoints of the path, lacking one neighbor, would have only two adjacent nodes in the preceding and subsequent layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A naive solution is to insert a additional edges between the boundary nodes, turning the layer into a cycle and the entire graph into a cylinder (with some special treatment of layer 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, realizing such a solution on the square would result in far too long edges between boundary nodes or require to, essentially, replicate each layer, effectively doubling the number of nodes and edges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead, we choose to replicate the boundary nodes only, which then provides the “missing” input to the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this increases the degree of the nodes next to the boundary nodes by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We cope with this by a general analysis allowing for the layers to be copies of an arbitrary base graph of minimum degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Figure 2 and Figure 3, we show the base graph and the connectivity of nodes between adjacent layers of our synchronization network in our assumed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Network Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We are given a simple connected base graph 𝐻 = (𝑉, 𝐸) of minimum degree 2 and diameter 𝐷 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑣,𝑤 ∈ 𝑉 , denote by 𝑑(𝑣,𝑤) ≤ 𝐷 the distance from 𝑣 to 𝑤 in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To derive the graph 𝐺 = (𝑉𝐺, 𝐸𝐺) we use for synchronization, for each ℓ ∈ N we create a copy 𝑉ℓ of 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denoting by (𝑣, ℓ) the copy of 𝑣 ∈ 𝑉 in 𝑉ℓ, we define 𝐸ℓ := {((𝑣, ℓ), (𝑤, ℓ + 1)) | {𝑣,𝑤} ∈ 𝐸 ∨ 𝑣 = 𝑤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now obtain 𝐺 by setting 𝑉𝐺 := � ℓ ∈N 𝑉ℓ and 𝐸𝐺 := � ℓ ∈N 𝐸ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' That is, for each layer ℓ ∈ N we have a copy Gradient TRIX 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Base graph 𝐻 used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Rather than using a cycle, which would result in a TRIX grid, we replicate the end nodes of a line to ensure a minimum degree of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Alternatively, one could use a line and exploit that the probability that one of the 𝑂(√𝑛) boundary nodes fails is 𝑜(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Layer structure of 𝐺 resulting from our choice of 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Most nodes have in- and out-degree 3, some 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' of 𝑣 ∈ 𝑉 , which has outgoing edges to the copies of itself and all its neighbors on layer ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Sine 𝑉𝐺 is a DAG, we refer to out-neighbors as successors and in-neighbors as predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fault Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' An unknown subset 𝐹 ⊂ 𝑉𝐺 is Byzantine faulty, meaning that these nodes may violate the protocol arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Edge faults are mapped to node faults, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', if edge ((𝑣, ℓ), (𝑤, ℓ +1)) is faulty, we instead consider (𝑣, ℓ) faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We impose the constraint that no node has two faulty predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Formally, for all ℓ ∈ N and 𝑣 ∈ 𝑉 , |({(𝑣, ℓ)} ∪ � {𝑣,𝑤}∈𝐸{(𝑤, ℓ)}) ∩ 𝐹 | ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' When analyzing the system under random faults, we will assume that each node fails independently with probability 𝑝 ∈ 𝑜(1/√𝑛), which ensures that the above constraint is met with probability 1 − 𝑜(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In addition, we impose the restriction that at most a constant number of such faulty nodes change their timing behavior between consecutive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Each node has the ability to broadcast pulse messages on its outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If node 𝑣ℓ ∈ 𝑉ℓ broadcasts at time 𝑡𝑣,ℓ, its successors receive its message at a (potentially different) time from [𝑡𝑣,ℓ + 𝑑 − 𝑢,𝑡𝑣,ℓ + 𝑑].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The maximum end-to-end delay 𝑑 includes any delay caused by computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Typically, the delay uncertainty 𝑢 is much smaller than 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As discussed above, we assume delays to be static, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', each edge 𝑒 = ((𝑣, ℓ), (𝑤, ℓ +1)) has an unknown, but fixed associated delay 𝛿𝑒 ∈ [𝑑 − 𝑢,𝑑] applied to each pulse sent from (𝑣, ℓ) to (𝑤, ℓ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that faulty nodes can send pulses at arbitrary times, without being required to broadcast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' even if physical node implementations disallow point-to-point communication, edge faults could still result in this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Local Clocks and Computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Each node is able to approximately measure the progress of time by means of a local time reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We model this by node (𝑣, ℓ) having query access to a hardware clock 𝐻𝑣,ℓ : R≥0 → R≥0 satisfying ∀𝑡 < 𝑡 ′ ∈ R≥0, 𝑡 ′ − 𝑡 ≤ 𝐻𝑣,ℓ (𝑡 ′) − 𝐻𝑣,ℓ (𝑡) ≤ 𝜗(𝑡 ′ − 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for some 𝜗 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' No known phase relation is assumed between the hardware clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The algorithm will use them exclusively to measure how much time passes between local events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Analogous to delays, we assume that hardware clock speeds are static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is justified in the same way as for delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 1This is an abuse of notation, since in a (roughly) square grid of 𝑛 := |𝑉𝐺 | nodes, we have Θ(√𝑛) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑛, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the size of the grid, will only play a role when making probabilistic statements, we opted for this more convenient notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 8 Christoph Lenzen and Shreyas Srinivas Computations are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, in addition to receiving a message, the hardware clock reaching a time value previously determined by the algorithm can also trigger computations and possibly broadcasting a pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Output and Skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The goal of the algorithm is to synchronize the pulses generated by correct nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We assume that correct nodes on layer 0 generate well-synchronized pulses at times 𝑡𝑘 𝑣,0 for 𝑘 ∈ N>0 at a frequency we control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Appendix A, we discuss how to realize this assumption in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' All other correct nodes generate pulses 𝑡𝑘 𝑣,ℓ, 𝑘 ∈ N>0, based on the pulse messages received from their predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our measure of quality is the worst-case local skew the algorithm guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We define the local skew as the largest offset between the 𝑘-th pulses of adjacent nodes on the same layer or pulses 𝑘 and 𝑘 + 1 of adjacent nodes on layers ℓ and ℓ + 1, whichever is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Formally, for ℓ ∈ N we define Lℓ := sup 𝑘 ∈N max {𝑣,𝑤}∈𝐸 (𝑣,ℓ),(𝑤,ℓ)∉𝐹 {|𝑡𝑘 𝑣,ℓ − 𝑡𝑘 𝑤,ℓ|}, Lℓ,ℓ+1 := sup 𝑘 ∈N max ((𝑣,ℓ),(𝑤,ℓ+1)) ∈𝐸ℓ (𝑣,ℓ),(𝑤,ℓ+1)∉𝐹 {|𝑡𝑘 𝑣,ℓ − 𝑡𝑘+1 𝑤,ℓ+1|}, and L := supℓ ∈N max{Lℓ, Lℓ,ℓ+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This deviates from the standard definition of the local skew: The definition is adjusted to pulse synchronization, which can be viewed as an essentially equivalent time-discrete variant of clock synchronization [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Between consecutive layers, we synchronize consecutive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' After initialization, which is complete once the first pulse propagated through the (in practice finite) grid, this is equivalent to a layer-dependent index shift of pulse numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3 ALGORITHM In this section, we discuss the pulse forwarding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We provide a simplified version of the algorithm that behaves identical so long as the predecessors of the executing node are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The full algorithm needs to handle the possibility that faulty nodes send multiple messages or none at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This complicates bookkeeping and loop control, distracting from the principles underlying the algorithm’s operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, we defer the full algorithm to Appendix B, where we show the equivalence to the simplified variant when there are no faulty predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Simplified Pulse Forwarding Algorithm The algorithm proceeds in iterations corresponding to pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In each iteration, node (𝑣, ℓ) (1) timestamps the arrival times of the pulses of its predecessors using its hardware clock, (2) determines a correction value C𝑣,ℓ based on these timestamps, and (3) forwards the pulse Λ − 𝑑 − C𝑣,ℓ time after receiving the pulse from 𝑣ℓ−1, measured by its hardware clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If all reception times are close to each other, then C𝑣,ℓ will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recalling that messages are in transit for roughly 𝑑 time, this translates to Λ being the nominal time for a pulse to propagate from layer ℓ − 1 to layer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We need to choose Λ large enough such that the above sequence can be always realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' That is, we need to consider how far apart the reception times of messages from the previous layer can be, and ensure that Λ − 𝑑 exceeds this value plus the resulting C𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assuming that this precondition holds, Algorithm 1 implements the above approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In each loop iteration, it initializes three reception times to ∞: 𝐻own, which stores the arrival time of the pulse from (𝑣, ℓ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the perspective of the simulated GCS algorithm, this reflects the state of the node 𝑣 ∈ 𝑉 simulated by (𝑣, ℓ), ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻min, which stores the minimum arrival time of a pulse from a neighbor 𝑤ℓ−1, 𝑤 ≠ 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This corresponds to the first pulse received from a neighbor 𝑤 of 𝑣 in 𝐺 in this iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 9 Algorithm 1 Simplified pseudocode for discrete GCS at node (𝑣, ℓ), ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As shown in Lemma 29, this code is equivalent to Algorithm 3 in the absence of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The parameters Λ and 𝜅 will be determined later, based on the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' loop 𝐻own,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻max := ∞ do if received pulse from (𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' ℓ − 1) then 𝐻own := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) if received pulse from first (𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' ℓ − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑤} ∈ 𝐸 then 𝐻min := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) if received pulse from last (𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' ℓ − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑤} ∈ 𝐸 then 𝐻max := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) until 𝐻own,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻max < ∞ C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ := min𝑠 ∈N{max{𝐻own − 𝐻max + 4𝑠𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅/2 if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ < 0 then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ := min{𝐻own − 𝐻min − 𝜅/2 + 2𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0} else if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ > 𝜗𝜅 then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ := max{𝐻own − 𝐻max − 𝜅/2 − 𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝜗𝜅} wait until 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) = 𝐻own + Λ − 𝑑 − C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ broadcast pulse 𝐻max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' which stores the maximum arrival time of a pulse from a neighbor 𝑤ℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑤 ≠ 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This corresponds to the last pulse received from a neighbor 𝑤 of 𝑣 in 𝐺 in this iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The do-until loop fills these variables with the correct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' At the heart of the algorithm lies the computation of 𝐶𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there were no faults, one could always compute Δ := min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 and then choose the closest value from the range [0,𝜗𝜅], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', set C𝑣,ℓ := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 Δ if Δ ∈ [0,𝜗𝜅], 0 if Δ < 0, and 𝜗𝜅 if Δ > 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To get intuition on this choice, observe that min𝑥 ∈R{max{𝐻own − 𝐻min − 𝑥, 𝐻own − 𝐻max + 𝑥}} is attained when 𝐻own − 𝐻max + 𝑥 = 𝐻own − 𝐻min − 𝑥, which is equivalent to 𝑥 = (𝐻max − 𝐻min)/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝐻own − Δ = (𝐻max − 𝐻min)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If timing was perfectly accurate, the reception times of the pulse messages could serve as exact proxies for the actual pulse forwarding times of the nodes on layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In iteration 𝑘, this would mean to generate the pulse at (𝑣, ℓ) faster if (𝑣, ℓ − 1) generated its pulse later than the average of min{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1} and max{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, any (𝑣, ℓ) for which 𝑡𝑘 𝑣,ℓ−1 − min{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1} > max{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1} − 𝑡𝑘 𝑣,ℓ−1 would choose 𝐶𝑣,ℓ > 0, attempting to reduce max{𝑣,𝑤}∈𝐸{|𝑡𝑘 𝑣,ℓ − 𝑡𝑘 𝑤,ℓ|} compared to max{𝑣,𝑤}∈𝐸{|𝑡𝑘 𝑣,ℓ−1 − 𝑡𝑘 𝑤,ℓ−1|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This can be viewed as trying to reduce the local skew by a greedy strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, this naive strategy fails to account for inaccuracies due to message delay uncer- tainty and drifting hardware clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Nonetheless, we follow this strategy up to deviations of 𝑂(𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The additional terms serve the following purposes: 10 Christoph Lenzen and Shreyas Srinivas Considering only discrete choices for 𝑥 ∈ 4𝜅N rather than arbitrary 𝑥 ∈ R is the key ingredient that makes the algorithmic approach succeed, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Essentially, this is necessary because there is no way to determine 𝑡𝑣ℓ−1,𝑘 − 𝑡𝑤ℓ−1,𝑘 precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Discretizing observed skews in units of 𝜅 ∈ Θ(𝑢 + (𝜗 − 1)(Λ − 𝑑)) enables a delicate strategy that alternates between overestimating skews to locally generate the next pulse earlier for the sake of “catching up” with others and underestimating skews to “wait” for others catch up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Substracting 𝜅/2 accounts for errors in measuring skews, which are caused by uncertainty in message delay and hardware clock speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To limit the damage that a faulty predecessor of (𝑣, ℓ) can do, we ensure that (𝑣, ℓ) generates its pulse without too large of a deviation from the median of 𝑡𝑣,ℓ−1, min{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1}, and max{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1} (plus the nominal offset of Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is achieved by permitting corrections 𝐶𝑣,ℓ < 0 if (𝑣, ℓ − 1) clearly generated its pulse earlier than min{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1} and 𝐶𝑣,ℓ > 𝜗𝜅 if it clearly generated its pulse later than max{𝑣,𝑤}∈𝐸{𝑡𝑘 𝑤,ℓ−1}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To further motivate the last point, recall that there can be at most one fault among the predecessors of (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A single faulty predecessor can only affect only one of the three values 𝐻own, 𝐻min, and 𝐻max: control 𝐻own arbitrarily, 𝐻min to be smaller than the minimum reception time from a correct node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, or 𝐻max to exceed the maximum reception time from correct nodes (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, ensuring that pulses are generated with only a small offset relative to median {𝐻own, 𝐻min, 𝐻max} + Λ − 𝑑 indeed limits the damage that a fault can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Achieving all of the desired properties is non-trivial, leading to the fairly involved choice of C𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It can be viewed as simultaneously implementing relaxed fast and slow conditions (as introduced in [15]), an additional jump condition required to make the GCS algorithm work under these relaxed fast and slow conditions, and the requirement to stick close to the median of predecessors’ pulse times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='2, we specify the (relaxed) slow and fast condition, as well as the jump condition, and show that the algorithm implements them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemmas 19 and 20 show that the algorithm also enforces deviates little from the time interval spanned by correct predecessors (offset by Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is some freedom in the choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For simplicity, we fix a good choice of 𝜅 and note that 𝑑 must satisfy a lower bound 𝐵 ∈ 𝑂(supℓ ∈N{Lℓ} +𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observe that this constraint simply means that the skew bounds are useful, as a skew that is of similar size as the maximum end-to-end delay requires to slow the system down substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, Λ must be at least 𝑑 +𝑂(supℓ ∈N{Lℓ}), which due to the previous constraint holds e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for the choice Λ = 2𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Formally, for a sufficiently large constant 𝐶,2 𝜅 := 2 � 𝑢 + � 1 − 1 𝜗 � (Λ − 𝑑) � , (1) Λ ≥ 𝐶𝜗(sup ℓ ∈N {Lℓ} + 𝑢) + 𝑑, and (2) 𝑑 ≥ 𝐶(𝜗(sup ℓ ∈N {Lℓ} + 𝑢) + 𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (3) Complete Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The complete algorithm cannot wait for messages from all predecessors to determine when to send its pulse, as a faulty node not sending its pulse then would deadlock all its descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As discussed above, the hardware clock time of the next pulse time does not deviate much from median {𝐻own, 𝐻min, 𝐻max} + Λ −𝑑, but does depend on max{𝐻min, 𝐻own, 𝐻max} in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we will prove that Lℓ−1 is small enough such that all pulse messages from correct nodes will be received in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, it is sufficient to wait until median {𝐻own, 𝐻min, 𝐻max}+𝜗Lℓ−1 (or later) according to 𝐻𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Provided that Λ − 𝑑 is large enough, this implies that any message for 2We do not attempt to optimize constants in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 11 computing C𝑣,ℓ missing is due to a fault;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' in fact, at the point in time when this becomes clear, C𝑣,ℓ is already determined, regardless of how late the message would arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The complete algorithm differs from Algorithm 1 by covering the case that a signal does not arrive in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Intuitively, one can treat the respective message arrival time (𝐻own or 𝐻max, 𝐻min is not possible) as ∞, while allowing such an ∞ to cancel out in substraction: If 𝐻own = ∞, then 𝐶𝑣,ℓ ∈ 𝐻own − 𝐻max − 𝑂(𝜅), and (𝑣, ℓ) will generate its pulse at local time 𝐻own + Λ − 𝑑 − 𝐶𝑣,ℓ ∈ 𝐻max + Λ − 𝑑 + 𝑂(𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻max = ∞ and 𝐻own ≥ 𝐻min, then 𝐶𝑣,ℓ ∈ 𝐻own − 𝐻min ± Θ(𝜅) and (𝑣, ℓ) will generate its pulse at local time 𝐻own + Λ − 𝑑 − 𝐶𝑣,ℓ ∈ 𝐻min + Λ − 𝑑 ± 𝑂(𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻max = ∞ and 𝐻own < 𝐻min, then 𝐶𝑣,ℓ ∈ [0, 2𝜅] and (𝑣, ℓ) will generate its pulse at local time 𝐻own + Λ − 𝑑 − 𝐶𝑣,ℓ ∈ 𝐻own + Λ − 𝑑 − 𝑂(𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that in all cases, the pulse is generated with an offset of Λ−𝑑 −Θ(𝜅) from the median reception time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The complete algorithm follows the above intuition, leveraging the fact that there is no need to wait indefinitely to determine that the third signal is late, and is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Last, but not least, it is of interest to make the pulse forwarding algorithm self-stabilizing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the design choice of propagating the clock signal from a single source along a DAG, this will immediately translate to the overall scheme being self-stabilizing, so long as the clock generation is self-stabilizing, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is straightforward, because one can assume that the signals from the previous layer are already well-synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, all that nodes need to do is to detect when all but possibly one (faulty) pulse signal arrive in close temporal proximity to determine when to clear their memory and start a new iteration of the main loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1, we sketch how this can be achieved using standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4 ANALYSIS We now analyze the pulse progagation scheme under the assumption that layer 0 generates well- synchronized pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We discuss a suitable method for achieving this in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our analysis proceeds along the following lines: (1) We show that, if the local skew is small enough compared to Λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', Equation (2) holds, all correct nodes execute their iterations as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' That is, each correct node on layer ℓ > 0 receives the 𝑘-th pulses of its correct predecessors in its 𝑘-th loop iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We then proceed under the assumption that this holds true, which will be justified retroactively once we establish that the local skew is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (2) Since delays and hardware clock speeds are (approximated as being) static, any (substantial) change in relative timing of consecutive pulses is due to faulty nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the task of bounding the local skew reduces to bounding the intra-layer skew Lℓ for a single pulse, since such a bound must take into account the full variability introduced by faulty nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This reasoning is deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (3) Based on potential functions, we analyze Lℓ in the absence of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The results entail not only bounded skew, but also that the potentials recover if they become unexpectedly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (4) We show that faulty nodes have limited impact on the potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From this and the above recovery property, we conclude that skews behave favorably also when there are faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As stated above, the first two steps of our line of reasoning are deferred to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main challenge is to bound Lℓ for a single pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the first step, we know that the 𝑘-th pulse at correct nodes depends only on the 𝑘-th pulses of their predecessors (Lemma 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, in the following fix 𝑘 and denote the 𝑘-th pulse time of correct (𝑣, ℓ) ∈ 𝑉𝐺 by 𝑡𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that for 𝑣,𝑤 ∈ 𝑉 , we denote by 𝑑(𝑣,𝑤) their distance in the base graph 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Our analysis is built around the following potential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 12 Christoph Lenzen and Shreyas Srinivas Definition 1 (Potential Functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑣,𝑤 ∈ 𝑉 and 𝑠, ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We define 𝜓𝑠 𝑣,𝑤(ℓ) := 𝑡𝑣,ℓ − 𝑡𝑤,ℓ − 4𝑠𝜅𝑑(𝑣,𝑤), Ψ𝑠 (ℓ) := max 𝑣,𝑤∈𝑉{𝜓𝑠 𝑣,𝑤(ℓ)}, 𝜉𝑠 𝑣,𝑤(ℓ) := 𝑡𝑣,ℓ − 𝑡𝑤,ℓ − (4𝑠 − 2)𝜅𝑑(𝑣,𝑤), and Ξ𝑠 (ℓ) := max 𝑣,𝑤∈𝑉{Ξ𝑠 𝑣,𝑤(ℓ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Bounding Ψ𝑠 (ℓ) readily translates to bounding Lℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If for 𝑠, ℓ ∈ N and some Ψ𝑠 ∈ R≥0 it holds that Ψ𝑠 (ℓ) ≤ Ψ𝑠, then Lℓ ≤ Ψ𝑠 + 4𝑠𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fix 𝑘 ∈ N and suppose that {𝑣,𝑤} ∈ 𝐸 maximizes |𝑡𝑣,ℓ − 𝑡𝑤,ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', assume that 𝑡𝑣,ℓ ≥ 𝑡𝑤,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since {𝑣,𝑤} ∈ 𝐸, we have that 𝑑(𝑣,𝑤) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑡𝑣,ℓ − 𝑡𝑤,ℓ| = 𝑡𝑣,ℓ − 𝑡𝑤,ℓ = 𝜓𝑠 𝑣,𝑤(ℓ) + 4𝑠𝜅 ≤ Ψ𝑠 (ℓ) + 4𝑠𝜅 ≤ Ψ𝑠 + 4𝑠𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑘 ∈ N is arbitrary, it follows that Lℓ ≤ Ψ𝑠 + 4𝑠𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ In summary, the goal of our analysis will be to bound Ψ𝑠 (ℓ) by a small value for some 𝑠 satisfying 4𝑠𝜅 ∈ 𝑂(𝑢 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first study the behavior of the algorithm if there are no faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, this will be tacitly assumed in all statements of this section, with the expection of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that by Lemma 29, this means that we may also tacitly assume that Algorithm 1 is run by all nodes in layers ℓ ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='4, we will then bound the impact of faulty layers on the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Basic Statements We first show three basic lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The first relates the local reception times of pulses to the actual sending times, bounding the error by 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0, set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and 𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑡𝑣,ℓ−1 − 𝑡max − 𝜅 ≤ 𝐻own − 𝐻max − 𝜅 2 ≤ 𝑡𝑣,ℓ−1 − 𝑡max 𝑡𝑣,ℓ−1 − 𝑡min − 𝜅 ≤ 𝐻own − 𝐻min − 𝜅 2 ≤ 𝑡𝑣,ℓ−1 − 𝑡min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove the first inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the second is shown analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑡 ′ 𝑣,ℓ−1 and 𝑡 ′ max denote the times when the pulse messages sent at time 𝑡𝑣,ℓ−1 and 𝑡max are received at 𝑣ℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the bounds on message delays, it follows that 𝑡𝑣,ℓ−1 + 𝑑 − 𝑢 ≤ 𝑡 ′ 𝑣,ℓ−1 ≤ 𝑡𝑣,ℓ−1 + 𝑑 and 𝑡max + 𝑑 − 𝑢 ≤ 𝑡 ′ max ≤ 𝑡max + 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑡𝑣,ℓ−1 − 𝑡max − 𝑢 ≤ 𝑡 ′ 𝑣,ℓ−1 − 𝑡 ′ max ≤ 𝑡𝑣,ℓ−1 − 𝑡max + 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using the bounds on hardware clock rates, we get that |𝑡 ′ 𝑣,ℓ−1 − 𝑡 ′ max − (𝐻own − 𝐻max)| ≤ (𝜗 − 1)|𝑡 ′ 𝑣,ℓ−1 − 𝑡 ′ max| ≤ (𝜗 − 1)(|𝑡𝑣,ℓ−1 − 𝑡max| + 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 13 Applying Equation (2), we infer that |𝑡𝑣,ℓ−1 − 𝑡max − (𝐻own − 𝐻max)| ≤ |𝑡 ′ 𝑣,ℓ−1 − 𝑡 ′ max − (𝐻own − 𝐻max)| + 𝑢 ≤ (𝜗 − 1)|𝑡𝑣,ℓ−1 − 𝑡max| + 𝜗𝑢 ≤ (𝜗 − 1)Lℓ−1 + 𝜗𝑢 ≤ (𝜗 − 1) �Λ − 𝑑 𝜗 − 𝑢 � + 𝜗𝑢 = � 1 − 1 𝜗 � (Λ − 𝑑) + 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, using Equation (1), we conclude that 𝑡𝑣,ℓ−1 − 𝑡max − 𝜅 ≤ 𝑡𝑣,ℓ−1 − 𝑡max − 2 �� 1 − 1 𝜗 � (Λ − 𝑑) + 𝑢 � ≤ 𝐻own − 𝐻max − 𝜅 2 ≤ 𝑡𝑣,ℓ−1 − 𝑡max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ The second lemma shows that corrections are not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑣 ∈ 𝑉 and ℓ ∈ N>0, C𝑣,ℓ ≤ Λ − 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Abbreviate Δ = min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Δ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Algorithm 1 sets C𝑣,ℓ ≤ min � 𝐻own − 𝐻min − 𝜅 2 + 2𝜅, 0 � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As Λ ≥ 𝑑 by Equation (2), the claim of the lemma holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0 ≤ Δ ≤ 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then, using the notation of Lemma 1, C𝑣,ℓ = Δ < min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 ≤ min 𝑠 ∈N � max{𝑡𝑣,ℓ−1 − 𝑡max + 4𝑠𝜅,𝑡𝑣,ℓ−1 − 𝑡min − 4𝑠𝜅} � ≤ min 𝑠 ∈N {max{Lℓ−1 + 4𝑠𝜅, Lℓ−1 − 4𝑠𝜅}} = Lℓ−1, which is smaller than Λ − 𝑑 by Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Δ > 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that then 𝜗𝜅 < Δ ≤ max{𝐻own − 𝐻max, 𝐻own − 𝐻min} − 𝜅 2 = 𝐻own − 𝐻max − 𝜅 2, 14 Christoph Lenzen and Shreyas Srinivas as 𝐻max ≥ 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, applying Lemma 2, C𝑣,ℓ = max � 𝐻own − 𝐻max − 𝜅 2 − 𝜅,𝜗𝜅 � ≤ 𝐻own − 𝐻max − 𝜅 2 ≤ 𝑡𝑣,ℓ−1 − 𝑡max ≤ Lℓ−1 < Λ − 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ The third lemma bounds the time difference between the pulses of (𝑣, ℓ − 1) and (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑣 ∈ 𝑉 and ℓ ∈ N>0 it holds that 𝑑 − 𝑢 + Λ − 𝑑 − C𝑣,ℓ 𝜗 ≤ 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 ≤ Λ − C𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑡 ′ 𝑣,ℓ−1 denote the time at which (𝑣, ℓ) receives the pulse sent by (𝑣, ℓ − 1) at time 𝑡𝑣,ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Inspecting the code of Algorithm 1, we see that 𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ = 𝐻𝑣,ℓ (𝑡 ′ 𝑣,ℓ−1) + Λ − 𝑑 − C𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since C𝑣,ℓ ≤ Λ −𝑑 by Lemma 2, it follows that 𝐻𝑣,ℓ (𝑡 ′ 𝑣,ℓ−1) ≥ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) and hence 𝑡𝑣,ℓ ≥ 𝑡 ′ 𝑣,ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using the bounds on message delays and hardware clock speeds, we get that 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 = 𝑡𝑣,ℓ − 𝑡 ′ 𝑣,ℓ−1 + 𝑡 ′ 𝑣,ℓ−1 − 𝑡𝑣,ℓ−1 ≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) − 𝐻𝑣,ℓ (𝑡 ′ 𝑣,ℓ−1) + 𝑑 = Λ − C𝑣,ℓ and 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 = 𝑡𝑣,ℓ − 𝑡 ′ 𝑣,ℓ−1 + 𝑡 ′ 𝑣,ℓ−1 − 𝑡𝑣,ℓ−1 ≥ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) − 𝐻𝑣,ℓ (𝑡 ′ 𝑣,ℓ−1) 𝜗 + 𝑑 − 𝑢 = Λ − 𝑑 − C𝑣,ℓ 𝜗 + 𝑑 − 𝑢, which can be rearranged into the claimed inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='2 The Slow, Fast, and Jump Conditions The key to bounding the local skew without faults is to find the right balance between two conflicting goals: choosing C𝑣,ℓ large enough to “catch up” to predecessors 𝑤ℓ−1 ≠ 𝑣ℓ−1 that generated their pulse earlier than 𝑣ℓ−1, but small enough to “wait” for predecessors 𝑤ℓ−1 ≠ 𝑣ℓ−1 that generated their pulse later than 𝑣ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The following condition captures what we need regarding the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 2 (Slow Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N, correct layers ℓ − 1 ∈ N, and 𝑣ℓ ∈ 𝑉ℓ \\ 𝐹, we require the slow condition SC(𝑠) := SC-1(𝑠) ∨ SC-2(𝑠) ∨ SC-3 to hold, where SC-1(𝑠) : C𝑣,ℓ 𝜗 ≤ 𝑡𝑣,ℓ−1 − max {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} + 4𝑠𝜅 SC-2(𝑠) : C𝑣,ℓ 𝜗 ≤ 𝑡𝑣,ℓ−1 − min {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − 4𝑠𝜅 SC-3: C𝑣,ℓ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 15 This can be viewed as a variant of the slow condition from [15], adjusted to our setting by quantifying by how much 𝑣ℓ may safely shift the timing of its pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main conceptual difference to [15] is that we relax the slow condition by adding SC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In what follows, we drop 𝑠 from the notation when it is clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N and (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0, SC(𝑠) holds at (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemma 29, we prove the claim for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and 𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑣,ℓ ≤ 0, SC-3 is trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, assume that C𝑣,ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Abbreviate Δ = min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 = max{𝐻own − 𝐻max + 4𝑠min𝜅, 𝐻own − 𝐻min − 4𝑠min𝜅} − 𝜅 2, where 𝑠min ∈ N is an index for which the minimum is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Δ ≤ 𝜗𝜅, then C𝑣,ℓ = Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, C𝑣,ℓ = max � 𝐻own − 𝐻max − 𝜅 2 − 𝜅,𝜗𝜅 � ≤ max{Δ,𝜗𝜅} = Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Either way, we get that C𝑣,ℓ/𝜗 < C𝑣,ℓ ≤ Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own − 𝐻max + 4𝑠min𝜅 ≥ 𝐻own − 𝐻min − 4𝑠min𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then for 𝑠 ∈ N, 𝑠 ≥ 𝑠min, by Lemma 1 we have that Δ ≤ 𝐻own − 𝐻max + 4𝑠min𝜅 − 𝜅 2 ≤ 𝑡𝑣,ℓ−1 − 𝑡max + 4𝑠𝜅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', SC-1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now consider 𝑠 ∈ N, 𝑠 < 𝑠min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝐻own −𝐻max −𝜗𝑢 + 4𝑠𝜅 < 𝐻own −𝐻max − 𝜗𝑢 + 4𝑠min𝜅 ≤ Δ, but the minimum is attained at index 𝑠min, we must have that Δ ≤ 𝐻own − 𝐻min − 4𝑠𝜅 − 𝜅 2 ≤ 𝑡𝑣,ℓ−1 − 𝑡min − 4𝑠𝜅, where the second step again applies Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, SC-2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own − 𝐻max + 4𝑠min𝜅 < 𝐻own − 𝐻min − 4𝑠min𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, we analogously infer that SC-1 holds for 𝑠 > 𝑠min and SC-2 holds for 𝑠 ≤ 𝑠min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ The fast condition is the counterpart to Definition 2 addressing the need to “catch up” to neighbors that are ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 3 (Fast Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N>0, correct layers ℓ − 1 ∈ N>0, and 𝑣ℓ ∈ 𝑉ℓ \\ 𝐹, we require the fast condition FC(𝑠) := FC-1(𝑠) ∨ FC-2(𝑠) ∨ FC-3 to hold, where FC-1(𝑠) : C𝑣,ℓ ≥ 𝑡𝑣,ℓ−1 − max {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} + (4𝑠 − 2)𝜅 + 𝜅 FC-2(𝑠) : C𝑣,ℓ ≥ 𝑡𝑣,ℓ−1 − min {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − (4𝑠 − 2)𝜅 + 𝜅 FC-3: C𝑣,ℓ ≥ 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This can be viewed as a variant of the fast condition from [15], adjusted to our setting by quantifying by how much 𝑣ℓ may safely shift the timing of its pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main conceptual difference to [15] is that we relax the fast condition by adding FC-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In addition, note that there is an additive term of 𝜅 that does not change sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Its purpose is to account for the fact that our simulation of the GCS algorithm from [18] operates in discrete time steps corresponding to the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The continuous versions of the GCS algorithm in [14, 15, 18] can choose this term arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In contrast, we need it to exceed the maximum error in time measurement accumulated in a step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We remark that, in principle, one could choose this term 16 Christoph Lenzen and Shreyas Srinivas 𝑣 𝑡𝑣 − 4𝑠𝜅𝑑(𝑣,𝑤) 𝑤 𝑡𝑤 − (4𝑠 − 2)𝜅𝑑(𝑣,𝑤) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Slow condition (left) and fast condition (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC(𝑠) is tailored to ensuring that max𝑤∈𝑉 {𝜓𝑠𝑣,𝑤(ℓ)} (the length of the green arrow) cannot grow quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Nodes 𝑤 with C𝑤,ℓ ≤ 0 (SC-3 holds) cannot apply a correction pushing them below the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑤,ℓ > 0, then both SC-1 and SC-2 will ensure that there is a neighbor 𝑥 of 𝑤 such that the offset of 𝑡𝑤,ℓ−1 − C𝑤,ℓ/𝜗 to the black line does not exceed the one of 𝑡𝑥,ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In other words, SC ensures that the blue arrows indicating C𝑤,ℓ/𝜗 do not reach below the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This means that any increase of max𝑤∈𝑉 {𝜓𝑠𝑣,𝑤(ℓ)} is caused by delay and clock speed variation, which in turn is bounded by 𝜅/2 per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Similarly, FC(𝑠) is tailored to ensuring that max𝑣∈𝑉 {𝜉𝑠𝑣,𝑤(ℓ)} (the length of the green arrow), if positive, decreases by at least 𝜅/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To ensure this, C𝑤,ℓ (indicated by blue arrows) must be large enough to reach below the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is achieved by FC(𝑠) having an additional “slack” term of 𝜅, which overcomes the “loss” of 𝜅/2 due to uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' different from 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, since both need to meet the same lower bound of 𝑢 + (1 − 1/𝜗)(Λ − 𝑑), there is no asymptotic gain in introducing a separate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑠 ∈ N and (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0, FC(𝑠) holds at (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemma 29, we prove the claim for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and 𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑣,ℓ ≥ 𝜗𝜅, trivially FC-3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, assume that C𝑣,ℓ < 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Abbreviate Δ = min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 = max{𝐻own − 𝐻max + 4𝑠min𝜅, 𝐻own − 𝐻min − 4𝑠min𝜅} − 𝜅 2, where 𝑠min ∈ N is an index for which the minimum is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Δ ≥ 0, then C𝑣,ℓ = Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, C𝑣,ℓ = min � 𝐻own − 𝐻min − 𝜅 2 + 2𝜅, 0 � ≥ Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Either way, we get that C𝑣,ℓ ≥ Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑠 ∈ N, 𝑠 ≤ 𝑠min, by Lemma 1 and Equation (1) it holds that Δ ≥ 𝐻own − 𝐻max + 4𝑠𝜅 − 𝜅 2 ≥ 𝑡𝑣,ℓ−1 − 𝑡max + (4𝑠 − 2)𝜅 + 𝜅, proving that FC-1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑠 ∈ N, 𝑠 > 𝑠min, by Lemma 1 and Equation (1) we get that Δ ≥ 𝐻own − 𝐻min − 4(𝑠 − 1)𝜅 − 𝜅 2 ≥ 𝑡𝑣,ℓ−1 − 𝑡min − (4𝑠 − 2)𝜅 + 𝜅, showing that FC-2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Our relaxation of the slow and fast conditions adds a substantial complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From the per- spective of the time-continuous variant of the algorithm in [15], we now allow for arbitrarily large clock “jumps,” rather than bounded clock rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In our discrete version, the rate bound from [15] corresponds to C𝑣,ℓ ∈ [0,𝜗𝜅].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Without this additional constraint, the slow and fast conditions are insufficient to bound skews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 17 𝑣ℓ+2 𝑣ℓ+1 𝑣ℓ 𝑣ℓ+2 𝑣ℓ+1 𝑣ℓ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the left, it is shown how skews increase without JC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' While SC(0) disallows that (𝑣, ℓ) speeds up its pulse by more than the equivalent of (𝑣, ℓ − 1) matching the earliest pulse of any (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, FC permits that a node (𝑣, ℓ) with slow (𝑣, ℓ − 1) to “overshoot,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', C𝑣,ℓ (shown as blue arrow) gets large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This results in an amplifying oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the right, the same scenario is shown with JC in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' JC forces the corrections to stop 𝜅 before the earliest or latest neighbor, respectively, resulting in a dampened oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is illustrated in Figure 5, showing an execution that satisfies SC and FC, but suffers from skews that grow without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key issue is that adjacent nodes could “jump” in opposite directions, resulting in an oscillatory behavior in which measurement errors accumulate indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To avoid this kind of behavior, we add an additional condition that “dampens” such oscillations, yet limits by how much a faulty predecessor can cause an increase in skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 4 (Jump Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all correct layers ℓ − 1 ∈ N>0 and 𝑣ℓ ∈ 𝑉ℓ \\ 𝐹, we require the jump condition JC := JC-1 ∨ JC-2 ∨ JC-3 to hold, where JC-1: 𝜅 < C𝑣,ℓ 𝜗 ≤ 𝑡𝑣,ℓ−1 − max {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − 𝜅 JC-2: 0 > C𝑣,ℓ ≥ 𝑡𝑣,ℓ−1 − min {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} + 𝜅 JC-3: 0 ≤ C𝑣,ℓ 𝜗 ≤ 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that layer ℓ − 1 ∈ N and 𝑣ℓ ∈ 𝑉ℓ are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then JC holds at 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemma 29, we prove the claim for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Set 𝑡min := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and 𝑡max := min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0 ≤ C𝑣,ℓ ≤ 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then JC-3 is satisfied trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 18 Christoph Lenzen and Shreyas Srinivas C𝑣,ℓ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 1 and Equation (1), then C𝑣,ℓ = 𝐻own − 𝐻min − 𝜅 2 + 2𝜅 ≥ 𝑡𝑣,ℓ−1 − 𝑡min + 𝜅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', JC-2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑣,ℓ > 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 1, then C𝑣,ℓ = 𝐻own − 𝐻max − 𝜅 2 − 𝜅 ≤ 𝑡𝑣,ℓ−1 − 𝑡max − 𝜅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', JC-3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='3 Bounding Ψ𝑠 in the Absence of Faults With the conditions established, we are ready to study how Ψ𝑠 (ℓ) evolves in the fault-free setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The main technical challenge in bounding Ψ𝑠 lies in performing the induction step from 𝑠 − 1 ∈ N to 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We will argue that for Ψ𝑠 ( ¯ℓ ) to be large for some ¯ℓ, Ξ𝑠 (ℓ ) must have been large for some ℓ < ¯ℓ, with an additive term growing with ¯ℓ − ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑠 ∈ N>0 and layers ℓ ≤ ¯ℓ, it holds that Ψ𝑠 ( ¯ℓ ) ≤ max � 0, Ξ𝑠 (ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 � + ( ¯ℓ − ℓ ) · 𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Intuitively, we intend to argue that if Ψ𝑠 ( ¯ℓ ) is large, so must be Ξ𝑠 (ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Tracing back the cause for this, we show that in every step, we have that Ξ𝑠 (ℓ − 1) is larger than Ξ𝑠 (ℓ) by at least 𝜅/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since Ξ𝑠 ( ¯ℓ ) ≥ Ψ𝑠 ( ¯ℓ ), as 𝜓𝑠 𝑣,𝑤(ℓ) ≥ 𝜉𝑠 𝑣,𝑤(ℓ) for all 𝑣, 𝑤, 𝑠, and ℓ, this yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To formalize that Ξ𝑠 (ℓ) must have been decreasing steadily, we seek to show that the minimal layer ℓ for which there are nodes 𝑣ℓ,𝑤ℓ ∈ 𝑉 satisfying that 𝜉𝑠 𝑣ℓ,𝑤ℓ (ℓ) is large enough is ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, we identify nodes 𝑤 and 𝑣 – either 𝑤ℓ and 𝑣ℓ themselves or neighbors of them – which cause the large skew on layer ℓ by a having a large skew on layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is done based on SC(𝑠) and FC(𝑠), with JC kicking in for the special case that 𝑤 = 𝑣ℓ and 𝑣 = 𝑤ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A key obstacle is that if 𝑤 is a neighbor of 𝑤ℓ, this results in a larger difference in skew than if 𝑣 is a neighbor of 𝑣ℓ, namely 4𝑠𝜅 versus (4𝑠 − 2)𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, when 𝑤 is closer to 𝑣ℓ than 𝑤ℓ, we “lose” 2𝜅 relative to the skew bound on layer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑑(𝑣 ¯ℓ,𝑤 ¯ℓ) many steps, we can compensate for this based on the initial skew between 𝑣 ¯ℓ and 𝑤 ¯ℓ, but not more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To address this, essentially we need to show that for any additional steps “towards” 𝑣ℓ there will be a corresponding step “away” from 𝑣ℓ, on which we “gain” additional 2𝜅 relative to the skew bound on the layer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If corrections were always positive, this would be straightforward: Steps towards 𝑣ℓ would also be steps towards 𝑣 ¯ℓ, and upon 𝑤ℓ = 𝑣 ¯ℓ we would reach a contradiction to the skew bounds shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Unfortunately, negative corrections foreclose this simple argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To address this, we introduce a third “prover” node 𝑝ℓ, where 𝑝 ¯ℓ = 𝑣 ¯ℓ, which never increases its distance to 𝑤ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' if 𝑝ℓ performs a negative correction, then 𝑝 is a neighbor of 𝑝ℓ that is closer to 𝑤ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We then can infer that 𝑝 ≠ 𝑤 from the skew bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A major complication this approach faces is the special case 𝑝 = 𝑤ℓ and 𝑤 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Again, JC kicks in to show that we have sufficiently large skew between 𝑝 and 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, now 𝑝 lies “behind” 𝑤 from the perspective of 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A later reversal of this situation by repeating the case that 𝑝 = 𝑤ℓ and 𝑤 = 𝑝ℓ results in 𝑤 being farther away from 𝑣ℓ, yet 𝑑(𝑝,𝑤) = 𝑑(𝑝ℓ,𝑤ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The proof covers this case by adding an additional (4𝑠 − 2)𝜅 to the skew bound if the above situation occured an odd number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Finally, we seek to avoid the case that 𝑣 = 𝑝ℓ and 𝑝 = 𝑣ℓ for analogous reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Fortunately, here we can exploit that the skew bound between 𝑣ℓ and 𝑤ℓ is stronger than the one between 𝑝ℓ and 𝑤ℓ, meaning that we can simply choose 𝑝 = 𝑣 instead in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the proof, we do so Gradient TRIX 19 whenever 𝑣 lies on the path connecting 𝑝ℓ and 𝑤ℓ that we maintain to keep track of hop counts in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that the statement of Theorem 1 is false for minimal ¯ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', there are 𝑣 ¯ℓ and 𝑤 ¯ℓ such that 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ > ( ¯ℓ − ℓ ) · 𝜅 2 (4) and 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ > Ξ𝑠 (ℓ ) − ( ¯ℓ − ℓ ) · 𝜅 2 − 𝜅 (5) and there is no smaller ¯ℓ′ for which this applies for some pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let ℓ ∈ [ℓ, ¯ℓ] be minimal such that are 𝑣ℓ, 𝑝ℓ,𝑤ℓ ∈ 𝑉 , a path 𝑄ℓ in 𝐻 from 𝑝ℓ to 𝑣ℓ, and a path 𝑃ℓ in 𝐻 from 𝑝ℓ to 𝑤ℓ with the following properties: (P1) 𝑤ℓ ≠ 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P2) 𝑤ℓ ≠ 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P3) 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅|𝑃ℓ| ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − ℓ) · 𝜅 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P4) Denote by |𝑃ℓ| and |𝑄ℓ| the length of 𝑃ℓ and 𝑄ℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With the shorthand Δℓ := � |𝑃ℓ| + |𝑄ℓ| − 1 if 𝑃ℓ and 𝑄ℓ have the same first edge |𝑃ℓ| + |𝑄ℓ| else, it holds that 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅Δℓ ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ) · 𝜅 2 + 2𝜅|𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P5) If 𝑣ℓ ∈ 𝑃ℓ, then 𝑝ℓ = 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To see that such an index must indeed exist, let 𝑝 ¯ℓ := 𝑣 ¯ℓ, 𝑃 ¯ℓ be a shortest path in 𝐻 from 𝑝 ¯ℓ to 𝑤 ¯ℓ, and 𝑄 ¯ℓ := (𝑝 ¯ℓ) = (𝑣 ¯ℓ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the 0-length path from 𝑝 ¯ℓ to 𝑣 ¯ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This choice satisfies (P1) and (P2), because Ψ𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) ≠ 0 implies that 𝑣 ¯ℓ ≠ 𝑤 ¯ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P4), because 𝑡𝑣 ¯ℓ,¯ℓ − 𝑡𝑤 ¯ℓ,¯ℓ − (4𝑠 − 2)𝜅Δℓ = 𝑡𝑣 ¯ℓ,¯ℓ − 𝑡𝑤 ¯ℓ,¯ℓ − (4𝑠 − 2)𝜅|𝑃 ¯ℓ| = 𝜓𝑣 ¯ℓ,𝑤 ¯ℓ + 2𝜅|𝑃 ¯ℓ|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' and (P3) and (P5), because 𝑝 ¯ℓ = 𝑣 ¯ℓ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑡𝑝 ¯ℓ,¯ℓ = 𝑡𝑣 ¯ℓ,¯ℓ and Δ¯ℓ = |𝑃 ¯ℓ|) and (P4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 2 proves that in fact ℓ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that 𝑑(𝑣ℓ,𝑤ℓ) ≤ � |𝑃ℓ| + |𝑄ℓ| − 2 if 𝑃ℓ and 𝑄ℓ share the first edge |𝑃ℓ| + |𝑄ℓ| else ≤ Δℓ 20 Christoph Lenzen and Shreyas Srinivas and that |𝑃ℓ| ≥ 1 due to (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, (P4) yields that Ξ𝑠 (ℓ ) ≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅𝑑(𝑣ℓ,𝑤ℓ) ≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅Δℓ ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ) · 𝜅 2 + 2𝜅|𝑃ℓ| ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ) · 𝜅 2 + 2𝜅, contradicting Equation (5) and completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ The remainder of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='3 is dedicated to proving Corollary 2, which is the missing step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, until the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='3 we consider the setting of the proof of Theorem 1 and assume for contradiction that ℓ > ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We take note of some straightforward implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For any fixed index ℓ, we have the following implications: (P3) ⇒ (P1) (P4) ⇒ (P2) (𝑣ℓ = 𝑝ℓ∧ (P4)) ⇒ (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, 𝜓𝑠 𝑣ℓ,𝑤ℓ (ℓ) − ( ¯ℓ − ℓ) · 𝜅 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove each implication separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From (P3), 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ > 4𝑠𝜅|𝑃ℓ| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This implies 𝑡𝑝ℓ,ℓ > 𝑡𝑤ℓ,ℓ and hence 𝑤ℓ ≠ 𝑝ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Δℓ ≥ 0, |𝑃ℓ| ≥ 0, and 4𝑠 − 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, (P4) and Equation (4) imply that 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ ≥ 𝜓𝑠 𝑣ℓ,𝑤ℓ ( ¯ℓ ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that 𝑤ℓ ≠ 𝑣ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑣ℓ = 𝑝ℓ, then 𝑡𝑣ℓ,ℓ = 𝑡𝑝ℓ,ℓ, |𝑄ℓ| = 0, and Δℓ = |𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, (P4) implies that 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − (4𝑠 − 2)𝜅|𝑃ℓ| ≥ 𝜓𝑠 𝑣ℓ,𝑤ℓ (¯𝑙) + ( ¯ℓ − ℓ) · 𝜅 2 + 2𝜅|𝑃ℓ| ≥ 𝜓𝑠 𝑣ℓ,𝑤ℓ (¯𝑙) − ( ¯ℓ − ℓ) · 𝜅 2 + 2𝜅|𝑃ℓ|, which can be rearranged to yield (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ A Step in the Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now identify nodes that are suitable for taking the role of 𝑣ℓ, 𝑝ℓ, and 𝑤ℓ on layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' These are either the nodes themselves or neighbors of them in 𝐻, where FC(𝑠), SC(𝑠), and JC serve to relate respective pulse times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is a node 𝑣 ∈ 𝑉 such that 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅Δ𝑣 − 𝜅, where Δ𝑣 = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 0 and 𝑣 = 𝑣ℓ, −1 and {𝑣, 𝑣ℓ} is the last edge of 𝑄ℓ or the first edge of 𝑃ℓ, or 1 and {𝑣ℓ, 𝑣} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 5, 𝑣ℓ obeys the fast condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus one of three things is true for 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' FC-1(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, let 𝑣 = arg max{𝑥,𝑣ℓ }∈𝐸{𝑡𝑥,ℓ−1} and bound 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ max {𝑥,𝑣ℓ }∈𝐸 � 𝑡𝑥,ℓ−1 � − (4𝑠 − 2)𝜅 − 𝜅 = 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅 − 𝜅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑣 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 21 FC-2(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, let {𝑣, 𝑣ℓ} be the last edge of 𝑄ℓ if |𝑄ℓ| ≠ 0 or the first edge of 𝑃ℓ otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the latter is feasible, because then 𝑣ℓ = 𝑝ℓ, and |𝑃ℓ| ≠ 0 due to (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We get that 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ min {𝑥,𝑣ℓ }∈𝐸 � 𝑡𝑥,ℓ−1 � + (4𝑠 − 2)𝜅 − 𝜅 ≤ 𝑡𝑣,ℓ−1 + (4𝑠 − 2)𝜅 − 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the claim of the lemma holds with Δ𝑣 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' FC-3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣ℓ,ℓ−1 − 𝜅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is a node 𝑤 ∈ 𝑉 such that 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 ≥ 𝑡𝑤,ℓ−1 + 4𝑠𝜅Δ𝑤, where Δ𝑤 = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 0 and 𝑤 = 𝑤ℓ, −1 and {𝑤,𝑤ℓ} is the last edge of 𝑃ℓ, or 1 and {𝑤ℓ,𝑤} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 4, 𝑤ℓ satisfies SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We make a case distinction based on which one of SC-1, SC-2, and SC-3 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC-1(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let {𝑤,𝑤ℓ} be the last edge of 𝑃ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' by (P1), |𝑃ℓ| ≠ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', this edge exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,𝑙 𝜗 ≥ max {𝑥,𝑤ℓ }∈𝐸{𝑡𝑥,ℓ−1} − 4𝑠𝜅 ≥ 𝑡𝑤,ℓ−1 − 4𝑠𝜅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑤 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC-2(𝑠) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, let 𝑤 = arg min{𝑥,𝑣ℓ }∈𝐸{𝑡𝑥,ℓ−1} and bound 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ, ℓ 𝜗 ≥ min {𝑥,𝑤ℓ }∈𝐸{𝑡𝑥,ℓ−1} + 4𝑠𝜅 = 𝑡𝑤,ℓ−1 + 4𝑠𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the lemma holds with Δ𝑤 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' SC-3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑡𝑤ℓ,ℓ−1 − C𝑤,ℓ ≥ 𝑡𝑤ℓ,ℓ−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the claim of the lemma holds with Δ𝑤 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There is a node 𝑝 ∈ 𝑉 such that 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ ≤ � 𝑡𝑝,ℓ−1 and 𝑝 = 𝑝ℓ, or 𝑡𝑝,ℓ−1 − 𝜅 and {𝑝ℓ, 𝑝} is the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If C𝑝ℓ,ℓ ≥ 0, the claim holds with 𝑝 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, suppose that C𝑝ℓ,ℓ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let {𝑝ℓ, 𝑝} be the first edge of 𝑃ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' such an edge exists, as by (P1) we have that 𝑝ℓ ≠ 𝑤ℓ and hence |𝑃ℓ| ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 6, 𝑝ℓ satisfies JC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As C𝑝ℓ,ℓ < 0, JC-2 must apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that C𝑝ℓ,ℓ ≥ 𝑡𝑝ℓ,ℓ−1 − min {𝑥,𝑝ℓ }∈𝐸 � 𝑡𝑥,ℓ−1 � + 𝜅 ≥ 𝑡𝑝ℓ,ℓ−1 − 𝑡𝑝,ℓ−1 + 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Rearranging terms, the desired inequality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 22 Christoph Lenzen and Shreyas Srinivas In the following, let (𝑣, 𝑝,𝑤) be the triple of nodes guaranteed by Lemmas 7 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by ◦ concatenation of paths, by prefix(𝑅,𝑥) the prefix of path 𝑅 ending at node 𝑥 ∈ 𝑅, and by suffix(𝑅,𝑥) the suffix of path 𝑅 starting at node 𝑥 ∈ 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑝′ = � 𝑣 if 𝑣 lies on suffix(𝑃ℓ, 𝑝), 𝑝 else, 𝑃 := � prefix(𝑃ℓ,𝑤) if 𝑤 lies on 𝑃ℓ, 𝑃ℓ ◦ (𝑤ℓ,𝑤) else, 𝑃 ′ := � suffix(𝑃, 𝑝′) if 𝑝′ lies on 𝑃, (𝑝′,𝑤) else, 𝑄 := � prefix(𝑄ℓ, 𝑣) if 𝑣 lies on 𝑄ℓ, 𝑄ℓ ◦ {𝑣ℓ, 𝑣} else, 𝑄 ′ := � suffix(𝑄, 𝑝′) if 𝑝′ lies on 𝑄, (𝑝′, 𝑝ℓ) ◦ 𝑄 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For notational convenience, in analogy to Δℓ we also define Δ := � |𝑃 ′| + |𝑄 ′| − 1 if 𝑃 ′ and 𝑄 ′ have the same first edge |𝑃 ′| + |𝑄 ′| else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We will show that this construction satisfies properties (P1) to (P5) for layer ℓ − 1 with 𝑣ℓ−1 = 𝑣, 𝑝ℓ−1 = 𝑝′, 𝑤ℓ−1 = 𝑤, 𝑃ℓ−1 = 𝑃 ′, and 𝑄ℓ−1 = 𝑄 ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' this will constitute the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we first point out that indeed 𝑃 ′ and 𝑄 ′ are paths in 𝐻 from 𝑝′ to 𝑤 and 𝑣, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, we first cover the special case that 𝑝′ does not lie on 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ does not lie on 𝑃, then 𝑝′ = 𝑤ℓ and either 𝑤 = 𝑝ℓ or 𝑝′ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 9, 𝑝 lies on the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, if 𝑝′ = 𝑝, 𝑝′ lies on 𝑃 unless prefix(𝑃ℓ,𝑤) does not contain this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 8, this can only happen if the first edge of 𝑃ℓ is also the last edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑃ℓ = (𝑝ℓ,𝑤ℓ) = (𝑤, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to consider the case that 𝑝′ ≠ 𝑝, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑝′ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Again, we use that all edges but the last of 𝑃ℓ are also contained in 𝑃 by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑝′ = 𝑣 = 𝑤ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃 ′ is a path in 𝐻 from 𝑝′ to 𝑤 and 𝑄 ′ is a path in 𝐻 from 𝑝′ to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To show that 𝑃 ′ is a path from 𝑝′ to 𝑤, note that by Lemma 8, 𝑃 is a path in 𝐻, which by definition ends at 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if 𝑃 ′ = suffix(𝑃, 𝑝′), 𝑃 ′ is a path from 𝑝′ to 𝑤 in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, by Observation 3, 𝑝′ = 𝑤ℓ, and {𝑝′,𝑤} = {𝑤ℓ,𝑤} ∈ 𝐸 by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To show that 𝑄 ′ is a path from 𝑝′ to 𝑣, note that by Lemma 7, 𝑄 is a path in 𝐻, which by definition ends at 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ = 𝑝, by Lemma 9 𝑄 ′ is also a path in 𝐻, which by definition begins at 𝑝′ and has the same endpoint as 𝑄, which is 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the other hand, if 𝑝′ = 𝑣, suffix(𝑄, 𝑝′) = suffix(𝑄, 𝑣) = (𝑣), which is the 0-length path from 𝑝′ = 𝑣 to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Proving the Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To prove Corollary 2, we establish that the tuple (𝑣, 𝑝′,𝑤, 𝑃 ′,𝑄′) satisfies properties (P1) to (P5) for layer ℓ − 1, contradicting the minimality of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Observation 4, indeed 𝑃 ′ and 𝑄 ′ are paths from 𝑣 to 𝑤 and 𝑝′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the following, we will repeatedly use this fact and the property that {𝑥ℓ,𝑥} ∈ 𝐸 for 𝑥 ∈ {𝑣,𝑤, 𝑝} whenever 𝑥 ≠ 𝑥ℓ, without explicitly invoking Observation 4 and Lemmas 7 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first rule out the special case that 𝑣 = 𝑤ℓ and 𝑤 = 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 23 Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The case that 𝑣 = 𝑤ℓ and 𝑤 = 𝑣ℓ is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that 𝑣 = 𝑤ℓ and 𝑤 = 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We use (P4), Lemma 3, and Lemma 8 to bound −C𝑤,ℓ ≥ 𝑡𝑤,ℓ − 𝑡𝑤,ℓ−1 − Λ = 𝑡𝑣ℓ,ℓ − (𝑡𝑤,ℓ−1 − 4𝑠𝜅) − Λ − 4𝑠𝜅 ≥ 𝑡𝑣ℓ,ℓ − � 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 � − Λ − 4𝑠𝜅 = 𝑡𝑣ℓ,ℓ − � 𝑡𝑤ℓ,ℓ−1 + 𝑑 − 𝑢 + Λ − 𝑑 − C𝑤ℓ,ℓ 𝜗 � − 𝜅 2 − 4𝑠𝜅 ≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 − 𝜅 2 ≥ 𝜓𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by JC, it holds that 𝑡𝑤,ℓ−1 ≤ 𝑡𝑤ℓ,ℓ−1 + C𝑤,ℓ − 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that by (P1), |𝑃ℓ| ≠ 0 and hence |𝑃ℓ|, Δℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by (P4) and Equation (4) 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − ℓ)𝜅 2 ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑤ℓ,ℓ ≤ 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then by Lemma 3 4𝑠𝜅 < 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ = 𝑡𝑤,ℓ − 𝑡𝑤ℓ,ℓ ≤ 𝑡𝑤,ℓ−1 − C𝑤,ℓ − � 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 � + 𝑢 + � 1 − 1 𝜗 � (Λ − 𝑑) ≤ 𝑢 + � 1 − 1 𝜗 � (Λ − 𝑑) < 𝜅, which is a contradiction, because 𝑠 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑤ℓ,ℓ > 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By JC, it follows that 𝑡𝑤ℓ,ℓ−1 ≥ 𝑡𝑤,ℓ−1 + C𝑤ℓ,ℓ 𝜗 + 𝜅, yielding by Lemma 3 that 𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 = 𝑡𝑤ℓ,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑤,ℓ−1 + C𝑤ℓ,ℓ 𝜗 + 𝜅 − (𝑡𝑤ℓ,ℓ−1 + C𝑤,ℓ − 𝜅) = 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ − � 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 � + 2𝜅 ≥ 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ + 2𝜅 − 𝜅 2 > 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ + 𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 24 Christoph Lenzen and Shreyas Srinivas Recall that by (P1), |𝑃ℓ| ≠ 0 and hence |𝑃ℓ|, Δℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, 𝑑(𝑣,𝑤) = 𝑑(𝑤ℓ,𝑤) ≤ 1, since by Lemma 8 𝑤 is either 𝑤ℓ or a neighbor of 𝑤ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, (P4) implies that 𝜓𝑠 𝑣,𝑤(ℓ − 1) = 𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 − 4𝑠𝜅𝑑(𝑣,𝑤) > 𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅|𝑃ℓ| + 𝜅 2 ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − (ℓ − 1))𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑣 and 𝑤 satisfy Equation (4) and Equation (5) with index ¯ℓ replaced by index ℓ − 1 < ¯ℓ, contradicting the minimality of ¯ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Next, we prove a helper lemma relating 𝑡𝑤ℓ,ℓ and 𝑡𝑤,ℓ−1 by a stronger bound than Lemma 8 for the special case that 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This follows similar reasoning as the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, it does not yield an immediate contradiction, as we need to rely on the weaker bound provided by (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ, then 𝑡𝑤ℓ,ℓ − 𝑡𝑤,ℓ−1 > 𝑑 − 𝑢 + Λ − 𝑑 𝜗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We use (P3) and Lemma 3 to bound −C𝑤,ℓ ≥ 𝑡𝑤,ℓ − 𝑡𝑤,ℓ−1 − Λ = 𝑡𝑝ℓ,ℓ − (𝑡𝑤,ℓ−1 − 4𝑠𝜅) − Λ − 4𝑠𝜅 ≥ 𝑡𝑝ℓ,ℓ − � 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 � − Λ − 4𝑠𝜅 = 𝑡𝑝ℓ,ℓ − � 𝑡𝑤ℓ,ℓ−1 + 𝑑 − 𝑢 + Λ − 𝑑 − C𝑤ℓ,ℓ 𝜗 � − 𝜅 2 − 4𝑠𝜅 ≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 − 𝜅 2 ≥ 𝜓𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by JC, it holds that 𝑡𝑤,ℓ−1 ≤ 𝑡𝑤ℓ,ℓ−1 − 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑤ℓ,ℓ ≤ 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑡𝑤ℓ,ℓ − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑤ℓ,ℓ − 𝑡𝑤ℓ,ℓ−1 + 𝜅 ≥ 𝑑 − 𝑢 + Λ − 𝑑 − C𝑤ℓ,ℓ 𝜗 + 𝜅 ≥ 𝑑 − 𝑢 + Λ − 𝑑 𝜗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' C𝑤ℓ,ℓ > 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By JC, it follows that 𝑡𝑤ℓ,ℓ−1 ≥ 𝑡𝑤,ℓ−1 + C𝑤ℓ,ℓ 𝜗 + 𝜅, Gradient TRIX 25 yielding that 𝑡𝑤ℓ,ℓ − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑤ℓ,ℓ − 𝑡𝑤ℓ,ℓ−1 + C𝑤ℓ,ℓ 𝜗 + 𝜅 > 𝑑 − 𝑢 + Λ − 𝑑 𝜗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Using Lemma 11, we establish (P4) for the special case of 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this entails that 𝑤 is closer to 𝑝ℓ, yet 𝑃 is not shorter than 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is accounted for by the case distinction in the definition of Δℓ, which covers the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ, then (P4) holds for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by Δ𝑣 ∈ {−1, 0, 1} the value such that 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅Δ𝑣 − 𝜅 according to Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemmas 3 and 11, 𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 > 𝑡𝑣,ℓ−1 − 𝑡𝑤ℓ,ℓ + 𝑑 − 𝑢 + Λ − 𝑑 𝜗 ≥ 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − 𝑡𝑤ℓ,ℓ + 𝑑 − 𝑢 + Λ − 𝑑 𝜗 ≥ 𝑡𝑣ℓ,ℓ − Λ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − 𝑡𝑤ℓ,ℓ + 𝑑 − 𝑢 + Λ − 𝑑 𝜗 =𝑡𝑣ℓ,ℓ − 𝑡𝑤ℓ,ℓ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 2 ≥ (4𝑠 − 2)𝜅(Δℓ + Δ𝑣) +𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − (ℓ − 1))𝜅 2 + 𝜅𝑠|𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We claim that Δ ≤ Δℓ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that plugging this into the above inequality yields 𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ (4𝑠 − 2)𝜅Δ +𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) + ( ¯ℓ − (ℓ − 1))𝜅 2 + 𝜅𝑠|𝑃 ′|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P4) for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, proving the above claim will complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To show the claim, we first note that 𝑃 ′ = (𝑝′,𝑤) = (𝑤ℓ, 𝑝ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑤 = 𝑝ℓ, by Lemma 8 we also have that 𝑃ℓ = (𝑝ℓ,𝑤ℓ) = (𝑤, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, |𝑃ℓ| = |𝑃 ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃ℓ and 𝑄ℓ share the first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that |𝑄ℓ| ≥ 2, as otherwise 𝑣ℓ = 𝑤ℓ, contradicting (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑣 = 𝑝′, then |𝑄 ′| = |𝑄| = |(𝑣)| = 0 ≤ |𝑄ℓ| − 2 ≤ |𝑄ℓ| + Δ𝑣 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, the first edge of𝑄 is the first edge of𝑄ℓ and thus 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This edge is {𝑝ℓ,𝑤ℓ} = {𝑝ℓ, 𝑝′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑄 ′| = | suffix(𝑄, 𝑝′)| ≤ |𝑄| − 1 = |𝑄ℓ| + Δ𝑣 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Either way, we get that Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 − 1 = Δℓ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃ℓ and 𝑄ℓ do not share the first edge, but 𝑃 ′ and 𝑄 ′ do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Δ = |𝑃 ′| + |𝑄 ′| − 1 ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 − 1 = Δℓ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑃ℓ and 𝑄ℓ do not share the first edge and neither do 𝑃 ′ and 𝑄 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As the first (and only) edge of 𝑃 ′ is {𝑝′,𝑤} = {𝑝′, 𝑝ℓ}, this entails that 𝑄 ′ = suffix(𝑄, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – | suffix(𝑄, 𝑝′)| ≤ |𝑄| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Δ = |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄| − 1 ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 26 Christoph Lenzen and Shreyas Srinivas – | suffix(𝑄, 𝑝′)| = |𝑄| and 𝑣ℓ ≠ 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑝′ is the last node on 𝑄, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', 𝑣 = 𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As by Observa- tion 4 𝑄 ′ is a path from 𝑝′ to 𝑣, it follows that |𝑄 ′| = 0 < |𝑄ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Δ = |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – | suffix(𝑄, 𝑝′)| = |𝑄| and 𝑣ℓ = 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑤 = 𝑝ℓ and 𝑝′ = 𝑣 = 𝑤ℓ as in the previous subcase, this contradicts Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Before proceeding to the case that 𝑣 ≠ 𝑤ℓ or 𝑤 ≠ 𝑣ℓ, we prove another helper statement ruling out the specific case that 𝑣 ≠ 𝑝′ = 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It is not possible that 𝑣 ≠ 𝑝′ = 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that 𝑣 ≠ 𝑝′ = 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑝′ = 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemmas 8 and 9 yield that 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 ≥ 𝑡𝑤,ℓ−1 − 4𝑠𝜅 and 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ ≤ 𝑡𝑝′,ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using (P3) and Lemma 3, it follows that 0 = 𝑡𝑝′,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ − � 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 � − 4𝑠𝜅 ≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 4𝑠𝜅 − 𝜅 2 ≥ 𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 2 > 0, arriving at the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ We now establish (P4) for the case that 𝑣 ≠ 𝑤ℓ or 𝑤 ≠ 𝑣ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑝′ ≠ 𝑤ℓ or 𝑤 ≠ 𝑝ℓ, then (P4) holds for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by Δ𝑤, Δ𝑣 ∈ {−1, 0, 1} the values such that 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ ≥ 𝑡𝑤,ℓ−1 + 4𝑠𝜅Δ𝑤 𝑡𝑣ℓ,ℓ−1 − C𝑣ℓ,ℓ ≤ 𝑡𝑣,ℓ−1 − (4𝑠 − 2)𝜅Δ𝑣 − 𝜅 according to Lemmas 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using (P4) and Lemma 3, we bound 𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑣ℓ,ℓ−1 − C𝑣,ℓ 𝜗 + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − (𝑡𝑤ℓ,ℓ−1 − C𝑤,ℓ − 4𝑠𝜅Δ𝑤) ≥ 𝑡𝑣ℓ,ℓ + (4𝑠 − 2)𝜅Δ𝑣 + 𝜅 − 𝑡𝑤ℓ,ℓ + 4𝑠𝜅Δ𝑤 − 𝜅 2 ≥ (4𝑠 − 2)𝜅(Δℓ + Δ𝑣 + Δ𝑤) + ( ¯ℓ − (ℓ − 1))𝜅 2 + 𝜅𝑠 (|𝑃ℓ| + Δ𝑤) ≥ (4𝑠 − 2)𝜅(Δℓ + Δ𝑣 + Δ𝑤) + ( ¯ℓ − (ℓ − 1))𝜅 2 + 𝜅𝑠|𝑃 ′|, where the last step exploits that |𝑃 ′| = | suffix(𝑃, 𝑝′)| ≤ |𝑃| ≤ |𝑃ℓ| + Δ𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We claim that Δ ≤ Δℓ + Δ𝑣 + Δ𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proving this claim will complete the proof, as by the above inequality then 𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ (4𝑠 − 2)𝜅Δ + ( ¯ℓ − (ℓ − 1))𝜅 2 + 𝜅𝑠|𝑃 ′|, Gradient TRIX 27 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P4) for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Observation 3 and the prerequisites of the lemma, 𝑃 ′ = suffix(𝑃, 𝑝′) or 𝑝′ = 𝑣 = 𝑤ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To cover the possibility that 𝑃 ′ = suffix(𝑃, 𝑝′), we distinguish several cases: 𝑝′ = 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑃 ′ = 𝑃 and 𝑄 ′ = 𝑄, as 𝑝′ is the first node of both 𝑃 and 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑃 ′| + |𝑄 ′| = |𝑃| + |𝑄| ≤ |𝑃ℓ| + |𝑄ℓ| + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ do not share their first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Δ ≤ |𝑃 ′| + |𝑄 ′| = |𝑃ℓ| + |𝑄ℓ| + Δ𝑤 + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ, 𝑄ℓ, and 𝑄 ′ share the same first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 13, 𝑤 ≠ 𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 𝑃 ′ = 𝑃 ≠ (𝑝′), which means that 𝑃ℓ and 𝑃 ′ have the same first edge, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑄 ′ and 𝑃 ′ have the same first edge as well, and Δ = |𝑃 ′| + |𝑄 ′| − 1 = |𝑃ℓ| + |𝑄ℓ| − 1 + Δ𝑤 + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ have the same first edge, but 𝑄 ′ does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑄ℓ ≠ (𝑝ℓ), we have that 𝑣ℓ ≠ 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By (P5), this implies that 𝑣ℓ ∉ 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, 𝑣ℓ cannot be part of the first edge of 𝑄ℓ and |𝑄ℓ| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑝′ = 𝑝ℓ, 𝑄 and 𝑄 ′ both start with 𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 𝑄 ′ is a prefix of 𝑄ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, 𝑄ℓ has the same first edge as 𝑃ℓ, while 𝑄 ′ does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, |𝑄 ′| = 0 ≤ |𝑄ℓ| + Δ𝑣 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Δ = |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + |𝑄ℓ| − 1 + Δ𝑤 + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑣 = 𝑝′ ≠ 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑄 ′ = (𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, by the prerequisites of the lemma, 𝑃 ′ = suffix(𝑃, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑝′ ≠ 𝑝ℓ, we have that | suffix(𝑃, 𝑝′)| ≤ |𝑃| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By construction, |𝑄 ′| ≤ |𝑄| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Overall, Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃| − 1 + |𝑄| + 1 = |𝑃ℓ| + Δ𝑤 + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝑣 ≠ 𝑝′ ≠ 𝑝ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝑝′ = 𝑝 and by Lemma 9 {𝑝ℓ, 𝑝′} is the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑃 ′| = | suffix(𝑃, 𝑝′)| ≤ |𝑃| − 1 ≤ |𝑃ℓ| + Δ𝑤 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ do not share their first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + Δ𝑤 + |𝑄ℓ| + Δ𝑣 = Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – 𝑃ℓ and 𝑄ℓ share their first edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑣 ≠ 𝑝′, 𝑄 has the same first edge as 𝑄ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', {𝑝ℓ, 𝑝′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, |𝑄 ′| = | suffix(𝑄, 𝑝′| = |𝑄| − 1 ≤ |𝑄ℓ| + Δ𝑣 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Δ ≤ |𝑃 ′| + |𝑄 ′| ≤ |𝑃ℓ| + Δ𝑤 + |𝑄ℓ| + Δ𝑣 − 2 < Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to consider the case that 𝑃 ′ ≠ suffix(𝑃, 𝑝′) and 𝑝′ = 𝑣 = 𝑤ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then |𝑄 ′| = (𝑣) and |𝑃 ′| = |(𝑝′,𝑤)| = 1, implying that Δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By (P2), 𝑣ℓ ≠ 𝑤ℓ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Δ𝑣 = 1, then Δ = 1 ≤ Δℓ ≤ Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 7, the remaining case is that Δ𝑣 = −1 and {𝑣, 𝑣ℓ} is the last edge of 𝑄ℓ or the first edge of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 10, it is impossible that 𝑣 = 𝑤ℓ, so this edge must be the last one of 𝑄ℓ and distinct from the first one of 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, by the prerequisites of the lemma, 𝑝ℓ ≠ 𝑤, so it must hold that |𝑃ℓ| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Overall, either |𝑄ℓ| ≥ 2 and Δ = 1 ≤ |𝑃ℓ| + |𝑄ℓ| − 3 ≤ Δℓ − 2 = Δℓ + Δ𝑤 + Δ𝑣, or |𝑄ℓ| = 1 and 𝑄ℓ and 𝑃ℓ do not share the first edge, yielding Δ = 1 ≤ |𝑃ℓ| + |𝑄ℓ| − 2 = Δℓ − 2 = Δℓ + Δ𝑤 + Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P4) and (P2) hold for 𝑣, 𝑝′, 𝑤, |𝑃 ′|, |𝑄 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 28 Christoph Lenzen and Shreyas Srinivas Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Follows from Lemma 12, Lemma 14, and Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ It remains to prove (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P3) holds for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑣 = 𝑝′, the statement readily follows from Corollary 1 and Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, assume that 𝑣 ≠ 𝑝′ and hence 𝑝′ = 𝑝 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by Δ𝑤 ∈ {−1, 0, 1} the value such that 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ ≥ 𝑡𝑤,ℓ−1 + 4𝑠𝜅Δ𝑤 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ ≤ 𝑡𝑝′,ℓ−1 according to Lemmas 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using (P3) and Lemma 3, it follows that 𝑡𝑝′,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ − � 𝑡𝑤ℓ,ℓ−1 − C𝑤ℓ,ℓ 𝜗 � + Δ𝑤4𝑠𝜅 ≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ + 4𝑠𝜅Δ𝑤 − 𝜅 2 ≥ 4𝑠𝜅(|𝑃ℓ| + Δ𝑤) +𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝑃 ′ = suffix(𝑃, 𝑝′), then |𝑃 ′| ≤ |𝑃| ≤ |𝑃ℓ| + Δ𝑤 and (P3) for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1 readily follows from the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, by the assumption that 𝑣 ≠ 𝑝′ and Observation 3, it holds that 𝑝′ = 𝑤ℓ and 𝑤 = 𝑝ℓ, and |𝑃 ′| = |𝑃ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using Lemmas 3 and 11 together with (P3), we arrive at 𝑡𝑝′,ℓ−1 − 𝑡𝑤,ℓ−1 ≥ 𝑡𝑝′,ℓ−1 − 𝑡𝑤ℓ,ℓ + � 𝑑 − 𝑢 + Λ − 𝑑 𝜗 � ≥ 𝑡𝑝ℓ,ℓ−1 − C𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ + � 𝑑 − 𝑢 + Λ − 𝑑 𝜗 � ≥ 𝑡𝑝ℓ,ℓ − 𝑡𝑤ℓ,ℓ − 𝜅 2 ≥ 4𝑠𝜅|𝑃ℓ| +𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 2 ≥ 4𝑠𝜅|𝑃 ′| +𝜓𝑠 𝑣 ¯ℓ,𝑤 ¯ℓ ( ¯ℓ ) − ( ¯ℓ − (ℓ − 1))𝜅 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P3) for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Finally, using these results it is not hard to show that (P5) is satisfied as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (P5) holds for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that 𝑣 lies on 𝑃 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 1, 𝑣 ≠ 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if 𝑃 ′ = (𝑝′,𝑤), 𝑣 = 𝑝′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', (P5) holds for 𝑣, 𝑝′, |𝑃 ′|, and layer ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Otherwise, 𝑃 ′ = suffix(𝑃, 𝑝′), implying that 𝑣 lies on suffix(𝑃, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝑣 ≠ 𝑤, this implies that 𝑣 lies on 𝑃ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assuming for contradiction that 𝑣 ≠ 𝑝′ = 𝑝, by Lemma 9 we have that prefix(𝑃, 𝑝′) = prefix(𝑃ℓ, 𝑝′), which equals either (𝑝ℓ) = (𝑝′) or (𝑝ℓ, 𝑝′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, the above entails that 𝑣 actually lies on suffix(𝑃ℓ, 𝑝′) = suffix(𝑃ℓ, 𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As then 𝑝′ = 𝑣, this is a contradiction and we must indeed have that 𝑝′ = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the proof of Theorem 1, it must hold that ℓ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assuming for contradiction that ℓ > ℓ, Corollary 1, Lemmas 15 and 16, and Observation 2 show that layer ℓ − 1 also satisfies the properties (P1) to (P5) for some 𝑣ℓ−1, 𝑝ℓ−1,𝑤ℓ−1, and paths 𝑃ℓ−1, 𝑄ℓ−1, contradicting the minimality of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Bounding Skews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With our machinery for bounding Ψ𝑠 in place, it remains to perform the induction on 𝑠 ∈ N>0 to wrap things up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To anchor the induction at 𝑠 = 1, we exploit that Ψ1(ℓ) ≤ Ξ1(ℓ)+2𝜅𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Ψ1(ℓ) ≤ � Ξ1(0) if ℓ < 4Ξ1(0)/𝜅 4𝜅𝐷 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that 𝜅 = 2(𝑢 + (1 − 1/𝜗)(Λ − 𝑑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Ξ1(ℓ) ≤ Ψ1(ℓ) + 2𝜅𝐷 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Theorem 1, we thus have for any ℓ ≤ ¯ℓ that Ψ1( ¯ℓ ) ≤ max � 0, Ξ1(ℓ ) − ( ¯ℓ − ℓ + 1)𝜅 � + ( ¯ℓ − ℓ )𝜅 2 ≤ max � 0, Ψ1(ℓ ) + 2𝜅𝐷 − ( ¯ℓ − ℓ + 1)𝜅 � + ( ¯ℓ − ℓ )𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, we have that Ψ1(ℓ) ≤ � max � 4𝜅𝐷, Ξ1(0) � if ℓ < 8𝐷 max � 4𝜅𝐷, Ψ1(ℓ − 8𝐷) − 2𝜅𝐷 � else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By induction on 𝑘 ∈ N, we thus have that Ψ1(ℓ) ≤ max{4𝜅𝐷, Ξ1(0) − 2𝑘𝜅𝐷} for all ℓ ∈ [8𝑘𝐷, 8(𝑘 + 1)𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim of the lemma follows by noting that ℓ ≥ 4Ξ1(0)/𝜅 results in 𝑘 ≥ Ξ1(0)/(2𝜅𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Note that this lemma shows that Ψ1 self-stabilizes [5] within 𝑂(Ξ1(0)/𝜅) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We remark that a more careful analysis reveals a bound on Ψ1(ℓ) that converges to 2𝜅𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We confine ourselves to stating this result for the small input skew that we guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If L0 ≤ 4𝜅, then Ψ1(ℓ) ≤ 2𝜅𝐷 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Ξ1(0) = max 𝑣,𝑤∈𝑉{𝑡𝑣,0 − 𝑡𝑤,0 − 2𝜅𝑑(𝑣,𝑤)} ≤ max 𝑣,𝑤∈𝑉{(L0 − 2𝜅)𝑑(𝑣,𝑤)} ≤ (L0 − 2𝜅)𝐷 ≤ 2𝜅𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By replacing 8𝐷 with 4𝐷 in the induction from the proof of Lemma 17, we get that Ψ1(ℓ) ≤ � max � 2𝜅𝐷, Ξ1(0) � if ℓ < 4𝐷 max � 2𝜅𝐷, Ψ1(ℓ − 4𝐷) � else, implying a uniform bound of Ψ1(ℓ) ≤ 2𝜅𝐷 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ For the sake of completeness, we also infer that supℓ ∈N{Ψ0(ℓ)}, also referred to as the global skew in the literature, is in 𝑂(𝑢 + (1 − 1/𝜗)(Λ −𝑑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Provided that Λ ∈ 𝑂(𝑑 +𝑢/(𝜗 − 1)), this bound is asymptotically optimal [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If L0 ≤ 4𝜅, then Ψ0(ℓ) ≤ 6𝜅𝐷 ∈ 𝑂(𝑢 + (1 − 1/𝜗)(Λ − 𝑑)) for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Follows from Corollary 3, the fact that Ψ0(ℓ) ≤ Ψ1(ℓ) + 4𝜅𝐷, and the choice of 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ In order to bound the local skew, we now turn to attention to Ψ𝑠 (ℓ) for 𝑠 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 30 Christoph Lenzen and Shreyas Srinivas Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For some 𝑠 ∈ N, 𝑠 > 0, suppose that Ψ𝑠−1(ℓ) ≤ Ψ𝑠−1 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Ψ𝑠 (ℓ) ≤ � Ξ𝑠 (0) + Ψ𝑠−1 2 if ℓ < Ψ𝑠−1/𝜅 Ψ𝑠−1 2 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Recall that 𝜅 = 2(𝑢 + (1 − 1/𝜗)(Λ − 𝑑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For ℓ < Ψ𝑠−1/𝜅, by Theorem 1 with ¯ℓ = ℓ and ℓ = 0 we have that Ψ𝑠 (ℓ) ≤ Ξ𝑠 (0) + 𝜅ℓ 2 ≤ Ξ𝑠 (0) + Ψ𝑠−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that Ξ𝑠 (ℓ) ≤ Ψ𝑠−1(ℓ) ≤ Ψ𝑠−1 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, for ℓ ≥ Ψ𝑠−1/𝜅 by Theorem 1 with ¯ℓ = ℓ and ℓ = ℓ − ⌊Ψ𝑠−1/𝜅⌋ we have that Ψ𝑠 (ℓ) ≤ max � 0, Ξ𝑠 � ℓ − � Ψ𝑠−1 𝜅 �� − �� Ψ𝑠−1 𝜅 � + 1 � 𝜅 � + � Ψ𝑠−1 𝜅 � 𝜅 2 ≤ Ψ𝑠−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Using this lemma, we can bound the local skew by 𝑂(𝜅(1 + log 𝐷)) = 𝑂((𝑢 + (1 − 1/𝜗)(Λ − 𝑑))(1 + log 𝐷)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there are no faults, then Lℓ ≤ 4𝜅(2 + log 𝐷) for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 27, L0 ≤ 4𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 3, Ψ1(ℓ) ≤ 2𝜅𝐷 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the assumption that L0 ≤ 4𝜅, for all 𝑠 > 1 we have that Ξ𝑠 (0) = max 𝑣,𝑤∈𝑉{𝑡𝑣,0 − 𝑡𝑤,0 − (4𝑠 − 2)𝜅𝑑(𝑣,𝑤)} ≤ max 𝑣,𝑤∈𝑉{(L0 − 6𝜅)𝑑(𝑣,𝑤)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, inductive use of Lemma 18 yields that Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, Ψ⌊log 𝐷⌋ ≤ 8𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim now follows by Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Moreover, in addition we obtain the following self-stabilization property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If for 𝑠,𝑠′ ∈ N, 𝑠 ≤ 𝑠′, we have that Ψ𝑠 (ℓ) ≤ Ψ𝑠 for all ℓ ≥ ℓ ∈ N, then for ℓ ≥ ℓ Lℓ ≤ � 4𝑠𝜅 + Ψ𝑠 if ℓ ≤ ℓ < ℓ + 2Ψ𝑠/𝜅 and 4𝑠′𝜅 + Ψ𝑠 2𝑠′−𝑠 if ℓ ≥ ℓ + 2Ψ𝑠/𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Inductive use3 of Lemma 18 yields for 𝑠′ ≥ 𝑠 and ℓ ≥ ℓ + �𝑠′ 𝜎=𝑠+1 Ψ𝑠/(2𝜎−𝑠𝜅) that Ψ𝑠′ ≤ Ψ𝑠 2𝑠′−𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since the sum forms a geometric series, this in particular applies to all ℓ ≥ ℓ + 2Ψ𝑠/𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim now follows by applying Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='4 Bounding Skews in the Presence of Faults To analyze how skews evolve with faults, we relate the setting with faults to the bounds we have for a fault-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key property the algorithm guarantees is that, up to an additive 2𝜅, the pulse time is within the interval spanned by the correct predecessors’ pulse times plus Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first show this for the case that for some node (𝑣, ℓ), (𝑣, ℓ − 1) is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 3As is, the lemma applies only if ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, the algorithm and hence all statements are invariant under shifting indices by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 31 Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that the only faulty predecessor of (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, is (𝑣, ℓ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote 𝑡min := min {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} and 𝑡max := max {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑡min + Λ − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡max + Λ + 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the assumption of the lemma, for all {𝑣,𝑤} ∈ 𝐸, (𝑤, ℓ − 1) ∉ 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We have that 𝐻own − 𝐻max = min 𝑠 ∈N {𝐻own − 𝐻max + 4𝑠𝜅} ≤ min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} ≤ max{𝐻own − 𝐻max, 𝐻own − 𝐻min} = 𝐻own − 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, abbreviating Δ = min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2, it holds that 𝐻own − 𝐻max − 𝜅 2 ≤ Δ ≤ 𝐻own − 𝐻min − 𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Taking into account the adjustments in case Δ ∉ [0,𝜗𝜅] and using that 𝐻min ≤ 𝐻max we get that 𝐻own − 𝐻max − 3𝜅 2 ≤ C𝑣,ℓ ≤ 𝐻own − 𝐻min + 3𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, the local time 𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ at which (𝑣, ℓ) generates its pulse satisfies 𝐻min + Λ − 𝑑 − 3𝜅 2 ≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) ≤ 𝐻max + Λ − 𝑑 + 3𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻min > 𝐻𝑣,ℓ (𝑡𝑣,ℓ), we have that 𝑡min − 𝑡𝑣,ℓ ≤ 𝐻min − 𝐻𝑣,ℓ (𝑡𝑣,ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Applying the lower bound of 𝑑 − 𝑢 on message delay and Equation (1), we get that 𝑡𝑣,ℓ ≥ 𝑡min + 𝑑 − 𝑢 + Λ − 𝑑 − 3𝜅 2 > 𝑡min + Λ − 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If 𝐻min ≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ), the bounds on message delays and hardware clock drift together with Equa- tion (1) yield that 𝑡𝑣,ℓ ≥ 𝑡min + 𝑑 − 𝑢 + Λ − 𝑑 − 3𝜅/2 𝜗 > 𝑡min + Λ − 3𝜅 2 − 𝑢 − � 1 − 1 𝜗 � (Λ − 𝑑) = 𝑡min + Λ − 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Concerning the upper bound on 𝑡𝑣,ℓ, note that because 𝑡𝑣,ℓ is increasing in 𝐻𝑣,ℓ (𝑡𝑣,ℓ), to bound 𝑡𝑣,ℓ from above we may assume that 𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻max + Λ − 𝑑 + 3𝜅 2 > 𝐻max, where the last step uses Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, 𝑡𝑣,ℓ − 𝑡max ≤ 𝐻𝑣,ℓ (𝑡𝑣,ℓ) − 𝐻max + 𝑑 ≤ Λ + 3𝜅 2 < Λ + 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 32 Christoph Lenzen and Shreyas Srinivas Similar reasoning covers the case that for some (𝑣, ℓ) ∈ 𝑉ℓ and {𝑣,𝑤} ∈ 𝐸, (𝑤, ℓ − 1) is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that for (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, (𝑣, ℓ − 1) is not faulty, and at most one predecessor is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denoting 𝑡min := min ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 (𝑤,ℓ−1)∉𝐹 {𝑡𝑤,ℓ−1} and 𝑡max := max ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 (𝑤,ℓ−1)∉𝐹 {𝑡𝑤,ℓ−1}, then 𝑡min + Λ − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡max + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 3, C𝑣,ℓ ≥ 0 implies that 𝑡𝑣,ℓ − 𝑡min ≤ 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 ≤ Λ, while C𝑣,ℓ ≤ 𝜗𝜅 yields that 𝑡𝑣,ℓ − 𝑡max ≥ 𝑡𝑣,ℓ − 𝑡𝑣,ℓ−1 ≥ 𝑑 − 𝑢 + Λ − 𝑑 𝜗 − 𝜅 ≥ Λ − 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to show the upper bound on 𝑡𝑣,ℓ if C𝑣,ℓ < 0 and the lower bound if C𝑣,ℓ > 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider first the case that C𝑣,ℓ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, C𝑣,ℓ = 𝐻own − 𝐻min − 𝜅 2 + 2𝜅 > 𝐻own − 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that 𝐻𝑣,ℓ (𝑡𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ ≤ 𝐻min + Λ − 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Noting that the reception time of the first message from a predecessor is bounded from above by the reception time of the message from a correct predecessor, we conclude that 𝑡𝑣,ℓ ≤ 𝑡min + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now consider the case that C𝑣,ℓ > 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consequently, C𝑣,ℓ = 𝐻own − 𝐻max − 𝜅 2 − 𝜅 > 𝐻own − 𝐻max − 𝜗𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that the local time 𝐻 at which (𝑣, ℓ) generates its pulse satisfies that 𝐻 = 𝐻own + Λ − 𝑑 − C𝑣,ℓ ≥ 𝐻max + Λ − 𝑑 + 𝜗𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Noting that the reception time of the latest message from a predecessor is bounded from below by the reception time of the latest message from a correct predecessor, by Equation (1) we conclude that 𝑡𝑣,ℓ ≥ 𝑡max + 𝑑 + Λ − 𝑑 𝜗 > 𝑡𝑣,ℓ−1 + Λ − 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote 𝑡min := min ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 (𝑤,ℓ−1)∉𝐹 {𝑡𝑤,ℓ−1} and 𝑡max := max ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 (𝑤,ℓ−1)∉𝐹 {𝑡𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝑡min + Λ − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡max + Λ + 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Immediate from Lemmas 19 and 20 and the assumption that no node has more than one faulty predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Using this result, we can bound the impact of a fault in layer ℓ − 1 on successors via the skew bounds of close-by nodes on layer ℓ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' we exploit that all bounds we show would in fact also apply to the faulty node if it was correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose for a node (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, that one of its predecessors is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, assume that in an execution that differs only in that the faulty predecessor of (𝑣, ℓ) is correct, it holds that max{𝑣,𝑤}∈𝐸{|𝑡𝑣,ℓ−1 − 𝑡𝑤,ℓ−1|} ≤ 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then in the execution with the predecessor being faulty, the pulse time of (𝑣, ℓ) differs by at most 2𝐵 + 4𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by standard variables values in the execution without the predecessor being faulty and by primed variables values in the one where it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, for node (𝑣, ℓ) ∈ 𝑉ℓ \\ 𝐹 𝑡min := min ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1{𝑡𝑤,ℓ−1}, 𝑡max := max ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 {𝑡𝑤,ℓ−1}, 𝑡 ′ min := min ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 (𝑤,ℓ−1)∉𝐹 {𝑡𝑤,ℓ−1}, and 𝑡 ′ max := max ((𝑤,ℓ−1),(𝑣,ℓ)) ∈𝐸ℓ−1 (𝑤,ℓ−1)∉𝐹 {𝑡𝑤,ℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' denote the earliest and latest pulsing times of (correct) predecessors without and with faults on layer ℓ − 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observe that 𝑡𝑣,ℓ−1 − 𝐵 ≤ 𝑡min ≤ 𝑡min′ ≤ 𝑡max′ ≤ 𝑡max ≤ 𝑡𝑣,ℓ−1 + 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, Corollary 5 (applied to both executions) shows that 𝑡𝑣,ℓ−1 − 𝐵 − 2𝜅 ≤ 𝑡𝑣,ℓ ≤ 𝑡𝑣,ℓ−1 + 𝐵 + 2𝜅 and 𝑡𝑣,ℓ−1 − 𝐵 − 2𝜅 ≤ 𝑡 ′ 𝑣,ℓ ≤ 𝑡𝑣,ℓ−1 + 𝐵 + 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Finally, we observe that such a “time shift” propagates without further increase, so long as there are no faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, a subtlety here is that this is only true for our bounds on timing: a change in timing might leave more time for drift of the local clock to accumulate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' since our worst-case bounds include the maximum time error that can possibly be accumulated from drift (so long as local skews do not become exceedingly large), this is already accounted for in the bound provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, we obtain the following generalized variant of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that for 𝑣 ∈ 𝑉 and ℓ ∈ N>0 the predecessors of (𝑣, ℓ) are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If we shift the pulse times of these predecessors by at most 𝛿 ∈ R, where Equation (2) still holds for the shifted times, then 𝑑 − 𝑢 + Λ − 𝑑 − C𝑣,ℓ 𝜗 − 𝛿 ≤ 𝑡 ′ 𝑣,ℓ − 𝑡𝑣,ℓ−1 ≤ Λ − C𝑣,ℓ + 𝛿, where 𝑡 ′ 𝑣,ℓ denotes the pulse time of (𝑣, ℓ) in the execution with the shifts applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Pulse times are increasing as functions of pulse times of predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, in order to maximize or minimize 𝑡 ′ 𝑣,ℓ, we need to maximize or minimize the predecessors’ pulse times, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Shifting all predecessors’ pulse times uniformly by 𝛿 also shifts 𝑡 ′ 𝑣,ℓ by 𝛿 relative to 𝑡𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The statement now follows analogously to the proof of Lemma 3, carrying the uniform shift through all inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 34 Christoph Lenzen and Shreyas Srinivas With these tools in place, we can conclude that skews do not grow arbitrarily in the face of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there are at most 𝑓 faulty nodes in the system and none in layer 0, then Lℓ ∈ 𝑂(5𝑓 𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove by induction on the number 𝑖 ≤ 𝑓 of layers ℓ > 0 with faults that the skew is bounded by 𝐵𝑖 := 4𝜅(2 + log 𝐷)5𝑖 �𝑖 𝑗=0 5−𝑗 ∈ 𝑂(5𝑓 𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 6, L0 ≤ 𝜅/2 < 4𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, if there are no faults in layers ℓ > 0, by Theorem 2 we have that Lℓ ≤ 𝐵0 := 4𝜅(2 + log 𝐷) for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume that we completed step 𝑖 ∈ N and that ℓ𝑖+1 is the next layer where faults need to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then we have that for all ℓ ≤ ℓ𝑖+1 that Lℓ′ ≤ 𝐵𝑖 = 4𝜅(2 + log 𝐷)5𝑓 �𝑖 𝑗=0 5−𝑗 both before and after adding the faults on layer 𝑖 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 21, it follows that pulsing times on layer ℓ𝑖+1 + 1 do not change by more than 2𝐵𝑖 + 4𝜅 due to the addition of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 22, this extends to all bounds4 we compute on pulse times in layers ℓ > ℓ𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝐷 ≥ 1 and thus log 𝐷 ≥ 0, we get that the local skew in step 𝑖 + 1 is bounded by 5𝐵𝑖 + 4𝜅 = 4𝜅(2 + log 𝐷)5𝑖+1 𝑖∑︁ 𝑗=0 5−𝑗 + 4𝜅 ≤ 4𝜅(2 + log 𝐷)5𝑖+1 𝑖+1 ∑︁ 𝑗=0 5−𝑗 = 𝐵𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Bounding Skews with Uniform Fault Distribution The bound in Theorem 4, which is exponential in 𝑓 , seems to suggest that the system can only support a very small number of faults or the local skew explodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we have not yet taken into account that the starting point of our entire approach is the assumption that faults are sufficiently sparse, meaning that it is highly unlikely that many of them cluster together in a way that causes an exponential pile-up of local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This enables the self-stabilization properties of the algorithm to prevent such a build-up altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In the following, assume that each node fails uniformly and independently with probability 𝑜(𝑛−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is the largest probability of error we can support while guaranteeing that no node has more than one faulty predecessor with probability 1 − 𝑜(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' A key observation is that this entails that within a fairly large distance of 𝑛1/12, no node has more than a constant number of faulty nodes that can influence it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now formalize and show this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 5 (Distance-𝛿 Ancestors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For node (𝑣, ℓ) ∈ 𝑉ℓ and 𝛿 ∈ N, its distance-𝛿 ancestors are all nodes (𝑤, ℓ′) ∈ 𝑉𝐺 \\ {(𝑣, ℓ)} such that there is a (directed) path of length at most 𝛿 from (𝑤, ℓ′) to (𝑣, ℓ) in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 6 (Distance-𝛿 𝑘-faulty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Node (𝑣, ℓ) ∈ 𝑉ℓ, ℓ ∈ N>0 is distance-𝛿 𝑘-faulty if 𝑘 ∈ N is minimal such that there are at most 𝑘 faulty nodes among the distance-((𝑘 + 1)𝛿) ancestors of (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that 𝛿 ≤ 𝑛1/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If nodes fail independently with probability 𝑝 ∈ 𝑜(1/√𝑛), then with probability 1 − 𝑜(1) all nodes are distance-𝛿 𝑘-faulty for 𝑘 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In order to be distance-𝛿 𝑘-faulty for 𝑘 > 2, a node must have at least 3 faults among its distance-(3𝛿) ancestors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The number of these ancestors is bounded by (3𝛿)2 ∈ 𝑂(𝑛1/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝑝 ∈ 𝑜(1/√𝑛), the probability for this to happen is bounded by 𝑂(𝑝3�𝑛1/6 3 �) = 𝑂(𝑝3√𝑛) ⊂ 𝑜(1/𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The claim follows by applying a union bound over all 𝑛 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 4Due to drifting hardware clocks, this does not apply to the pulse times themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we rely on Lemma 3 to prove our bounds in the absence of faults, and this is covered by Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 35 We can exploit this to control how much skews grow as the result of faults much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that Ψ𝑠 (ℓ) ≤ 𝐵𝑠,ℓ and Lℓ ≤ 𝐵 for all layers ℓ ≥ ℓ and 𝑠 ∈ N, where ℓ, ℓ ∈ N, if there are no faults in these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If no node in a layer ℓ ≥ ℓ has more than 2 faulty nodes among its distance-(ℓ − ℓ ) ancestors, then Ψ𝑠 (ℓ) ≤ 𝐵𝑠,ℓ + 12𝐵 + 24𝜅 for all ℓ ≥ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We examine by how much adding faults on layers ℓ ≥ ¯ℓ might affect pulsing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For ℓ ≥ ¯ℓ and (𝑣, ℓ) ∈ 𝑉ℓ, denote by 𝑓𝑣,ℓ ∈ {0, 1, 2} the number of faulty distance-(ℓ − ¯ℓ) ancestors of (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑓𝑣,ℓ = 0, there is no change in 𝑡𝑣,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑓𝑣,ℓ > 0, consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If (𝑣, ℓ) has no faulty predecessor, then by Lemma 22, 𝑡𝑣,ℓ is changed at most by the maximum shift that any of its predecessors undergoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On the other hand, if (𝑣, ℓ) does have a faulty predecessor, then 𝑓𝑣,ℓ > 𝑓𝑤,ℓ−1 for all correct predecessors of (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, by Lemma 21 we can bound shifts by 𝐵𝑓𝑣,ℓ , where 𝐵0 := 0 and 𝐵𝑓 +1 := 2(𝐵 + 𝐵𝑓 ) + 4𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By assumption, 𝑓𝑣,ℓ ≤ 2 and hence the maximum shift is bounded by 𝐵2 = 6𝐵 + 12𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Ψ𝑠 (ℓ) ≤ 𝐵𝑠,ℓ + 2𝐵2 = 𝐵𝑠,ℓ + 12𝐵 + 24𝜅, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Together with Lemma 23, Observation 5 shows that skews do not increase by more than a constant factor within 𝑛1/12 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we need to handle a total of Θ(√𝑛) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, we slice up the task into chunks of 𝑛1/12 layers and leverage the self-stabilization properties of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For simplicity, in the following we assume that 𝑛1/12 is integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As we prove asymptotic bounds, this does not affect the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Definition 7 (Slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Slice 𝑖 ∈ N>0 consists of layers ℓ ∈ [(𝑖 − 1)𝑛1/12,𝑖𝑛1/12 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that there are no more than 𝑛5/12 slices, because the nodes are arranged in square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Due to the duplication of nodes on layer 0 and the boundary nodes on layers ℓ > 0, the number of slices is actually 𝑛5/12 − Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As our next step towards a probabilistic skew bound, we prove that if the local skew remains bounded, then for levels 𝑠 that are not too large, Ψ𝑠 remains almost as small as without faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' First, we show a loose bound that naively accumulates shifts slice by slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that L0 ≤ 4𝜅, each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2, and Lℓ ≤ 𝐵 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then for each 𝑠 ∈ N and layer ℓ in slice 𝑖 ∈ N>0, we have that Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷 + 𝑖(12𝐵 + 24𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume first that there are no faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this case, analogously to the proof of Theorem 2, we get that Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷 for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now we “add” faults inductively slice by slice, by Lemma 23 each time increasing the bound on Ψ𝑠 (ℓ) by 12𝐵 + 24𝜅 for all slices 𝑗 ≥ 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ For larger values of 𝑠, 22−𝑠𝜅𝐷 ≪ 𝑛1/12, meaning that this naive bound is insufficient to show that Ψ𝑠 (ℓ) does not increase much compared to the fault-free setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we can take things much further by leveraging Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that L0 ≤ 4𝜅, each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2, and Lℓ ≤ 𝐵 ∈ 𝑜(𝑛1/12𝜅/log 𝐷) for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 36 Christoph Lenzen and Shreyas Srinivas Then for5 𝑠 ∈ N>0, 𝑠 ≤ log 𝐷 − log(𝐵/𝜅) − 2 log log 𝐷, it holds that Ψ𝑠 (ℓ) ≤ Ψ𝑠 ∈ (1 + 𝑜(1))22−𝑠𝜅𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that 𝐷 ∈ Θ(𝑛1/2) and hence log log 𝐷 ∈ 𝜔(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, the prerequisites of the lemma ensure that 𝑛5/12(𝐵 + 𝜅) ∈ 𝑜(𝜅𝐷/log 𝐷) and 𝐵 + 𝜅 ∈ 𝑜(Ψ𝑠−1/log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, we may fix a suitable 𝜀 ∈ 𝑜(1) such that 𝑛5/12(12𝐵 + 24𝜅) ≤ 𝜀 log 𝐷 · 2𝜅𝐷 and �� Ψ𝑠−1 𝑛5/12𝜅 � + 1 � (12𝐵 + 24𝜅) ≤ 𝜀 4 log 𝐷 · Ψ𝑠−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We claim that if 𝑛 is sufficiently large such that 𝜀 ≤ 1, we have that Ψ𝑠 (ℓ) ≤ Ψ𝑠 := 22−𝑠𝜅𝐷 · � 1 + 𝜀𝑠 log 𝐷 � , which we show by induction on 𝑠 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the base case of 𝑠 = 1, note that there are no more than 𝑛5/12 slices, yielding by Lemma 24 that Ψ1(ℓ) ≤ 2𝜅𝐷 + 𝑛5/12(12𝐵 + 24𝜅) ≤ � 1 + 𝜀 log 𝐷 � 2𝜅𝐷, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', indeed Ψ1(ℓ) ≤ Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now assume that the claim holds for 𝑠 − 1 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then, by Lemma 24 and the induction hypothesis, for layers ℓ in slices 𝑖 ≤ ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉, we have that Ψ𝑠 (ℓ) ≤ 22−𝑠𝜅𝐷 + � Ψ𝑠−1 𝑛1/12𝜅 � (12𝐵 + 24𝜅) < Ψ𝑠−1 2 + �� Ψ𝑠−1 𝑛1/12𝜅 � + 1 � (12𝐵 + 24𝜅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For a layer ℓ in a slice 𝑖 > ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉, assume first that we add only faults in slices 𝑗 < 𝑖 − ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, we can apply Lemma 18, shifting layer indices such that “layer 0” is the first layer of slice 𝑖 − ⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In this setting, we thus have that Ψ𝑠 (ℓ) ≤ Ψ𝑠−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We now apply Lemma 23 inductively to slices 𝑗 ∈ [𝑖−⌈(Ψ𝑠−1/(𝑛1/12𝜅)⌉,𝑖], adding in total (⌈Ψ𝑠−1/(𝑛1/12𝜅)⌉+ 1)(12𝐵 + 24𝜅) to the bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', Ψ𝑠 (ℓ) ≤ Ψ𝑠−1 2 + �� Ψ𝑠−1 𝑛1/12𝜅 � + 1 � (12𝐵 + 24𝜅) ≤ �1 2 + � 𝜀 4 log 𝐷 �� Ψ𝑠−1 = �1 2 + � 𝜀 4 log 𝐷 �� 22−(𝑠−1)𝜅𝐷 · � 1 + 𝜀(𝑠 − 1) log 𝐷 � = 22−𝑠𝜅𝐷 · � 1 + 𝜀(𝑠 − 1/2) log 𝐷 + 𝜀2 2 log2 𝐷 � ≤ 22−𝑠𝜅𝐷 · � 1 + 𝜀𝑠 log 𝐷 � , where the last step assumes that 𝑛 is large enough so that 𝜀 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ 5If 𝐷 = 1, we assume the upper bound on 𝑠 to be negative and the claim is vacuously true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that we are making an asymptotic statement in 𝑛 and that 𝐷 grows with 𝑛, so this case is actually of no concern here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 37 Our goal is to bound Ψ⌊log 𝐷⌋ by 𝑂(𝜅 log 𝐷), since by Observation 1 this implies a bound of 𝑂(𝜅 log 𝐷) on the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, we will use the above lemma with 𝐵 ∈ 𝑂(𝜅 log 𝐷), which gets us within 𝑂(log log 𝐷) levels of our “target” level ⌊log 𝐷⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To bridge this remaining gap, we exploit that the time required for stabilizing the remaining 𝑂(log log 𝐷) levels after a fault-induced increase of skews takes only log𝑂 (1) 𝐷 = log𝑂 (1) 𝑛 ⊂ 𝑜(𝑛1/12) layers, since the involved potentials are bounded by 𝑜(𝜅𝑛1/12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that L0 ≤ 4𝜅 and each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then Lℓ ∈ 𝑂(𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that the claim is false, and let ¯ℓ ∈ N>0 be minimal such that Lℓ is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, for layers ℓ < ¯ℓ, we may assume that Lℓ ≤ 𝐶𝜅 log 𝐷 for a sufficiently large constant 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider 𝑠 = ⌊log 𝐷 − log(𝐵/𝜅) − 2 log log 𝐷 − log𝐶⌋ − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Lemma 25, for all ℓ ∈ N, ℓ < ¯ℓ it holds that Ψ𝑠 (ℓ) ∈ Ψ𝑠 := (1 + 𝑜(1))22−𝑠𝜅𝐷 ⊆ �1 4 + 𝑜(1) � log3 𝐷, which for sufficiently large 𝑛 is smaller than ⌊log3 𝐷⌋/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In fact, this bound also applies to layer ¯ℓ, since the pulsing times of nodes on layer ¯ℓ depend only on the behavior of nodes on layer ¯ℓ − 1 and the delays of messages sent to nodes on layer ¯ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now assume that 𝑛 is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This ensures that log3 𝐷 ≤ 𝑛1/12, implying by the prerequisites of the lemma that each node is distance-(log3 𝐷) 𝑘-faulty for 𝑘 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider adjacent correct nodes (𝑣, ℓ), (𝑤, ℓ) ∈ 𝑉ℓ \\ 𝐹 for any ℓ ∈ N, ℓ ≤ ¯ℓ, and {𝑣,𝑤} ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We first show that distance-(log3 𝐷) 0-faulty nodes satisfy that 𝑡𝑣,ℓ − 𝑡𝑤,ℓ ∈ (4 + 𝑜(1))𝜅(2 + log 𝐷) ⊂ 𝑂(𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (6) Since faults that are not among the ancestry of a node cannot affect its pulse time, this follows by applying Theorem 3 with ℓ = ℓ − ⌊(log3 𝐷)⌋ ≤ ℓ − 2Ψ𝑠 and 𝑠′ := ⌊log 𝐷⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To extend this to distance-(log3 𝐷) 𝑘-faulty nodes for 𝑘 ∈ {1, 2}, we show by induction on 𝑘 ∈ {0, 1, 2} that such nodes have their pulse time shifted by no more than 𝑂(𝜅 log 𝐷) relative to an execution in which they are distance-(log3 𝐷) 0-faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The base case of 𝑘 = 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To perform the step from 𝑘 − 1 ∈ {0, 1} to 𝑘, assume towards a contradiction that there is a node (𝑣, ℓ) with a larger shift, on some minimal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Now consider a distance-(log3 𝐷) 𝑘-faulty node (𝑣, ℓ) ∈ 𝑉ℓ \\ 𝐹, ℓ ≤ ¯ℓ, whose predecessors are all correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There must be a distance-(log3 𝐷) ancestor of (𝑣, ℓ) that is faulty, since otherwise (𝑣, ℓ) would be distance-(log3 𝐷) 0-faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Let 𝑑 be the minimal distance in which there is a faulty ancestor of (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then all ancestors of (𝑣, ℓ) in distance 𝑑 are distance-(log3 𝐷) 𝑘′-faulty for 𝑘′ < 𝑘, as otherwise (𝑣, ℓ) would be 𝑘′-faulty for some 𝑘′ > 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider an ancestor of (𝑣, ℓ) in distance 𝑑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If its predecessors are all correct, by the induction hypothesis and Lemma 22 their pulse time is shifted by 𝑂(𝜅 log 𝐷) relative to an execution in which they are distance distance-(log3 𝐷) 0-faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If there is a faulty predecessor, we infer this from the induction hypothesis, Equation (6), and Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='6 If 𝑑 > 1, we now inductively apply Lemma 22 until having extended this bound to all ancestors of (𝑣, ℓ) within distance 𝑑 − 1 and finally (𝑣, ℓ) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is a contradiction to (𝑣, ℓ) violating the claimed bound on the shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 6Here the constants in the 𝑂-notation change, while Lemma 22 maintains the bound used in its prerequisites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since we perform only two inductive steps, we do not need to keep track of how much the constants increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 38 Christoph Lenzen and Shreyas Srinivas We conclude that indeed shifts are bounded by 𝑂(𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' From this and Equation (6), it im- mediately follows that L ¯ℓ ∈ 𝑂(𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝐶 is sufficiently large, for sufficiently large 𝑛 this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We conclude that Lℓ ∈ 𝑂(𝜅 log 𝐷) for all ℓ ∈ N, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Putting these results together, we arrive the desired bound on the local skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1 − 𝑜(1), Lℓ ∈ 𝑂(𝜅 log 𝐷) for all ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 6, with probability 1 − 𝑜(1) it holds that L0 ≤ 𝜅/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Observation 5, with probability 1 − 𝑜(1) each node is distance-𝑛1/12 𝑘-faulty for 𝑘 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By a union bound, both events occur concurrently with probability 1 − 𝑜(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, the claim follows by applying Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Gradient TRIX 39 REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Bailey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Clocks Getting Skewed Up, March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 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204–218, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [16] Clock Synchronisation and Adversarial Fault Tolerance, 2021, retrieved on 04 Jan 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='mpi-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='de/ fileadmin/inf/d1/teaching/summer21/csaft/reading-material-ch09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='org/wiki/Transistor_count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Xanthopoulos, editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Clocking in Modern VLSI Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Springer US, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 40 Christoph Lenzen and Shreyas Srinivas A GENERATING SYNCHRONIZED INPUTS In this appendix we describe a method for generating well synchronised pulses at layer 0, at a rate of roughly one pulse per Λ time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' There are several ways of approaching this task, but even when aiming for a fault-tolerant solution, this is an easy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The reason is that we merely need to maintain a small local skew on a line topology, with no alternative propagation paths to neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since our goal is to handle an independent probability of 𝑝 ∈ 𝑜(𝑛−1/2) of node failures, in fact we can simply exploit that at most √𝑛 nodes are required on layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We provide a trivial scheme that is suitable for our specific setting of the base graph 𝐺 being a line (with replicated endpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Algorithm 2 Pulse forwarding algorithm for nodes (𝑖, 0), 𝑖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' , 𝐷};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' node (0, 0) is the clock source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The parameter Λ is as described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻 := ∞ loop do if received pulse from (𝑖 − 1, 0) then 𝐻 := 𝐻𝑖,0(𝑡) until 𝐻𝑖,0(𝑡) = 𝐻 + Λ − 𝑑 broadcast pulse to (𝑖 + 1, 0) and successors on layer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For 𝑘 ∈ N, assume that the clock source at node (0, 0) generates its 𝑘-th pulse at time (𝑘 − 1)Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If all nodes on layer 0 are correct, the scheme given in the above algorithm generates pulses with local skew L0 ≤ 𝜅/2 and 𝑡𝑘 𝑖,0 ∈ [(𝑘 + 𝑖 − 1)Λ − 𝑖𝜅/2, (𝑘 + 𝑖 − 1)Λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, it stabilizes after transient faults within time 𝐷Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider first the case that there are no transient faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We prove the statement by induction on 𝑖 ∈ N, where the base case is covered by the assumptions on node 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the step from 𝑖 − 1 ∈ N to 𝑖, we perform an induction over the pulse number 𝑘 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The induction hypothesis is that pulses 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' ,𝑘 − 1 have been generated in accordance with the claim of the lemma and the first 𝑘 − 1 loop iterations at node 𝑖 have been completed by the time the 𝑘-th pulse message from node 𝑖 − 1 arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that we can use 𝑘 = 0 as base case for this induction, for which the claim is vacuously true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the step from 𝑘 − 1 ∈ N to 𝑘, denote by 𝑡 ′ 𝑖−1,𝑘 ∈ [𝑡𝑖−1,𝑘 + 𝑑 − 𝑢,𝑡𝑖−1,𝑘 + 𝑑] the reception time of the pulse message from node (0,𝑖 − 1) at node (0,𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the bounds on hardware clock rates, Equation (1), and the induction hypothesis of the induction on 𝑖, node (0,𝑖) generates its 𝑘-th pulse at time 𝑡𝑖,𝑘 ∈ � 𝑡𝑖−1,𝑘 + 𝑑 − 𝑢 + Λ − 𝑑 𝜗 ,𝑡𝑖−1,𝑘 + Λ � ⊆ � 𝑡𝑖−1,𝑘 + Λ − 𝜅 2,𝑡𝑖−1,𝑘 + Λ � ⊆ � (𝑘 + 𝑖 − 1)Λ − 𝑖𝜅 2 , (𝑘 + 𝑖 − 1)Λ � , unless it receives another pulse message from (𝑖 − 1, 0) before doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This, however, is not the case, since we assume that message delays and hardware clock rates do not vary over time, entailing that these reception times lie Λ time apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='7 7Note that a separation of Λ − 𝑑 time would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The slack of 𝑑 means that small changes in timing between pulses are unproblematic, which we exploit in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 41 It remains to show the claimed bound on stabilization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' To this end, observe that the only state information that nodes maintain is 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' On reception of a pulse message, this state is overwritten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This will remove spurious state from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We would like to argue that the above induction can therefore be performed as-is, meaning that the system has stabilized by the time each node has generated its first pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, there is a subtlety: it could happen that a spurious message that is still in transit at time 0 overwrites the state of node (1, 0) after it received the first message from (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Node (1, 0) then behaves as if the first message of (0, 0) arrived later, at the exact same time as the spurious message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Because also such a spurious message is delivered within at most 𝑑 time, we can re-interpret this as a longer delay of still at most 𝑑 of the first message sent by node (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this modification reduces the difference between the reception times of the first and second pulse from node (0, 0) at node (1, 0) by up to 𝑢, but the separation remains at least Λ − 𝑢 ≥ Λ − 𝑑, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the second message is not received before (1, 0) generates its first pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We can apply the same scheme to nodes 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' , 𝐷, resulting in the desired bound on the stabilization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' L0 ≤ 𝜅/2 with probability 1 − 𝑜(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It is self-stabilizing with stabilization time Λ𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We remark that for a general base graph 𝐺, ensuring a small local skew is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, so long as |𝑉 | is small enough such that faults on layer 0 occur with probability 𝑜(1), one is free to fall back on a non-fault-tolerant GCS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This achieves L0 ∈ 𝑂(𝜅 log 𝐷), which does not increase the asymptotic local skew bound of the pulse forwarding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 42 Christoph Lenzen and Shreyas Srinivas B FULL PULSE FORWARDING ALGORITHM Algorithm 3 Discrete GCS at node (𝑣, ℓ), ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The parameters Λ, and 𝜅 will be determined later, based on the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' loop 𝐻min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻max := ∞ for {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑤} ∈ 𝐸 do 𝑟𝑤 := 0 do if received pulse from 𝑣ℓ−1 and 𝐻own = ∞ then 𝐻own := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) if for some {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑤} ∈ 𝐸 received pulse from (𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' ℓ − 1) and 𝑟𝑤 = 0 then if 𝑟𝑤′ = 0 for all {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑤 ′} ∈ 𝐸 then 𝐻min := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) 𝑟𝑤 := 1 if 𝑟𝑤′ = 1 for all {𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝑤 ′} ∈ 𝐸 then 𝐻max := 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) until 𝐻min < ∞ and 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) ≥ min{𝐻max + 𝜅/2 + 𝜗𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 2𝐻own − 𝐻min + 2𝜅)} if 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅 then wait until 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) = 𝐻max + 3𝜅/2 + Λ − 𝑑 else C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ := min𝑠 ∈N{max{𝐻own − 𝐻max + 4𝑠𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅/2 if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ < 0 then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ := min {𝐻own − 𝐻min + 3𝜅/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 0} else if C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ > 𝜗𝜅 then C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ := max {𝐻own − 𝐻max − 3𝜅/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='𝜗𝜅} wait until 𝐻𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ (𝑡) = 𝐻own + Λ − 𝑑 − C𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='ℓ broadcast pulse A basic requirement for the algorithm to work correctly is that (𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' ℓ) receives the 𝑘-th pulses of all correct predecessors within its 𝑘-th iteration of the main loop of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For all 𝑘 ∈ N and (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, node (𝑣, ℓ) receives the 𝑘-th pulses of all correct predecessors within its 𝑘-th iteration of the main loop of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We show by induction on ℓ ∈ N>0 and 𝑘 ∈ N>0 that (𝑣, ℓ) broadcasts the 𝑘𝑡ℎ pulse after receiving the 𝑘-th pulse from all correct (𝑤, ℓ − 1) satisfying that ((𝑤, ℓ − 1), (𝑣, ℓ)) ∈ 𝐸, but before receiving the (𝑘 + 1)-th pulse from such a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, for all 𝑘 ≥ 2, 𝑡𝑘 𝑣,ℓ − 𝑡𝑘−1 𝑣,ℓ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the induction on ℓ, we use ℓ = 0 as base case, requiring only that nodes generate pulses at frequency 1/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As delays and clock speeds do not change, this holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' For the step from ℓ − 1 ∈ N to ℓ, we perform the induction on 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that the claim holds for all 𝑘′ < 𝑘 ∈ N>0 and consider the 𝑘-th iteration of the outer loop at (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then a message from each node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, has been received in the current loop iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the induction hypotheses for layer ℓ − 1 and pulse 𝑘 − 1, respectively, for correct such nodes this is the 𝑘-th pulse message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We need to show that the 𝑘-th message from (𝑣, ℓ − 1) is received in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' the induction hypothesis guarantees that it is not received too early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As the minimum degree of 𝐺 is 2, at Gradient TRIX 43 least one node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If (𝑣, ℓ − 1) is correct, too, it sent its pulse message at the latest at time 𝑡𝑤,ℓ−1 + Lℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By the bounds on message delay and clock speed, this message is received at a local time 𝐻 ≤ 𝐻max + 𝜗(Lℓ−1 + 𝑢) ≤ 𝐻max + Λ − 𝑑 < 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own −𝐻min +2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝐻min < ∞, also 𝐻own < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using that 𝐻min ≤ 𝐻max, we get that Δ := min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 ≤ max{𝐻own − 𝐻max, 𝐻own − 𝐻min} − 𝜅 2 ≤ 𝐻own − 𝐻min − 𝜅 2 and hence C𝑣,ℓ ≤ 𝐻own − 𝐻min + 3𝜅/2 ≤ 3𝜅/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It follows that 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) ≥ max{𝐻min, 𝐻own} + Λ − 𝑑 − 3𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – (𝑣, ℓ − 1) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then by the bounds on message delay and clock speed, for each correct (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, its 𝑘-th pulse message is received at a local time 𝐻 ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) ≤ 𝐻own + Λ − 𝑑 − 3𝜅 2 < 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ), where the last step uses Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – (𝑣, ℓ − 1) is faulty, implying that all (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then by the bounds on message delay and clock speed, for each correct (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, its 𝑘-th pulse message is received at a local time 𝐻 ≤ 𝐻min + Λ − 𝑑 − 3𝜅 2 < 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ), where we use that in order to guarantee that Λ − 𝑑 ≥ 𝜗(2Lℓ−1 + 𝑢) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', Equation (2)), this must also hold in an execution that differs by (𝑣, ℓ − 1) being correct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' in such an execution, we have that max {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − min {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} ≤ max {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − 𝑡𝑣,ℓ−1 + 𝑡𝑣,ℓ−1 − min {𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} ≤ 2Lℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It remains to show that (𝑣, ℓ) generates its pulse before receiving a (𝑘 + 1)-th pulse message from a correct predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (𝑣, ℓ − 1) is not faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then the earliest local time 𝐻 at which (𝑣, ℓ) has received a 𝑘-th pulse from a correct predecessor is bounded from below by 𝐻 ≥ 𝐻own − 𝜗(Lℓ−1 + 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As delays and clock speeds do not change, the earliest message reception time for a (𝑘 + 1)- th pulse from a correct predecessor is Λ time later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, it is sufficient to show that 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) ≤ 𝐻 + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish three subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅 and at local time 𝐻min a message from a correct predecessor (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, was received by (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 𝐻own + 𝜗(Lℓ−1 + 𝑢) + 2𝜅 ≥ 2𝐻own − 𝐻min + 2𝜅 ≥ 𝐻max + 𝜅 2 + 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' 44 Christoph Lenzen and Shreyas Srinivas and, by Equation (3), 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) = 𝐻max + 3𝜅 2 + Λ − 𝑑 ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) + 2𝜅 + Λ − 𝑑 ≤ 𝐻own − 𝜗(Lℓ−1 + 𝑢) ≤ 𝐻 + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅 and at local time 𝐻max a message from a correct predecessor (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, was received by (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 𝐻own + 𝜗(Lℓ−1 + 𝑢) ≥ 𝐻max and, by Equation (3), 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) = 𝐻max + 3𝜅 2 + Λ − 𝑑 ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) + 3𝜅 2 + Λ − 𝑑 ≤ 𝐻own − 𝜗(Lℓ−1 + 𝑢) ≤ 𝐻 + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own−𝐻min+2𝜅 and C𝑣,ℓ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Equation (3), then 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ ≤ 𝐻own + Λ − 𝑑 ≤ 𝐻 + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own − 𝐻min + 2𝜅 and C𝑣,ℓ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then C𝑣,ℓ = 𝐻own − 𝐻min + 3𝜅 2 and 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ = 𝐻min − 3𝜅 2 + Λ − 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Since 𝐻min is bounded from above by the earliest local reception time of a message from a correct node (𝑤, ℓ − 1), {𝑣,𝑤} ∈ 𝐸, we have that 𝐻min ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Equation (3), we conclude that 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) ≤ 𝐻own + 𝜗(Lℓ−1 + 𝑢) − 3𝜅 2 + Λ − 𝑑 < 𝐻 + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' (𝑣, ℓ − 1) is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then 𝐻 = 𝐻min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Checking all cases in a similar fashion, we see that 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) ≤ 𝐻max + 3𝜅 2 + Λ − 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Using that Equation (3) must also apply in an execution where (𝑣, ℓ − 1) is not faulty and hence max{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} − min{𝑣,𝑤}∈𝐸{𝑡𝑤,ℓ−1} ≤ 2Lℓ−1, it follows that 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) ≤ 𝐻max + 3𝜅 2 + Λ − 𝑑 ≤ 𝐻min + 2𝜗(Lℓ−1 + 𝑢) + 3𝜅 2 + Λ − 𝑑 ≤ 𝐻min + Λ ≤ 𝐻 + Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ We are now ready to show that Algorithm 3 is equivalent to Algorithm 1 in the absence of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 45 Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Suppose that for (𝑣, ℓ) ∈ 𝑉ℓ, ℓ > 0, and the predecessors of (𝑣, ℓ) are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then running Algorithm 1 instead of Algorithm 3 results in the same pulse times of node (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Assume towards a contradiction that the claim is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Denote by 𝑡𝑘 𝑣,ℓ and (𝑡𝑘 𝑣,ℓ)′ the pulse times of Algorithm 1 and Algorithm 3 in executions with identical delays, clock speeds, and behavior of faulty nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', let 𝑡𝑘 𝑣,ℓ be minimal with the property that 𝑡𝑘 𝑣,ℓ ≠ (𝑡𝑘 𝑣,ℓ)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Consider the 𝑘-th loop iteration of Algorithm 3 at node (𝑣, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish cases according to why the inner loop terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 𝐻max + 𝜅/2 + 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then in Algorithm 1, we have that 𝐻own ≥ 𝐻max + 𝜅 2 + 𝜗𝜅, implying that Δ := min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 ≥ min 𝑠 ∈N {𝐻own − 𝐻max + 4𝑠𝜅} − 𝜅 2 ≥ 𝐻own − 𝐻min − 𝜅 2 ≥ 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, Algorithm 1 computes C𝑣,ℓ = 𝐻own − 𝐻max − 3𝜅 2 and generates its 𝑘-th pulse at local time 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) = 𝐻max + Λ − 𝑑 − C𝑣,ℓ = 𝐻max + 3𝜅 2 + Λ − 𝑑 = 𝐻𝑣,ℓ ((𝑡𝑘 𝑣,ℓ)′), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The inner loop terminated because 𝐻𝑣,ℓ (𝑡) = 2𝐻own −𝐻min +2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As 𝐻min < ∞, also 𝐻own < ∞ for Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We distinguish two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – In Algorithm 1, we have Δ := min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Then the same holds in Algorithm 3, as there 𝐻max is either identical to that of Algorithm 1 of ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, both algorithms compute 𝐶𝑣,ℓ = min{𝐻own −𝐻min +3𝜅/2, 0} and subsequently 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) = 𝐻own + Λ − 𝑑 − C𝑣,ℓ = 𝐻𝑣,ℓ ((𝑡𝑘 𝑣,ℓ)′), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' – In Algorithm 1, we have Δ := min 𝑠 ∈N {max{𝐻own − 𝐻max + 4𝑠𝜅, 𝐻own − 𝐻min − 4𝑠𝜅}} − 𝜅 2 ≥ 0 Let 𝑠min ∈ N be such that Δ := max{𝐻own − 𝐻max + 4𝑠min𝜅, 𝐻own − 𝐻min − 4𝑠min𝜅} − 𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If Δ = 𝐻own − 𝐻min − 4𝑠min𝜅 − 𝜅/2, the fact that 𝐻own and 𝐻min are identical in both algorithms, while 𝐻max is either also identical or −∞ in Algorithm 3, again leads to the 46 Christoph Lenzen and Shreyas Srinivas contradiction 𝐻𝑣,ℓ (𝑡𝑘 𝑣,ℓ) = 𝐻𝑣,ℓ ((𝑡𝑘 𝑣,ℓ)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Hence, suppose that Δ = 𝐻own −𝐻max + 4𝑠min𝜅 −𝜅/2 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Therefore, 0 ≤ Δ = 𝐻own − 𝐻max + 4𝑠min𝜅 − 𝜅/2 ≤ max{𝐻own − 𝐻max + 4(𝑠min − 1)𝜅, 𝐻own − 𝐻min − 4(𝑠min − 1)𝜅} − 𝜅 2 = 𝐻own − 𝐻min − 4(𝑠min − 1)𝜅 − 𝜅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, 2𝐻own − 𝐻min + 2𝜅 ≥ 𝐻own + 4𝑠min𝜅 − 3𝜅 2 ≥ 𝐻max − 𝜅 < 𝐻max − 𝜅 2 − 𝜗𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is a contradiction, as then the inner loop in Algorithm 3 would have terminated at an earlier time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='1 Self-Stabilization Making Algorithm 3 self-stabilizing follows standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Accordingly, we confine ourselves to a brief high-level discussion of how this is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The pulse propagation algorithm can be implemented in a self-stabilizing way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' It stabilizes within 𝑂(√𝑛) pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The key observation is that self-stabilization can proceed layer by layer, where Corollary 6 shows that layer 0 stabilizes fast enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, we can assume that the correct nodes of the previous layer generate pulses at a stable frequency of Λ satisfying the skew bounds obtained in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This allows us to make sure that the timing of its listening loop aligns with the pulse signals from the previous layer: From all but one predecessor, the pulse signals must be received while the inner loop is running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Moreover, the inner loop will terminate within Λ time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Instead of restarting the inner loop dependent on the own generated pulse, we can instead start a loop iteration when receiving the first pulse after a quiet period of, say, Λ/10 (where too frequent pulses of a faulty predecessor are filtered out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As such a quiet period must occur by Equation (2), this will align the loop correctly with the 𝑘-th pulses of correct predecessors for some 𝑘 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Once the inner loop terminates, we look for the next quiet period, and start a new instance of the inner loop on reception of the next pulse from a predecessor, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Whenever the inner loop terminates correctly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', not due to a timeout, we also compute the time to generate the next pulse as in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' However, we do not wait until the pulse is generated before willing to start a new instance of the inner loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This way, we ensure that we do not miss the first pulse message of a correct predecessor for pulse 𝑘 + 1 in case the inner loop for pulse 𝑘 was misaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ C OBTAINING THE FINAL SKEW BOUNDS Recall that our model assumes that message delays and clock speeds do not vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If the behavior of faulty nodes is static, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', the timing of their output pulse messages is identical in each pulse as well, a stable input frequency of 1/Λ results in repeating the exact same message pattern with the same timing every 1/Λ time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' We can exploit this to bound Lℓ,ℓ+1 in terms of Lℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' If faulty nodes do not change the timing of their output pulses, then L ∈ 𝑂(𝜅 log 𝐷) with probability 1 − 𝑜(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Gradient TRIX 47 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Corollary 5, for correct (𝑣, ℓ + 1) ∈ 𝑉ℓ+1, ℓ ∈ N, min ((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ (𝑤,ℓ)∉𝐹 {𝑡𝑘 𝑤,ℓ} + Λ − 2𝜅 ≤ 𝑡𝑘 𝑣,ℓ+1 ≤ max ((𝑤,ℓ),(𝑣,ℓ)) ∈𝐸ℓ (𝑤,ℓ)∉𝐹 {𝑡𝑘 𝑤,ℓ} + Λ + 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Because the behavior of fault nodes does not change between pulses, a simple induction shows that 𝑡𝑘+1 𝑥,ℓ′ = 𝑡𝑘 𝑥,ℓ′ + Λ for all correct nodes (𝑥, ℓ′) ∈ 𝑉ℓ′, ℓ′ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In particular, min ((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ (𝑤,ℓ−1)∉𝐹 {𝑡𝑘+1 𝑤,ℓ } − 2𝜅 ≤ 𝑡𝑘 𝑣,ℓ+1 ≤ max ((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ (𝑤,ℓ)∉𝐹 {𝑡𝑘+1 𝑤,ℓ } + 2𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' By Theorem 5, Lℓ ∈ 𝑂(𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Note that this bound applies uniformly over all executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, even if (𝑣, ℓ) is faulty, using that its neighbors are within distance 2 of each other, it holds that min ((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ (𝑤,ℓ−1)∉𝐹 {𝑡𝑘+1 𝑤,ℓ } − max ((𝑤,ℓ),(𝑣,ℓ+1)) ∈𝐸ℓ (𝑤,ℓ)∉𝐹 {𝑡𝑘+1 𝑤,ℓ } ∈ 𝑂(𝜅 log 𝐷), by virtue of comparing to an execution in which (𝑣, ℓ) is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' As (𝑣, ℓ + 1) was an arbitrary correct node, the claim of the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □ It remains to argue that some variation can be sustained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' With probability 1−𝑜(1), L ∈ 𝑂(𝜅 log 𝐷) even when in each pulse (i) a constant number of faulty nodes change their output behavior and timing, (ii) link delays vary by up to 𝑛−1/2𝑢 log 𝐷, and (iii) hardware clock speeds vary by up to 𝑛−1/2(𝜗 − 1) log 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' The maximum length of a directed path in 𝐻 is bounded by 2√𝑛: at most 𝐷 ≤ √𝑛 hops in layer 0, followed by at most √𝑛 links from layer to layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' Thus, accumulating all changes in timing due to link delay and clock speed variation along a path results in a deviation of 𝑂((𝑢 + (𝜗 − 1)(Λ − 𝑑)) log 𝐷 = 𝑂(𝜅 log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' This is trivial for layer 0 and applies to pulse propagation through the layers as well, because our respective analysis relies on Corollary 5 and Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' In order to take into account a constant number of faulty nodes with arbitrary behavior, we reason analogously to the proof of Theorem 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=', rely on Corollary 5 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} +page_content=' □' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfbgxF/content/2301.05073v1.pdf'} diff --git a/4dFST4oBgHgl3EQfZzjU/content/tmp_files/2301.13793v1.pdf.txt b/4dFST4oBgHgl3EQfZzjU/content/tmp_files/2301.13793v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac55d0b56735e4f6f7f0a03fbb48e7984a260778 --- /dev/null +++ b/4dFST4oBgHgl3EQfZzjU/content/tmp_files/2301.13793v1.pdf.txt @@ -0,0 +1,881 @@ +Protecting the Texas power grid from tropical cyclones: +Increasing resilience by protecting critical lines +Julian Stürmer1,2, Anton Plietzsch1, Thomas Vogt1, Frank Hellmann1, Jürgen Kurths1,3,4, +Christian Otto1,*, Katja Frieler, Mehrnaz Anvari1,* +1Potsdam Institute for Climate Impact Research, Telegrafenberg A56, 14473 Potsdam, +Germany +2Institute for Theoretical Physics, TU Berlin, 10623 Germany +3Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany +4Institute of Physics, Humboldt Universität zu Berlin, 12489 Berlin, Germany +*Corresponding author(s): anvari@pik-potsdam.de; christian.otto@pik-potsdam.de +Abstract +The Texan electric network in the Gulf Coast of the United States is frequently hit by Tropical +Cyclones (TC) causing widespread power outages, a risk that is expected to substantially +increase under global warming. Here, we introduce a new approach of combining a +probabilistic line fragility model with a network model of the Texas grid to simulate the +temporal evolution of wind-induced failures of transmission lines and the resulting cascading +power outages from seven major historical hurricanes. The approach allows reproducing +observed supply failures. In addition, compared to a static approach, it provides a significant +advantage in identifying critical lines whose failure can trigger large supply shortages. We +show that protecting only 1% of total lines can reduce the likelihood of the most destructive +type of outages by a factor of between 5 and 20. The proposed modelling approach could +represent a tool so far missing to effectively strengthen the power grids against future +hurricane risks even under limited knowledge. +Keywords: Electric networks, Extreme weather events (hurricane), Cascading failures +Introduction +Modern societies depend heavily on reliable access to electricity. Power outages have the +potential to disrupt transportation and telecommunication networks, heating and health +systems, the cooling chain underpinning food delivery and more1–3. Depending on the cause +of power outages and the amount of physical damages to infrastructures, the recovery of +the electric network, and the social infrastructures dependent on it, often takes days or even +months4. Such outages are often driven by extreme weather events. In Norway +of all +overhead line failures are caused by extreme weather which involves strong winds, icing and +lightning strikes5. In February 2021 a winter storm in Texas led to outages that in turn caused +a breakdown of the gas supply and thus the heating sector6–8. Impacts are particularly + +devastating when it comes to tropical cyclones. In the summer months, the Gulf Coast and +the East Coast of the United States are frequently hit by tropical cyclones (TC) that entail +widespread outages and costs of billions of dollars. For example, hurricane Ike hitting +southeast Texas on September 13, 2008 destroyed around 100 towers holding high voltage +transmission lines and cut off electric power for between 2.8 and 4.5 million customers for +weeks to months9, 10. On August 29, 2021 hurricane Ida made landfall in Louisiana, and +destroyed major transmission lines delivering power into New Orleans, causing more than a +million customers to lose power11. +Resilience against line failures in power grids is usually discussed in terms of the N-1 (rarely +also N-2) security of the system, that is, the ability of the system to stay fully functional upon +the failure of one or two elements12. When a line fails, the power flow automatically +reroutes through the intact grid. To avoid overloads in the rerouting, relevant lines are +intentionally taken out of the grid. This secondary failures of lines can trigger a cascade13–19 +of additional failures. N-1 security asserts that single line failures do not trigger such +cascades. Significant secondary failures do occur in larger events and were, e.g., observed in +response to the software error leading to the U.S.-Canadian blackout on August 14th, 200320. +They are typically also induced by the widespread primary damages and line failures caused +by TCs. +The N-1 approach to system resilience does not scale to extreme weather events. The tens +or even hundreds of primary failures during events such as hurricanes can not be fully +mitigated by an electric network, because N-100 security is not realistic to achieve. N-1 +security is typically studied by simulating the reaction of the system to every possible failure +scenario. As the number of possible failure scenarios scales exponentially with the number +of failures, it is computationally infeasible to consider all possible such scenarios in larger +events. Initialising failure cascade models designed for N-1 studies with many initial failures +is challenging. +Here, we present an approach that solves these issues by temporally resolving the potential +damages induced by hurricanes and a stepwise application of a failure cascade model. This +approach particularly allows us to identify critical power lines whose protection could most +effectively reduce the risk of severe widespread power outages. Although the frequency of +severe hurricanes is expected to increase12–14, such an approach does not exist so far. +Main text +Our approach explicitly models the dynamical interplay of an extreme wind event with +the power grid. It temporally resolves both, the primary wind damages, and the cascades +and secondary failures that result from them. We will use this approach to study the +impact of massive TCs on the Texan power grid. Strong hurricanes, such as Harvey that + +made landfall on Texas and Louisiana in August 2017, can destroy more than hundreds +transmission lines in an electric grid (see Fig. 1(a)). These lines do not collapse +simultaneously, but over the hours or days the TC passage takes. Making use of the +chronological order of the line destructions, we divide each overall TC scenario into a +sequence of 5 minute long scenarios. In most of these individual steps, only one line fails. +We then solve individual scenarios by representing the Texan transmission network in a DC +power flow approximation with conservative load balancing assumptions (see Methods and +Supplementary Methods 3 and 4). This approach accounts for the ‘path dependency’ of the +solution: Everytime a line collapses, secondary failures can occur, but also control +mechanisms are immediately activated and try to bring back the energy balance to the +system and, consequently, mitigate the effect of the failure (see Supplementary Methods 4). +Later primary damages along the TC track then meet a partially destroyed, rebalanced grid. +Thus, the effect of later failures can be more or, even, less intense. It is the resilience of +these intermediate, partially destroyed states that ultimately decides whether the impact of +the TC is amplified by secondary failures. +Fig. 1: Probability distributions of primary line failures and final power outages (a) +Probability distribution of the total number of wind-induced line failures +as generated by +the probabilistic line fragility model for each of the seven recent hurricanes hitting Texas +(category in brackets behind the name). TCs are sorted according to the means of the +distributions which are indicated as solid vertical lines. (b) Probability distribution of the +associated total power outage +after TC passage. The inset highlights large cascading +failures that can also occur for the weaker hurricanes. The dashed vertical lines indicate the +reported power outages listed in the Supplementary Table 1 and the solid vertical lines +represent the means. See Methods section for the model parameters used in the +simulations. +Unfortunately, neither detailed information about the topology of the exposed power grid +nor about the exact power lines destroyed by the considered TC is publically accessible. So + +a +b +Harvey (4) +Ike (2) +Claudette (1) +Hanna (1) +Erin (TS) +Hermine (TS) +(anod)d +p(Np) +Pout +Laura (4) +μp +pout +0 +20 +40 +60 +80 +100 +120 +140 +0 +10 +20 +30 +40 +Np +pout [GW]here, we use a synthetic model of the Texan grid introduced by Bircheld et al21 (see +Supplementary Fig. 2 as well as Methods). +To represent the TCs impact on the energy supply we combine this grid model with a +probabilistic line destruction model (see Methods) forced by modelled historical wind fields +from seven different TCs (see Supplementary Supplementary Methods 2). The probabilistic +model provides the probability of line failure in terms of wind speeds and allows to generate +a large sample of temporally resolved realisations of line failure maps. In the default setting +considered here we assume a homogeneous base failure rate for all transmission lines. This +is our main adjustable parameter and is tuned to reproduce observed power outages (see +Fig. 1(b) and Supplementary Methods 5). The TCs are selected to cover several different +types of trajectories and intensities and particularly include storms that continue to move +westward after landfall and affect the southern and western parts of Texas such as Hurricane +Claudette, Tropical Storm Erin, and Hurricane Hanna, contrary to most hurricanes that are +steered northward by the Coriolis effect before western parts of Texas are reached22. +Core result +While the number of primary line failures follows a Poisson binomial distribution, the +derived distribution of outages is heavily multimodal for all storm tracks with the potential +of large +to +outages (see Fig. 1(b)). These large damages turn out to not +accumulate gradually over the course of the hurricane but occur suddenly in one or few time +steps (see Fig. 2(b)). This sudden increase in outages is induced by cascading line failures +taking the Houston and a weakly connected North-Western section of the grid offline (see +Fig. 2(d) and Fig. 3). +Figure 3 shows what damage patterns correspond to the various modes of the outage +distribution. The disconnection of the North-West occurs due to the non-local effects of +cascading failures in areas not directly affected by high wind speeds. For example, hurricanes +Harvey and Hanna never reach this region, but cause a considerable probability of outages +affected by Harvey (Fig. 3(a)-(c)), but also due to non-local cascades as seen for Hanna (Fig. +3(h) and (i)). As the most populous city in Texas and a major load centre, the disconnection +of Houston from the electrical networks causes the disconnection of a huge number of +consumers from the electrical network and, consequently, the overproduction of generators +located in the west of Texas, which have key roles to provide the required energy in Houston +(see Supplementary Methods 5 and Supplementary Fig. 8). Interestingly, the northern part +of the electric grid is never impacted by outages caused by these three hurricanes. Same +figures for other hurricanes have been shown in Supplementary Fig. 7. + +Figure 2: Simulation of hurricane-induced cascading failures in the Texan electric grid (a) +The schematic variation of the supplied load in an electric grid before (pre-hurricane), during +(hurricane phase) and after (restoration phase) a hurricane is loosely based on ERCOT's23. +The total power outage +after a hurricane has passed, and the total energy +(red +area) that was not supplied are measures for the severity of an outage scenario. (b) +Summary of all realisations of power outage trajectories simulated for hurricane Claudette +(see Methods section and Supplementary Methods 5 for specification of the model +parameters). Trajectories shown in red come in two types, those that aggregate damages +gradually over time (Type I in the figure) and those that include a large cascade (Type II). The +distribution of cascade sizes is multimodal and we use an empirical threshold of +to define large cascades (see Supplementary Fig. 9). (c) and (d) show +respectively the state of the power grid at the beginning and the end of the hurricane. These +two states are shown in panel (b), for one realisation of primary line failures. Lines shown in +black were destroyed by the hurricane or deactivated due to the secondary effects, for the +other lines the relative line loading is shown, with red lines close to overload. In addition, +the panel includes the track and a snapshot of the windfields of hurricane Claudette in blue. +In the Supplement, we also provide a video of the simulation showing how the wind +damages spread along the passage of hurricane Claudette. + +b +a +Hurricane +Restorationphase +phase +Supplied load +60 +Type +ye +[GW] +Eout +anod +50 +p +d +40 +Time +60 +70 +80 +90 +100 +110 +t [h] +c +d +1.0 +40 +36 +pout = 0.0 GW +36 - +pout = 20.5 GW +[Pc = 67.i GW +P: = 46.6 GW +35 +0.8 +34 +34 +30 +25 +32 - +32 +0.6 +Windspeed [m/s] +Latitude [°] +Line Loading +20 +30 +30 - +0.4 +15 +28 +28 - +10 + 0.2 +5 +Inactive Parts +26 +26 - +- Hurricane Track +-104 +-102 +-100 +-98 +-96 +-94 +-104 +-102 +-100 +-98 +-96 +-94 +0.0 +0 +Longitude [°] +Longitude ["]noC +15GMFigure 4: Probability of line failure for different parts of the total power outage +distribution (a-i) Probability that the failure of a given power line is involved in three +different modes of the power outage distribution. The modes are indicated by the insets and +the exact range of considered power outages are shown below these insets. The +probabilities +are calculated as: number of realisations with a total outage within the +specified range in each figure where the considered line failed / number of total realisations. +The rows describe the probabilities for different hurricanes as indicated in the panel. Texan +electric grid with grid elements colored according to their respective outage probability. +The probability distributions shown in the insets are identical to the ones shown in Fig. 1(b) . + +a +b +p +Cluster of +pout E[6 GW, 12GW] +generators +pout E[23GW, 28GW] +pout E[34GW, 44GW] +Dallas +Austin +San Antonio +Generators +Houston +Generators +Generators +Corpus +Loads +Loads +Loads +Christi +Harvey +Harvey +Harvey +d +pout E[0GW +pout E[3GW, 12 GW] +pout E[15GW, 30GW] +Generators +Generators +Generators +Loads +Loads +Loads +Claudette +Claudette +Claudette +g +h +pout E[15GW, 30GW] +Generators +Generators +Generators +Loads +Loads +Loads +Hanna +Hanna +Hanna +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nodFor all seven hurricanes, the cascades play a major role in the total line failures associated +with the event (see Fig. 4 ). They are induced by the overload of remaining lines and the +isolation of grid elements, as well as the failure of islands with unavoidable overproduction. +Figure 3: The probability of primary damages and secondary failures induced by hurricane +Harvey In this plot the transmission lines are colored according to their high probability to +be directly damaged by Harvey (blue lines) or to be deactivated due to the secondary effect +of the hurricane (red lines). As expected the primary damages are located around the path +of Harvey. However, secondary failures can occur far away from the hurricane track, which is +related to the non-local effect of the primary damages in the power grid (see supplementary +Methods 5). In this plot, grey lines have a higher probability of remaining operational than +failing due to any reason. +Our results are not sensitive to the assumption of a homogeneous base failure rate as similar +characteristics are also derived when assuming randomised base failure rates (see +Supplementary Table 3). In addition, a temporal resolution of 5 minutes turned out to be +adequate as time steps where several lines fail are rare. At this resolution it is also +reasonable to assume that cascades of secondary failures have run their course before +further lines are destroyed by the hurricane24,25 (for further discussion regarding the +temporal resolution, see Supplementary Note 2). +Increasing Resilience +The fact that large cascades are triggered by the failure of specific lines suggests targeting +these lines for protection. To identify the critical lines that should be protected we define a + +36 +34 +32 +Latitude [°] +30 +28 +Most probable line status +Unaffected +26 +Primary Damage +Secondary Failure +-104 +-102 +-100 +-98 +-96 +-94 +Longitude [°]priority index as the probability that the wind-induced damage of this specific line triggers a +large cascade, that is, a cascade that increase the outage by more than 15 GW, averaged +over all seven hurricanes (see Fig. 2(b) and Eq. (4) in Methods). +As a baseline we also consider a conventional, static model (see Methods). The static index +of a line is the conditional probability of a large outage given that the line is damaged by a +TC. In both the co-evolution model and the static baseline (see Fig. 5(a) and (b)) the critical +lines are mostly located around Houston. +To estimate the reduction in power outages that can be reached by protecting critical lines, +we order them according to their priority index and evaluate the impact of the TC on the +system with the first one to twenty lines protected, e.g. by being replaced by underground +cables. It is worth noting that the co-evolution priority index value for most transmission +lines is zero. Only +of them have a value above +, and only +lines above +. By +protecting these +lines, large power outages and cascading failures are almost completely +prevented for smaller storms and dramatically reduced for the larger ones (see Fig. 5 and +Supplementary Fig.9). For the stronger hurricanes Harvey and Ike, the power outage +distributions are shifted from the second peak to the first peak with +(see +Fig. 5(c)). Protecting the lines one by one shows that the reduction of the largest power +outages improves smoothly, thus it is effective to protect up to twenty lines (see Fig. 5(c) and +(d)). While in the original system damage amplification was almost guaranteed, it rarely +occurs in the reinforced one. In summary +of total lines reinforced leads to a 5 to 20 time +reduction of the largest scale outages. The level of protection that can be reached by +protecting the lines according to the priority index derived from the co-evolution models is +generally higher than the protection of the same number of lines selected according to the +priority index derived by the static model (see panel (d) of Fig. 5). The static baseline also +identifies some of the most critical lines (see Supplementary 4), but additional protections +stop being effective after the first 6-10 lines (see Fig. 5(c) and (d)). This demonstrates that +the co-evolution model, with its detailed picture of the partially destroyed states, reveals +genuinely new and critical information for increasing the resilience of the system. +It is worth to mention that the results obtained from homogeneous base failure rates are +similar to the randomised ones (see Supplementary Methods 5 and Supplementary Table 3). + +moC +1OGMFigure 5: Level of risk reduction that can be reached by protecting power lines according to +the priority index: The co-evolution model against the static model (a)-(b) 20 lines of the +Texan power grid with the highest priority index (see Eq. Eq. (4)) obtained from the static +model (orange lines), the co-evolution model (blue lines), and both approaches (green lines). +The inset (b) shows a close-up view of Houston and Harris County, which contain most of the +critical lines. As seen in (b) the critical lines obtained from both models are located in the +same region, however, the co-evolutionary model identifies additional lines whose +protection has a dramatic effect on increasing resilience. (c) Power outage distributions of +hurricane Harvey in terms of the number of critical lines protected in both the co-evolution +(blue) and the static model (orange). The second peak in the power outage distribution is +strongly reduced as the number of protected lines increases. However, protecting lines +obtained from the static model does not increase the resilience of the power grid as much as +occurs in the co-evolution model. (d) Reduction of the large power outages obtained from +both models. For all three strong hurricanes, i.e. Harvey, Ike and Claudette, the reduction in +power outages is much greater in co-evolution model than the static one. + +b +a +31.0 +36 - +30.5 +34 - +32 +30.0 +Latitude [°] +Latitude [°] +30 +29.5 +28 +29.0 +20 most critical lines +coevolution method +26 - +static method +both methods +28.5 +-104 +-102 +-100 +-98 +-96 +-94 +-97.0 +96.5 +96.0 +95.5 +95.0 +94.5 +Longitude [°] +d +Longitude [°] +c +Staticmethod +1.0 +Coevolutionmethod +0 +Number of protected lines +0.8 +0.6 +10 +Method +0.4 +coevolution +static +0.2 +20 +Harvey +ike +Claudette +0.0 +0 +10 +20 +30 +40 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +p(pout) [GW] +Number of protected linesConclusion and outlook +The co-evolution model of the Texan power grid has been introduced as an efficient +approach to temporally resolve the line failures and secondary grid outages induced by TCs. +The model can resolve to considerable detail the way secondary failure cascades amplify the +impact of extreme events. Using this information it can be used to identify critical lines that +should be protected to effectively increase the system's resilience and prevent the most +severe outages. Our model goes significantly beyond the state of the art so far represented +by statistical and economic models that can only capture a static picture of the event and +the network24–29. We have seen that such static approaches do not easily identify all of the +critical lines during extended events. Their importance is only revealed by stepwise ‘tracking’ +the destruction of the system and associated power outages and overloads. We expect that +this co-evolution approach will also be a promising tool to understand and protect other +grids exposed to spatio-temporally extended extreme events. +The results of our study are in agreement with a recent TC related risk assessment for +Texas26. Combining our priority index with additional information about the cost of a +reinforcement of the considered lines could also enable the identification of the most cost +efficient way to reduce the probability of power outages above a critical limit to an intended +value (see Supplementary Methods 6). +While the model based on wind speeds and historical hurricane tracks already identified +crucial structures in the grid, the co-evolution approach could naturally be extended to more +sophisticated models and broader settings. One particularly important goal for future +research will be to drive the model with potential future storm tracks due to climate +change27. As the frequency of particularly strong TCs is expected to increase under global +warming (WGI contribution to the AR6), understanding what lines are critical in the face of +the weather of the next decades is crucial. Another important avenue of broadening the +model is to account for TC induced flooding (coastal flooding, pluvial or fluvial flooding) and +associated destructions. These may follow a different temporal pattern where the adequacy +of the approach proposed here has to be newly tested. This would also provide a first step +towards an assessment of genuine compound events in which several stresses for the grid +coincide. +Methods +Electric grid data of Texas +For the study we used the publicly available electric grid test case ACTIVSg200028, that +covers the area of the so-called ERCOT Interconnection, which supplies +percent of the +electricity demand in Texas29. The test case is synthetic but resembles fundamental + +properties of the real grid, such as the spatial distribution of power generation and +demand21. It encompasses +buses with geographic locations, +branches (both +transmission lines and transformers) and covers four different voltage levels. The test +case comes with all required electrical parameters ranging from the power injections of +buses to the power flow capacities of transmission lines and transformers. The flow +capacities +play a particularly important role for the simulation of cascading failures +as they determine the amount of power that can be transported by individual lines and +transformers without potentially damaging the equipment. +Historical hurricane data +Hurricane storm tracks are extracted from the International Best Track Archive for +Climate Stewardship (IBTrACS)30, 31 as time series of cyclone center coordinates along +with meteorological variables like maximum sustained wind speeds and minimum +pressure on a +h snapshot basis. For this study, a hand-picked selection of seven +historical storms is used (Supplementary Fig. 1 and 2) to cover several different types of +trajectories and intensities. Particularly, the selection also includes storms that continue +to move westward after landfall and affect the southern and western parts of Texas (see +Hurricane Claudette, Tropical Storm Erin, and Hurricane Hanna in Supplementary Fig. 2 +and the Supplementary Fig. 1), contrary to most hurricanes that are steered northward +by the Coriolis effect before western parts of Texas are reached22. From the track records, +we compute time series of wind fields within a radius of +km from the storm center +using the Holland model for surface winds, as implemented in the Python-package +CLIMADA32, 33, at a spatial resolution of +degrees (approximately +km) and a +temporal resolution of +minutes. The intensities of the considered storms are also +shown along the respective tracks in Supplementary Fig.1 while other properties of the +storms are listed in Supplementary Table 1. +Transmission line fragility model +To model wind-induced failures of transmission lines, we first differentiate between +overhead transmission lines and underground cables in the electric grid of Texas. +Following Birchfield et al., we analyse lines that are shorter than +km ( +miles) +and connect a total load of at least +MW as underground cables21. All other lines are +assumed to be overhead transmission lines. The latter are then divided into segments of +length +m, which corresponds to the average distance between transmission +towers in Texas34. Our fragility model assigns failure rates to individual line segments +according to + +where, +denotes the wind force acting on the line segment +for a given wind +speed +and is calculated according to the guidelines published by the American Society +of Civil Engineers35. The parameter +represents the inverse of the so-called time to +failure, which indicates how long a line segment can withstand a wind force equal to the +breaking force +. It is used as a free parameter to calibrate the model such that +historically +reported +power +outages +are +reproduced +in +our +simulations +(see +Supplementary Methods 5). The full wind force equation as well as the meaning and the +values of all parameters can be found in Supplementary Methods 2 and Supplementary +Table 2. In all figures shown in the main text, +. Using the failure rates +, we define the probability that a line segment +fails during the time interval +as +This failure probability is inspired by the line fragility model established by Winkler et al., +which assumes that the failure probability is proportional to the ratio of the wind force +and the breaking force36. However, in contrast to their model, we define the failure +probability +using a time-dependent failure rate +that allows us to take the time +evolution of a field into account. A line is removed from the test case if any of its line +segments fails during a time interval. It should be noted that multiple lines may be +destroyed in the same time step, meaning that they are removed from the network +simultaneously. According to Eq. (2), the probability of simultaneous failures increases +with time step size +. A discussion of the role of the time resolution can be found in +Supplementary Note 2. +Cascading failure model +Wind-induced line failures can trigger cascades of overload failures in the branches of +the electric grid. As cascading failures typically evolve on smaller time scales than the +temporal resolution +of the wind field, we can assume a time scale separation. When +the network topology is changed by a primary damage event, the power flows +on +the branches are rerouted using the DC power flow model +here, +are the net active power injections at the buses, +are the bus voltage angles +and +are the elements of the nodal susceptance matrix that comprises the network +topology. More details on the assumptions of the DC power flow model and the software +used can be found in Supplementary Methods 3. If the new state of the network exhibits +any overloaded branch ( +), they are deactivated and the process is repeated. +When the network reaches a state without overloads, the algorithm advances to the + +br +0.002 hnext primary damage event. When a load or generator gets disconnected or the grid is +split into several parts, the global active power balance (GAPB) has to be restored in each +network component. Motivated by a primary frequency control in real electric grids, we +adjust the outputs of generators uniformly, while respecting their output limits defined +in the data set. Whenever the generator limits do not allow to fully restore the GAPB, we +either conduct a uniform minimal load shedding or consider the blackout of the whole +network component in the case of an unavoidable overproduction. The details of the +algorithm are explained in Supplementary Methods 4. +Quantification of power outages +We use the following three different quantities to track the power outages arising in our +simulations: (i) +denotes the total supplied load at the end of each time step, i.e., +after the cascading algorithm finished, respectively. It is calculated by adding up the +demands of all connected loads across all islands that exist at the given time. Since our +co-evolution model assumes that cascading failures happen instantaneously, +represents a step function for each individual TC scenario as shown in Fig. 2(b). We have +simulated +scenarios for each hurricane. (ii) Any cascading failure that actually causes +a loss of supplied load results in a vertical transition of size +in +. One such +transition is annotated with +for the highlighted scenario in Fig. 2(b). (iii) All +cascading failures that are triggered in a given TC scenario lead to a final power outage +. The interesting statistics of +are +shown and discussed in Fig. 1(b) . +Identification of critical lines +We identify critical overhead transmission lines by means of a priority index defined for +each line +as +where +denotes the set of considered hurricanes (seven hurricanes in this study) and +is the probability of a large cascade being triggered by the wind-induced failure of +line +. More specifically, we call cascades large or belonging to type II if their +associated power outage +lies above an empirical threshold of +GW (indicated +as type II in Fig. 2(b) and Fig. 5(d)). Eq. (4) includes an averaging over all considered +hurricanes to discern lines that are critical for multiple hurricanes. This allows us to +propose line reinforcements that increase the resilience not only for a particular +hurricane. Some properties of the +most critical lines found in this study are listed in + +Dou +na +0GM.67.GVSupplementary Table 3. Fig. 5(a) and (b) shows the location of these lines and +demonstrates that reinforcing them indeed increases the resilience of the electric grid +substantially. More details of the critical lines and a possibility to incorporate economic +considerations into our analysis are discussed in Supplementary Methods 6. +Baseline Method +Here, we apply the static model as a baseline method. By static model, we mean that all +primary damages occur simultaneously and, then, the DC power model along with global +active power balance (see Supplementary Methods 6) are activated once to bring back +the energy balance in the system and to evaluate the total final power outages +. As +discussed in Supplementary Note 2 the final power outage distributions are independent +of the time resolution of the wind field, however the primary damages leading to large +outages, i.e. +to +, can be completely different ones. To indicate the +critical lines obtained from the static model, first, we separate all scenarios in which +. Then, we use Eq. (4) to calculate the priority index of the primary +damages leading to large cascades. The top +lines with the highest priority index have +been listed in Supplementary Table 5. As seen in this table, except for the six lines +highlighted in red, the other lines are completely different from lines obtained from the +co-evolution model. +Code availability +All code necessary to reproduce the findings in this work is openly available. The +time-dependent wind fields are computed using the open-source platform CLIMADA32, 33. +The implementation of the transmission line fragility and the DC power model is +available from https://gitlab.pik-potsdam.de/stuermer/itcpg.jl. +Data availability +The observed TCs from IBTrACS30, 31 are distributed under the permissive WMO open data +licence +through +the +IBTrACS +website +(https://www.ncei.noaa.gov/products/international-best-track-archive) +and +can +be +directly retrieved through the CLIMADA32, 33 platform. The electrical network data is +openly available from the Texas +A&M University’s electric grid test case repository +(https://electricgrids.engr.tamu.edu/electricgrid-test-cases/activsg2000/). +Acknowledgements +This project has received funding from the ConNDyNet2 project under grant no. +03EF3055F. This research has received funding from the German Academic Scholarship +Foundation and the German Federal Ministry of Education and Research (BMBF) under + +20 GM30 GMnoC +15GMthe research projects QUIDIC (01LP1907A) and SLICE (FKZ: 01LA1829A), and from the +CHIPS project, part of AXIS, an ERA-NET initiated by JPI Climate, funded by FORMAS +(Sweden), DLR/BMBF (Germany, grant no. 01LS1904A), AEI (Spain) and ANR (France) +with co-funding by the European Union (grant no. 776608). +Author Contribution +M. Anvari, F. Hellmann and C. Otto contributed to design and conceive the research. The +co-evolution model is designed and developed by M. Anvari, J. Stürmer, A. Plietzsch and +F. Hellmann. All simulations and data analyses of this work have been done by J. +Stürmer and under supervision of M. Anvari. All hurricane data have been provided by +T. Vogt during this research. 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Weather Rev., vol. 136, no. +9, pp. 3432–3445, 2008. +34. +E. B. Watson and A. H. Etemadi, Modeling Electrical Grid Resilience Under Hurricane +Wind Conditions With Increased Solar and Wind Power Generation, IEEE Trans Power +Syst, vol. 35, no. 2, pp. 929–937, Mar. 2020, doi: 10.1109/TPWRS.2019.2942279. +35. +C. J. Wong and M. D. Miller, Guidelines for electrical transmission line structural +loading, 2009. +36. +J. Winkler, L. Duenas-Osorio, R. Stein, and D. Subramanian, Performance assessment of +topologically diverse power systems subjected to hurricane events, Reliab. Eng. Syst. +Saf., vol. 95, no. 4, pp. 323–336, 2010. + diff --git a/4dFST4oBgHgl3EQfZzjU/content/tmp_files/load_file.txt b/4dFST4oBgHgl3EQfZzjU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6774d8414ee3c34d5513e4e5bfb2ec8ae248868 --- /dev/null +++ b/4dFST4oBgHgl3EQfZzjU/content/tmp_files/load_file.txt @@ -0,0 +1,688 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf,len=687 +page_content='Protecting the Texas power grid from tropical cyclones: Increasing resilience by protecting critical lines Julian Stürmer1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Anton Plietzsch1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Thomas Vogt1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Frank Hellmann1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Jürgen Kurths1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Christian Otto1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Katja Frieler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Mehrnaz Anvari1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='* 1Potsdam Institute for Climate Impact Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Telegrafenberg A56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 14473 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Germany 2Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' TU Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 10623 Germany 3Institute of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' University of Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 14476 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Germany 4Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Humboldt Universität zu Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 12489 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Germany Corresponding author(s): anvari@pik-potsdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='otto@pik-potsdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='de Abstract The Texan electric network in the Gulf Coast of the United States is frequently hit by Tropical Cyclones (TC) causing widespread power outages, a risk that is expected to substantially increase under global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Here, we introduce a new approach of combining a probabilistic line fragility model with a network model of the Texas grid to simulate the temporal evolution of wind-induced failures of transmission lines and the resulting cascading power outages from seven major historical hurricanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The approach allows reproducing observed supply failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In addition, compared to a static approach, it provides a significant advantage in identifying critical lines whose failure can trigger large supply shortages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' We show that protecting only 1% of total lines can reduce the likelihood of the most destructive type of outages by a factor of between 5 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The proposed modelling approach could represent a tool so far missing to effectively strengthen the power grids against future hurricane risks even under limited knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Keywords: Electric networks, Extreme weather events (hurricane), Cascading failures Introduction Modern societies depend heavily on reliable access to electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Power outages have the potential to disrupt transportation and telecommunication networks, heating and health systems, the cooling chain underpinning food delivery and more1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Depending on the cause of power outages and the amount of physical damages to infrastructures, the recovery of the electric network, and the social infrastructures dependent on it, often takes days or even months4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Such outages are often driven by extreme weather events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In Norway of all overhead line failures are caused by extreme weather which involves strong winds, icing and lightning strikes5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In February 2021 a winter storm in Texas led to outages that in turn caused a breakdown of the gas supply and thus the heating sector6–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Impacts are particularly devastating when it comes to tropical cyclones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In the summer months, the Gulf Coast and the East Coast of the United States are frequently hit by tropical cyclones (TC) that entail widespread outages and costs of billions of dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' For example, hurricane Ike hitting southeast Texas on September 13, 2008 destroyed around 100 towers holding high voltage transmission lines and cut off electric power for between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 million customers for weeks to months9, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' On August 29, 2021 hurricane Ida made landfall in Louisiana, and destroyed major transmission lines delivering power into New Orleans, causing more than a million customers to lose power11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Resilience against line failures in power grids is usually discussed in terms of the N-1 (rarely also N-2) security of the system, that is, the ability of the system to stay fully functional upon the failure of one or two elements12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' When a line fails, the power flow automatically reroutes through the intact grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' To avoid overloads in the rerouting, relevant lines are intentionally taken out of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This secondary failures of lines can trigger a cascade13–19 of additional failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' N-1 security asserts that single line failures do not trigger such cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Significant secondary failures do occur in larger events and were, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=', observed in response to the software error leading to the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='-Canadian blackout on August 14th, 200320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' They are typically also induced by the widespread primary damages and line failures caused by TCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The N-1 approach to system resilience does not scale to extreme weather events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The tens or even hundreds of primary failures during events such as hurricanes can not be fully mitigated by an electric network, because N-100 security is not realistic to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' N-1 security is typically studied by simulating the reaction of the system to every possible failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As the number of possible failure scenarios scales exponentially with the number of failures, it is computationally infeasible to consider all possible such scenarios in larger events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Initialising failure cascade models designed for N-1 studies with many initial failures is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Here, we present an approach that solves these issues by temporally resolving the potential damages induced by hurricanes and a stepwise application of a failure cascade model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This approach particularly allows us to identify critical power lines whose protection could most effectively reduce the risk of severe widespread power outages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Although the frequency of severe hurricanes is expected to increase12–14, such an approach does not exist so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Main text Our approach explicitly models the dynamical interplay of an extreme wind event with the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It temporally resolves both, the primary wind damages, and the cascades and secondary failures that result from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' We will use this approach to study the impact of massive TCs on the Texan power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Strong hurricanes, such as Harvey that made landfall on Texas and Louisiana in August 2017, can destroy more than hundreds transmission lines in an electric grid (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' These lines do not collapse simultaneously, but over the hours or days the TC passage takes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Making use of the chronological order of the line destructions, we divide each overall TC scenario into a sequence of 5 minute long scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In most of these individual steps, only one line fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' We then solve individual scenarios by representing the Texan transmission network in a DC power flow approximation with conservative load balancing assumptions (see Methods and Supplementary Methods 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This approach accounts for the ‘path dependency’ of the solution: Everytime a line collapses, secondary failures can occur, but also control mechanisms are immediately activated and try to bring back the energy balance to the system and, consequently, mitigate the effect of the failure (see Supplementary Methods 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Later primary damages along the TC track then meet a partially destroyed, rebalanced grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Thus, the effect of later failures can be more or, even, less intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It is the resilience of these intermediate, partially destroyed states that ultimately decides whether the impact of the TC is amplified by secondary failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1: Probability distributions of primary line failures and final power outages (a) Probability distribution of the total number of wind-induced line failures as generated by the probabilistic line fragility model for each of the seven recent hurricanes hitting Texas (category in brackets behind the name).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' TCs are sorted according to the means of the distributions which are indicated as solid vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (b) Probability distribution of the associated total power outage after TC passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The inset highlights large cascading failures that can also occur for the weaker hurricanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The dashed vertical lines indicate the reported power outages listed in the Supplementary Table 1 and the solid vertical lines represent the means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' See Methods section for the model parameters used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Unfortunately, neither detailed information about the topology of the exposed power grid nor about the exact power lines destroyed by the considered TC is publically accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' So a b Harvey (4) Ike (2) Claudette (1) Hanna (1) Erin (TS) Hermine (TS) (anod)d p(Np) Pout Laura (4) μp pout 0 20 40 60 80 100 120 140 0 10 20 30 40 Np pout [GW]here, we use a synthetic model of the Texan grid introduced by Bircheld et al21 (see Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2 as well as Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' To represent the TCs impact on the energy supply we combine this grid model with a probabilistic line destruction model (see Methods) forced by modelled historical wind fields from seven different TCs (see Supplementary Supplementary Methods 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The probabilistic model provides the probability of line failure in terms of wind speeds and allows to generate a large sample of temporally resolved realisations of line failure maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In the default setting considered here we assume a homogeneous base failure rate for all transmission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This is our main adjustable parameter and is tuned to reproduce observed power outages (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1(b) and Supplementary Methods 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The TCs are selected to cover several different types of trajectories and intensities and particularly include storms that continue to move westward after landfall and affect the southern and western parts of Texas such as Hurricane Claudette, Tropical Storm Erin, and Hurricane Hanna, contrary to most hurricanes that are steered northward by the Coriolis effect before western parts of Texas are reached22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Core result While the number of primary line failures follows a Poisson binomial distribution, the derived distribution of outages is heavily multimodal for all storm tracks with the potential of large to outages (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' These large damages turn out to not accumulate gradually over the course of the hurricane but occur suddenly in one or few time steps (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This sudden increase in outages is induced by cascading line failures taking the Houston and a weakly connected North-Western section of the grid offline (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2(d) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Figure 3 shows what damage patterns correspond to the various modes of the outage distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The disconnection of the North-West occurs due to the non-local effects of cascading failures in areas not directly affected by high wind speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' For example, hurricanes Harvey and Hanna never reach this region, but cause a considerable probability of outages affected by Harvey (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 3(a)-(c)), but also due to non-local cascades as seen for Hanna (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 3(h) and (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As the most populous city in Texas and a major load centre, the disconnection of Houston from the electrical networks causes the disconnection of a huge number of consumers from the electrical network and, consequently, the overproduction of generators located in the west of Texas, which have key roles to provide the required energy in Houston (see Supplementary Methods 5 and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Interestingly, the northern part of the electric grid is never impacted by outages caused by these three hurricanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Same figures for other hurricanes have been shown in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=" Figure 2: Simulation of hurricane-induced cascading failures in the Texan electric grid (a) The schematic variation of the supplied load in an electric grid before (pre-hurricane), during (hurricane phase) and after (restoration phase) a hurricane is loosely based on ERCOT's23." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The total power outage after a hurricane has passed, and the total energy (red area) that was not supplied are measures for the severity of an outage scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (b) Summary of all realisations of power outage trajectories simulated for hurricane Claudette (see Methods section and Supplementary Methods 5 for specification of the model parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Trajectories shown in red come in two types, those that aggregate damages gradually over time (Type I in the figure) and those that include a large cascade (Type II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The distribution of cascade sizes is multimodal and we use an empirical threshold of to define large cascades (see Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (c) and (d) show respectively the state of the power grid at the beginning and the end of the hurricane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' These two states are shown in panel (b), for one realisation of primary line failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Lines shown in black were destroyed by the hurricane or deactivated due to the secondary effects, for the other lines the relative line loading is shown, with red lines close to overload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In addition, the panel includes the track and a snapshot of the windfields of hurricane Claudette in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In the Supplement, we also provide a video of the simulation showing how the wind damages spread along the passage of hurricane Claudette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' b a Hurricane Restorationphase phase Supplied load 60 Type ye [GW] Eout anod 50 p d 40 Time 60 70 80 90 100 110 t [h] c d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 40 36 pout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 GW 36 - pout = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 GW [Pc = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='i GW P: = 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='6 GW 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='8 34 34 30 25 32 - 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='6 Windspeed [m/s] Latitude [°] Line Loading 20 30 30 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='4 15 28 28 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='2 5 Inactive Parts 26 26 - Hurricane Track 104 102 100 98 96 94 104 102 100 98 96 94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 0 Longitude [°] Longitude ["]noC 15GMFigure 4: Probability of line failure for different parts of the total power outage distribution (a-i) Probability that the failure of a given power line is involved in three different modes of the power outage distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The modes are indicated by the insets and the exact range of considered power outages are shown below these insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The probabilities are calculated as: number of realisations with a total outage within the specified range in each figure where the considered line failed / number of total realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The rows describe the probabilities for different hurricanes as indicated in the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Texan electric grid with grid elements colored according to their respective outage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The probability distributions shown in the insets are identical to the ones shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' a b p Cluster of pout E[6 GW, 12GW] generators pout E[23GW, 28GW] pout E[34GW, 44GW] Dallas Austin San Antonio Generators Houston Generators Generators Corpus Loads Loads Loads Christi Harvey Harvey Harvey d pout E[0GW pout E[3GW, 12 GW] pout E[15GW, 30GW] Generators Generators Generators Loads Loads Loads Claudette Claudette Claudette g h pout E[15GW, 30GW] Generators Generators Generators Loads Loads Loads Hanna Hanna Hanna 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 nodFor all seven hurricanes, the cascades play a major role in the total line failures associated with the event (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' They are induced by the overload of remaining lines and the isolation of grid elements, as well as the failure of islands with unavoidable overproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Figure 3: The probability of primary damages and secondary failures induced by hurricane Harvey In this plot the transmission lines are colored according to their high probability to be directly damaged by Harvey (blue lines) or to be deactivated due to the secondary effect of the hurricane (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As expected the primary damages are located around the path of Harvey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' However, secondary failures can occur far away from the hurricane track, which is related to the non-local effect of the primary damages in the power grid (see supplementary Methods 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In this plot, grey lines have a higher probability of remaining operational than failing due to any reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Our results are not sensitive to the assumption of a homogeneous base failure rate as similar characteristics are also derived when assuming randomised base failure rates (see Supplementary Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In addition, a temporal resolution of 5 minutes turned out to be adequate as time steps where several lines fail are rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' At this resolution it is also reasonable to assume that cascades of secondary failures have run their course before further lines are destroyed by the hurricane24,25 (for further discussion regarding the temporal resolution, see Supplementary Note 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Increasing Resilience The fact that large cascades are triggered by the failure of specific lines suggests targeting these lines for protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' To identify the critical lines that should be protected we define a 36 34 32 Latitude [°] 30 28 Most probable line status Unaffected 26 Primary Damage Secondary Failure 104 102 100 98 96 94 Longitude [°]priority index as the probability that the wind-induced damage of this specific line triggers a large cascade, that is, a cascade that increase the outage by more than 15 GW, averaged over all seven hurricanes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2(b) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (4) in Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As a baseline we also consider a conventional, static model (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The static index of a line is the conditional probability of a large outage given that the line is damaged by a TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In both the co-evolution model and the static baseline (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5(a) and (b)) the critical lines are mostly located around Houston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' To estimate the reduction in power outages that can be reached by protecting critical lines, we order them according to their priority index and evaluate the impact of the TC on the system with the first one to twenty lines protected, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' by being replaced by underground cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It is worth noting that the co-evolution priority index value for most transmission lines is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Only of them have a value above , and only lines above .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' By protecting these lines, large power outages and cascading failures are almost completely prevented for smaller storms and dramatically reduced for the larger ones (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5 and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' For the stronger hurricanes Harvey and Ike, the power outage distributions are shifted from the second peak to the first peak with (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Protecting the lines one by one shows that the reduction of the largest power outages improves smoothly, thus it is effective to protect up to twenty lines (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5(c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' While in the original system damage amplification was almost guaranteed, it rarely occurs in the reinforced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In summary of total lines reinforced leads to a 5 to 20 time reduction of the largest scale outages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The level of protection that can be reached by protecting the lines according to the priority index derived from the co-evolution models is generally higher than the protection of the same number of lines selected according to the priority index derived by the static model (see panel (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The static baseline also identifies some of the most critical lines (see Supplementary 4), but additional protections stop being effective after the first 6-10 lines (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5(c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This demonstrates that the co-evolution model, with its detailed picture of the partially destroyed states, reveals genuinely new and critical information for increasing the resilience of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It is worth to mention that the results obtained from homogeneous base failure rates are similar to the randomised ones (see Supplementary Methods 5 and Supplementary Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' moC 1OGMFigure 5: Level of risk reduction that can be reached by protecting power lines according to the priority index: The co-evolution model against the static model (a)-(b) 20 lines of the Texan power grid with the highest priority index (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (4)) obtained from the static model (orange lines), the co-evolution model (blue lines), and both approaches (green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The inset (b) shows a close-up view of Houston and Harris County, which contain most of the critical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As seen in (b) the critical lines obtained from both models are located in the same region, however, the co-evolutionary model identifies additional lines whose protection has a dramatic effect on increasing resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (c) Power outage distributions of hurricane Harvey in terms of the number of critical lines protected in both the co-evolution (blue) and the static model (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The second peak in the power outage distribution is strongly reduced as the number of protected lines increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' However, protecting lines obtained from the static model does not increase the resilience of the power grid as much as occurs in the co-evolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (d) Reduction of the large power outages obtained from both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' For all three strong hurricanes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Harvey, Ike and Claudette, the reduction in power outages is much greater in co-evolution model than the static one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' b a 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 36 - 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 34 - 32 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 Latitude [°] Latitude [°] 30 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 28 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 20 most critical lines coevolution method 26 - static method both methods 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 104 102 100 98 96 94 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='5 Longitude [°] d Longitude [°] c Staticmethod 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 Coevolutionmethod 0 Number of protected lines 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='6 10 Method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='4 coevolution static 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='2 20 Harvey ike Claudette 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='0 0 10 20 30 40 0 2 4 6 8 10 12 14 16 18 20 p(pout) [GW] Number of protected linesConclusion and outlook The co-evolution model of the Texan power grid has been introduced as an efficient approach to temporally resolve the line failures and secondary grid outages induced by TCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The model can resolve to considerable detail the way secondary failure cascades amplify the impact of extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=" Using this information it can be used to identify critical lines that should be protected to effectively increase the system's resilience and prevent the most severe outages." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Our model goes significantly beyond the state of the art so far represented by statistical and economic models that can only capture a static picture of the event and the network24–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' We have seen that such static approaches do not easily identify all of the critical lines during extended events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Their importance is only revealed by stepwise ‘tracking’ the destruction of the system and associated power outages and overloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' We expect that this co-evolution approach will also be a promising tool to understand and protect other grids exposed to spatio-temporally extended extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The results of our study are in agreement with a recent TC related risk assessment for Texas26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Combining our priority index with additional information about the cost of a reinforcement of the considered lines could also enable the identification of the most cost efficient way to reduce the probability of power outages above a critical limit to an intended value (see Supplementary Methods 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' While the model based on wind speeds and historical hurricane tracks already identified crucial structures in the grid, the co-evolution approach could naturally be extended to more sophisticated models and broader settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' One particularly important goal for future research will be to drive the model with potential future storm tracks due to climate change27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As the frequency of particularly strong TCs is expected to increase under global warming (WGI contribution to the AR6), understanding what lines are critical in the face of the weather of the next decades is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Another important avenue of broadening the model is to account for TC induced flooding (coastal flooding, pluvial or fluvial flooding) and associated destructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' These may follow a different temporal pattern where the adequacy of the approach proposed here has to be newly tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This would also provide a first step towards an assessment of genuine compound events in which several stresses for the grid coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Methods Electric grid data of Texas For the study we used the publicly available electric grid test case ACTIVSg200028, that covers the area of the so-called ERCOT Interconnection, which supplies percent of the electricity demand in Texas29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The test case is synthetic but resembles fundamental properties of the real grid, such as the spatial distribution of power generation and demand21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It encompasses buses with geographic locations, branches (both transmission lines and transformers) and covers four different voltage levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The test case comes with all required electrical parameters ranging from the power injections of buses to the power flow capacities of transmission lines and transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The flow capacities play a particularly important role for the simulation of cascading failures as they determine the amount of power that can be transported by individual lines and transformers without potentially damaging the equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Historical hurricane data Hurricane storm tracks are extracted from the International Best Track Archive for Climate Stewardship (IBTrACS)30, 31 as time series of cyclone center coordinates along with meteorological variables like maximum sustained wind speeds and minimum pressure on a h snapshot basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' For this study, a hand-picked selection of seven historical storms is used (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1 and 2) to cover several different types of trajectories and intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Particularly, the selection also includes storms that continue to move westward after landfall and affect the southern and western parts of Texas (see Hurricane Claudette, Tropical Storm Erin, and Hurricane Hanna in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2 and the Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1), contrary to most hurricanes that are steered northward by the Coriolis effect before western parts of Texas are reached22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' From the track records, we compute time series of wind fields within a radius of km from the storm center using the Holland model for surface winds, as implemented in the Python-package CLIMADA32, 33, at a spatial resolution of degrees (approximately km) and a temporal resolution of minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The intensities of the considered storms are also shown along the respective tracks in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='1 while other properties of the storms are listed in Supplementary Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Transmission line fragility model To model wind-induced failures of transmission lines, we first differentiate between overhead transmission lines and underground cables in the electric grid of Texas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Following Birchfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=', we analyse lines that are shorter than km ( miles) and connect a total load of at least MW as underground cables21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' All other lines are assumed to be overhead transmission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The latter are then divided into segments of length m, which corresponds to the average distance between transmission towers in Texas34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Our fragility model assigns failure rates to individual line segments according to where, denotes the wind force acting on the line segment for a given wind speed and is calculated according to the guidelines published by the American Society of Civil Engineers35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The parameter represents the inverse of the so-called time to failure, which indicates how long a line segment can withstand a wind force equal to the breaking force .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It is used as a free parameter to calibrate the model such that historically reported power outages are reproduced in our simulations (see Supplementary Methods 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The full wind force equation as well as the meaning and the values of all parameters can be found in Supplementary Methods 2 and Supplementary Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' In all figures shown in the main text, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Using the failure rates , we define the probability that a line segment fails during the time interval as This failure probability is inspired by the line fragility model established by Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=', which assumes that the failure probability is proportional to the ratio of the wind force and the breaking force36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' However, in contrast to their model, we define the failure probability using a time-dependent failure rate that allows us to take the time evolution of a field into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' A line is removed from the test case if any of its line segments fails during a time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It should be noted that multiple lines may be destroyed in the same time step, meaning that they are removed from the network simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (2), the probability of simultaneous failures increases with time step size .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' A discussion of the role of the time resolution can be found in Supplementary Note 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Cascading failure model Wind-induced line failures can trigger cascades of overload failures in the branches of the electric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As cascading failures typically evolve on smaller time scales than the temporal resolution of the wind field, we can assume a time scale separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' When the network topology is changed by a primary damage event, the power flows on the branches are rerouted using the DC power flow model here, are the net active power injections at the buses, are the bus voltage angles and are the elements of the nodal susceptance matrix that comprises the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' More details on the assumptions of the DC power flow model and the software used can be found in Supplementary Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' If the new state of the network exhibits any overloaded branch ( ), they are deactivated and the process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' When the network reaches a state without overloads, the algorithm advances to the br 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='002 hnext primary damage event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' When a load or generator gets disconnected or the grid is split into several parts, the global active power balance (GAPB) has to be restored in each network component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Motivated by a primary frequency control in real electric grids, we adjust the outputs of generators uniformly, while respecting their output limits defined in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Whenever the generator limits do not allow to fully restore the GAPB, we either conduct a uniform minimal load shedding or consider the blackout of the whole network component in the case of an unavoidable overproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The details of the algorithm are explained in Supplementary Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Quantification of power outages We use the following three different quantities to track the power outages arising in our simulations: (i) denotes the total supplied load at the end of each time step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=', after the cascading algorithm finished, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' It is calculated by adding up the demands of all connected loads across all islands that exist at the given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Since our co-evolution model assumes that cascading failures happen instantaneously, represents a step function for each individual TC scenario as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' We have simulated scenarios for each hurricane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (ii) Any cascading failure that actually causes a loss of supplied load results in a vertical transition of size in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' One such transition is annotated with for the highlighted scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (iii) All cascading failures that are triggered in a given TC scenario lead to a final power outage .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The interesting statistics of are shown and discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Identification of critical lines We identify critical overhead transmission lines by means of a priority index defined for each line as where denotes the set of considered hurricanes (seven hurricanes in this study) and is the probability of a large cascade being triggered by the wind-induced failure of line .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' More specifically, we call cascades large or belonging to type II if their associated power outage lies above an empirical threshold of GW (indicated as type II in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 2(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (4) includes an averaging over all considered hurricanes to discern lines that are critical for multiple hurricanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This allows us to propose line reinforcements that increase the resilience not only for a particular hurricane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Some properties of the most critical lines found in this study are listed in Dou na 0GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='GVSupplementary Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 5(a) and (b) shows the location of these lines and demonstrates that reinforcing them indeed increases the resilience of the electric grid substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' More details of the critical lines and a possibility to incorporate economic considerations into our analysis are discussed in Supplementary Methods 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Baseline Method Here, we apply the static model as a baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' By static model, we mean that all primary damages occur simultaneously and, then, the DC power model along with global active power balance (see Supplementary Methods 6) are activated once to bring back the energy balance in the system and to evaluate the total final power outages .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As discussed in Supplementary Note 2 the final power outage distributions are independent of the time resolution of the wind field, however the primary damages leading to large outages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' to , can be completely different ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' To indicate the critical lines obtained from the static model, first, we separate all scenarios in which .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Then, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' (4) to calculate the priority index of the primary damages leading to large cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The top lines with the highest priority index have been listed in Supplementary Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' As seen in this table, except for the six lines highlighted in red, the other lines are completely different from lines obtained from the co-evolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Code availability All code necessary to reproduce the findings in this work is openly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The time-dependent wind fields are computed using the open-source platform CLIMADA32, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The implementation of the transmission line fragility and the DC power model is available from https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='pik-potsdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='de/stuermer/itcpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Data availability The observed TCs from IBTrACS30, 31 are distributed under the permissive WMO open data licence through the IBTrACS website (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='ncei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='gov/products/international-best-track-archive) and can be directly retrieved through the CLIMADA32, 33 platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The electrical network data is openly available from the Texas A&M University’s electric grid test case repository (https://electricgrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='engr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content='edu/electricgrid-test-cases/activsg2000/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Acknowledgements This project has received funding from the ConNDyNet2 project under grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 03EF3055F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' This research has received funding from the German Academic Scholarship Foundation and the German Federal Ministry of Education and Research (BMBF) under 20 GM30 GMnoC 15GMthe research projects QUIDIC (01LP1907A) and SLICE (FKZ: 01LA1829A), and from the CHIPS project, part of AXIS, an ERA-NET initiated by JPI Climate, funded by FORMAS (Sweden), DLR/BMBF (Germany, grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 01LS1904A), AEI (Spain) and ANR (France) with co-funding by the European Union (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 776608).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Author Contribution M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Anvari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Hellmann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Otto contributed to design and conceive the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' The co-evolution model is designed and developed by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Anvari, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Stürmer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Plietzsch and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Hellmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' All simulations and data analyses of this work have been done by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' St\x7fürmer and under supervision of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Anvari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' All hurricane data have been provided by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Vogt during this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' All authors contributed to discussing and interpreting the results, and contributed to writing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Competing Interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Bialek, What Does the Power Outage on 9 August 2019 Tell Us about GB Power System, University of Cambridge, Energy Policy Research Group, Cambridge, Technical Report 2006, 2020.' metadata={'source': 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limiting global warming for tropical cyclone exposure, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Change, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 1–6, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFST4oBgHgl3EQfZzjU/content/2301.13793v1.pdf'} +page_content=' Birchfield, ACTIVSg2000: 2000-bus synthetic grid on footprint of Texas.' metadata={'source': 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Alvarez1 +Anima Anandkumar1,4 +1NVIDIA +2Meta AI, FAIR +3Fudan University +4Caltech +https://github.com/NVlabs/mask-auto-labeler +Figure 1. Examples of mask pseudo-labels generated by Mask Auto-Labeler on COCO. Only human-annotated bounding boxes are +used as supervision during training to obtain these results. +Abstract +We propose Mask Auto-Labeler (MAL), a high-quality +Transformer-based mask auto-labeling framework for in- +stance segmentation using only box annotations. MAL takes +box-cropped images as inputs and conditionally generates +their mask pseudo-labels.We show that Vision Transform- +ers are good mask auto-labelers. Our method significantly +reduces the gap between auto-labeling and human annota- +tion regarding mask quality. Instance segmentation models +trained using the MAL-generated masks can nearly match +the performance of their fully-supervised counterparts, re- +taining up to 97.4% performance of fully supervised mod- +els. The best model achieves 44.1% mAP on COCO in- +stance segmentation (test-dev 2017), outperforming state- +of-the-art box-supervised methods by significant margins. +Qualitative results indicate that masks produced by MAL +are, in some cases, even better than human annotations. +1. Introduction +Computer vision has seen significant progress over the +last decade. Tasks such as instance segmentation have made +it possible to localize and segment objects with pixel-level +accuracy. +However, these tasks rely heavily on expan- +sive human mask annotations. For instance, when creat- +ing the COCO dataset, about 55k worker hours were spent +on masks, which takes about 79% of the total annotation +time [1]. Moreover, humans also make mistakes. Human +annotations are often misaligned with actual object bound- +aries. On complicated objects, human annotation quality +tends to drop significantly if there is no quality control. Due +to the expensive cost and difficulty of quality control, some +other large-scale detection datasets such as Open Images [2] +and Objects365 [3], only contain partial or even no instance +segmentation labels. +In light of these limitations, there is an increasing in- +terest in pursuing box-supervised instance segmentation, +where the goal is to predict object masks from bounding +box supervision directly. Recent box-supervised instance +segmentation methods [4–8] have shown promising perfor- +mance. The emergence of these methods challenges the +long-held belief that mask annotations are needed to train +instance segmentation models. However, there is still a non- +negligible gap between state-of-the-art approaches and their +fully-supervised oracles. +Our contributions: To address box-supervised instance +segmentation, we introduce a two-phase framework consist- +ing of a mask auto-labeling phase and an instance segmenta- +tion training phase (see Fig. 2). We propose a Transformer- +based mask auto-labeling framework, Mask Auto-Labeler +(MAL), that takes Region-of-interest (RoI) images as inputs +1 +arXiv:2301.03992v1 [cs.CV] 10 Jan 2023 + +50 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +600100 +200 +300 +400 +0 +100 +200 +30050 +100 +150 +200 +250 +300 +350 +400 +100 +200 +300 +400 +500 +60050 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +600100 +200 +300 +400 +0 +100 +200 +300 +400 +500 +600100 +200 +300 +400 +0 +100 +200 +300 +400 +500 +600100 +PEPSI +200 +300 +400 +0 +100 +200 +3000 +100 +200 +300 +400 +0 +100 +200 +300 +400 +500 +6000 +50 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +600100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +4000 +50 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +600Box-supervised +Loss +Cropped +Regions +Supervised +Mask Loss +Mask Labels +MAL +Generate +Masks +Phase 1: Mask Auto-labeling +Inst Seg +Image +Phase 2: Instance Segmentation Training +Masks +Figure 2. +An overview of the two-phase framework of box- +supervised instance segmentation. For the first phase, we train +Mask Auto-Labeler using box supervision and conditionally gen- +erate masks of the cropped regions in training images (top). We +then train the instance segmentation models using the generated +masks (bottom). +and conditionally generates high-quality masks (demon- +strated in Fig. 1) within the box. Our contributions can be +summarized as follows: +• Our two-phase framework presents a versatile design +compatible with any instance segmentation architecture. +Unlike existing methods, our framework is simple and +agnostic to instance segmentation module designs. +• We show that Vision Transformers (ViTs) used as image +encoders yield surprisingly strong auto-labeling results. +We also demonstrate that some specific designs in MAL, +such as our attention-based decoder, multiple-instance +learning with box expansion, and class-agnostic training, +crucial for strong auto-labeling performance. Thanks to +these components, MAL sometimes even surpasses hu- +mans in annotation quality. +• Using MAL-generated masks for training, instance seg- +mentation models achieve up to 97.4% of their fully +supervised performance on COCO and LVIS. Our re- +sult significantly narrows down the gap between box- +supervised and fully supervised approaches. We also +demonstrate the outstanding open-vocabulary general- +ization of MAL by labeling novel categories not seen +during training. +Our method outperforms all the existing state-of-the- +art box-supervised instance segmentation methods by large +margins. This might be attributed to good representations +of ViTs and their emerging properties such as meaningful +grouping [9], where we observe that the attention to objects +might benefit our task significantly (demonstrated in Fig. +6). We also hypothesize that our class-agnostic training de- +sign enables MAL to focus on learning general grouping +instead of focusing on category information. Our strong re- +sults pave the way to remove the need for expensive human +annotation for instance segmentation in real-world settings. +2. Related work +2.1. Vision Transformers +Transformers were initially proposed in natural language +processing [10]. +Vision Transformers [11] (ViTs) later +emerged as highly competitive visual recognition models +that use multi-head self-attention (MHSA) instead of con- +volutions as the basic building block. These models are re- +cently marked by their competitive performance in many vi- +sual recognition tasks [12]. We broadly categorize existing +ViTs into two classes: plain ViTs, and hierarchical ViTs. +Standard Vision Transformers. Standard ViTs [11] are +the first vision transformers. Standard ViTs have the sim- +plest structures, which consist of a tokenization embedding +layer followed by a sequence of MHSA layers. However, +global MHSA layers can be heavy and usually face signif- +icant optimization issues. To improve their performance, +many designs and training recipes are proposed to train +ViTs in data-efficient manners [9,13–19]. +Hierarchical Vision Transformers. Hierarchical Vision +Transformers [12,20–22] are pyramid-shaped architectures +that aim to benefit other tasks besides image classification +with their multi-scale designs. On top of plain ViTs, these +ViTs [20,21] separate their multi-head self-attention layers +into hierarchical stages. Between the stages, there are spa- +tial reduction layers, such as max-pooling layers. These ar- +chitectures are usually mixed with convolutional layers [23] +and often adopt efficient self-attention designs to deal with +long sequence lengths. +2.2. Instance segmentation +Instance segmentation is a visual recognition task that +predicts the bounding boxes and masks of objects. +Fully supervised instance segmentation. In this setting, +both bounding boxes and instance-level masks are provided +as the supervision signals. Early works [24–27] follow a +two-stage architecture that generates box proposals or seg- +mentation proposals in the first stage and then produces the +final segmentation and classification information in the sec- +ond stage. Later, instance segmentation models are broadly +divided into two categories: some continue the spirit of +the two-stage design and extend it to multi-stage architec- +tures [28, 29]. +Others simplify the architecture and pro- +pose one-stage instance segmentation, e.g., YOLACT [30], +SOLO [31, 32], CondInst [33], PolarMask [34, 35]. Re- +cently, DETR and Deformable DETR [36, 37] show great +potential of query-based approaches in object detection. +Then, methods like MaxDeepLab [38], MaskFormer [39], +2 + +0 +100 +200 +300 +400 +0 +100 +200 +300 +400 +500 +600… +… +… +MHSA +FFN +FFN +MaxPool +FC +* +… +… +MHSA +FFN +FFN +* +Pos. Bags +Neg. Bags +Average +Multiple Instance Learning Loss ++ +- ++ + ++ +++ +-- -- ++ +++ + ++ +- - +-- - +Self Training +EMA +EMA +Conditional Random Fields Loss +𝑋! +Neighbors 𝑋" +Mean Field Algorithm +MaxPool +FC +𝐸 +𝐷 +𝐷! +𝐸! +𝑉! +𝑉 +𝐾! +𝐾 +Task Network +Teacher Network +Figure 3. Overview of MAL architecture. We visualize the architecture of Mask Auto-Labeler. Mask Auto-Labeler takes cropped images +as inputs. Mask Auto-Labeler consists of two symmetric networks, Task Network and Teacher Network. Each network contains the image +encoder E(or Et), and the mask decoder D(or Dt). We use the exponential moving average (EMA) to update the weights of the teacher +network. We apply multiple instance learning (MIL) loss and conditional random fields (CRFs) loss. The CRF loss takes the average mask +predictions of the teacher network and the task network to make the training more stable and generate refined masks for self-training. +PanopticSegFormer [40], Mask2Former [41] and Mask +DINO [42] are introduced along this line and have pushed +the boundary of instance segmentation. On the other hand, +the instance segmentation also benefits from more power- +ful backbone designs, such as Swin Transformers [12, 22], +ViTDet [43], and ConvNeXt [44]. +Weakly supervised instance segmentation. There are two +main styles of weakly supervised instance segmentation: +learning with image-level and box-level labels. The former +uses image-level class information to perform instance seg- +mentation [45–49], while the latter uses box-supervision. +Hsu et al. [4] leverages the tight-box priors. Later, Box- +Inst [5] proposes to leverage color smoothness to improve +accuracy. Besides that, DiscoBox [7] proposes to leverage +both color smoothness and inter-image correspondence for +the task. Other follow-ups [6,8] also leverage tight-box pri- +ors and color smoothness priors. +2.3. Deep learning interpretation +The interest in a deeper understanding of deep net- +works has inspired many works to study the interpreta- +tion of deep neural networks. +For example, Class Ac- +tivation Map (CAM) [50] and Grad-CAM [51] visualize +the emerging localization during image classification train- +ing of convolutional neural networks (CNNs). This abil- +ity has also inspired much weakly-supervised localization +and shows deep connections to general weakly-supervised +learning, which partly motivates our decoder design in this +paper. DINO [9] further shows that meaning visual group- +ing emerges during self-supervised learning with ViTs. In +addition, FAN [52] shows that such emerging properties in +ViTs are linked to their robustness. +3. Method +Our work differs from previous box-supervised instance +segmentation frameworks [4–8] that simultaneously learns +detection and instance segmentation. We leverage a two- +phase framework as visualized in Fig. 2, which allows us to +have a network focused on generating mask pseudo-labels +in phase 1, and another network focused on learning in- +stance segmentation [24, 28, 41, 43] in phase 2. Our pro- +posed auto-labeling framework is used in phase 1 to gener- +ate high-quality mask pseudo-labels. +We propose this two-phase framework because it brings +the following benefits: +• We can relax the learning constraints in phase 1 and +focus only on mask pseudo-labels. Therefore, in this +phase, we can take Region-of-interest (RoI) images in- +stead of untrimmed images as inputs. This change al- +lows us to use a higher resolution for small objects and +a strong training technique mentioned in Sec. 3.1, which +helps improve the mask quality. +• We can leverage different image encoders and mask de- +coders in phases 1 and 2 to achieve higher performance. +We empirically found that phases 1 and 2 favor different +architectures for the image encoders and mask decoders. +See the ablation study in Tab. 3 and 4. +• We can use MAL-generated masks to directly train the +most fully supervised instance segmentation models in +phase 2. This makes our approach more flexible than +previous architecture-specific box-supervised instance +segmentation approaches [4–8]. +As phase 2 follows the previous standard pipelines, +which do not need to be re-introduced here, we focus on +introducing phase 1 (MAL) in the following subsections. +3 + +0 +50 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +6003.1. RoI input generation +Most +box-supervised +instance +segmentation +ap- +proaches [4–7] are trained using the entire images. +However, we find that using RoI images might have +more benefits in box-supervised instance segmentation. +Moreover, we compare two intuitive sampling strategies +of RoI images to obtain foreground and background pixels +and explain the better strategy, box expansion, in detail. +Benefits of using RoI inputs. There are two advantages of +using RoI images for inputs. First, using the RoI images +as inputs is naturally good for handling small objects be- +cause no matter how small the objects are, the RoI images +are enlarged to avoid the issues caused by low resolution. +Secondly, having RoI inputs allows MAL to focus on learn- +ing segmentation and avoid being distracted from learning +other complicated tasks, e.g., object detection. RoI sam- +pling strategy. The sampling strategy should ensure both +positive and negative pixels are included. We present two +straightforward sampling strategies: +• The first strategy is to use bounding boxes to crop the +images for positive inputs. We crop the images using +randomly generated boxes containing only background +pixels for negative inputs. MAL does not generate good +mask pseudo-labels with cropping strategy. We observe +that the networks tend to learn the trivial solution (all +pixels are predicted as either foreground or background). +• The second is to expand the bounding boxes randomly +and include background pixels, where negative bags are +chosen from the expanded rows and columns. We visu- +alize how we define positive/negative bags in Fig. 3 and +explain the detail in Sec. 3.3. This detailed design is +critical to make MAL work as it prevents MAL from +learning trivial solutions. Without this design, the gen- +erated masks tend to fill the entire bounding box. +Box expansion specifics. Given an untrimmed image Iu ∈ +RC×Hu×W u and the bounding box b = (x0, y0, x1, y1) in- +dicating the x, y coordinates of the top-left corners and the +bottom-right corners. To obtain background pixels, we ran- +domly expand the bounding box b to b′ = (xc + βx(x0 − +xc), yc +β′ +x(y0 −yc), xc +βy(x1 −xc), yc +β′ +y(y1 −yc)), +where xc = (x0 + x1)/2, yc = (y0 + y1)/2. To gener- +ate random values of βx, β′ +x, βy, β′ +y, we randomly generate +θx, θy ∈ [0, θ] for x- and y-direction, where θ is the upper +bound of box expansion rate. Next, we randomly generate +βx ∈ [0, θx] and βy ∈ [0, θy]. In the end, we assign β′ +x as +θx−βx and β′ +y as θy −βy. Finally, we use b′ to crop the im- +age and obtain trimmed image It. We conduct the ablation +study for θ in Tab. 5. At last, We resize the trimmed image +It to the size of C × Hc × W c as the input image Ic. +Class +Tokens +k +q +v +Transformer +Layer +* +(a) +(b) +(c) +(d) +Figure 4. (a) The fully connected decoder (b) The fully convolu- +tional Decoder (c) The attention-based decoder (used in MAL) (d) +The query-based Decoder. +3.2. MAL architecture +MAL can be divided into two symmetric networks: the +task network and the teacher network. The task network +consists of an image encoder denoted as E, and a mask de- +coder denoted as D, demonstrated in Fig. 3. The architec- +ture of the teacher network is identical to the task network. +We denote the segmentation output of the task network and +the teacher network as m, mt ∈ {0, 1}N, respectively. +Image encoder. We use Standard ViTs [11] as the image +encoder and drop the classification head of Standard ViTs. +We compare different image encoders in Sec. 4.4. We also +try feature pyramid networks on top of Standard ViTs, e.g., +FPN [53], but it causes a performance drop. Similar con- +clusions were also found in ViTDet [43]. +Mask decoder. For the mask decoder D, we use a simple +attention-based network inspired by YOLACT [30], which +includes an instance-aware head K and a pixel-wise head +V , where D(E(I)) = K(E(I)) · V (E(I)), and “ · ” repre- +sents the inner-product operator. +For the instance-aware head K, we use a max-pooling +layer followed by a fully connected layer. The input chan- +nel dimension of K is equivalent to the output channel di- +mension of E. The output channel dimension of K is 256. +For the pixel-wise head V , we use four sequential convo- +lutional layers. Each is followed by a ReLU layer. Between +the second and the third convolutional layer, we insert a bi- +linear interpolation layer to increase the feature resolution +by 2. The input channel dimension is equivalent to the out- +put channel dimension of E. We use 256 dimensions for +hidden channels and output channels. We also compare dif- +ferent design choices of mask decoders in Sec. 4.5. +Exponential moving average (EMA) teacher. Instead of +training the teacher network directly, we leverage exponen- +tial moving averages (EMA) to update the parameters in the +teacher network using the parameters in the task network +similar to MOCO [54]. The goal of using EMA Teacher +is to eliminate the loss-explosion issues in training since +optimizing Standard Vision Transformers is usually non- +trivial [13, 14, 16]. We do not observe any significant per- +formance drop or improvement on DeiT-small-based MAL +after removing the teacher network. However, it makes the +training more stable when we use larger-scale image en- +coders in MAL, e.g. ViT-MAE-Base [13]. +4 + +3.3. Losses +We use Multiple Instance Learning Loss Lmil and Con- +ditional Random Field Loss Lcrf as the box-supervised loss: +L = αmilLmil + αcrfLcrf +(1) +Multiple Instance Learning Loss. The motivation of the +Multiple Instance Segmentation is to exploit the priors of +tight-bounding box annotations. +After the student network produces the output m, we ap- +ply the Multiple Instance Learning (MIL) Loss on the out- +put mask m. We demonstrate the process in Fig. 3. +We denote mi,j as the mask score at the location i, j in +the image Ic. We define each pixel as an instance in the +MIL loss. Inspired by BBTP [4], we treat each row or col- +umn of pixels as a bag. We determine whether a bag is pos- +itive or negative based on whether it passes a ground-truth +box. We define the bags as B, and each bag Bi contains a +row or column of pixels. Additionally, we define the label +for each bag g, and each label gi corresponds to a bag Bi. +Therefore, we use the max pooling as the reduction func- +tion and dice loss [55]: +Lmil = 1 − +2 � +i gi · max{Bi}2 +� +i max{Bi}2 + � +i g2 +i +(2) +Conditional Random Field Loss. The goal of CRF loss +is to refine the mask prediction by imposing the smooth- +ness priors via energy minimization. Then, we leverage this +refined mask as pseudo-labels to self-train the mask predic- +tion in an online-teacher manner. We use the average mask +prediction ma = 1 +2(m + mt) as the mask prediction to be +refined for more stable training. +Next, we define a random field X = {X1, ..., XN}, +where N = Hc × W c is the size of cropped image and +each Xi represents the label that corresponds to a pixel in +Ic, therefore we have X ∈ {0, 1}N, meaning the back- +ground or the foreground. We use l ∈ {0, 1}N to represent +a labeling of X minimizing the following CRF energy: +E(l|ma, Xc) = µ(X|ma, Ic) + ψ(X|Ic), +(3) +where µ(X|ma, Ic) represents the unary potentials, which +is used to align Xi and ma +i since we assume that most of +the mask predictions are correct. Meanwhile, ψ(X|Ic) rep- +resents the pairwise potential, which sharpens the refined +mask. Specifically, we define the pairwise potentials as: +ψ(X|Ic) = +� +i∈{0..N−1}, +j∈N (i) +ω exp(−|Ic +i − Ic +j |2 +2ζ2 +)[Xi ̸= Xj], (4) +where N(i) represents the set of 8 immediate neighbors to +Xi as shown in Fig. 3. Then, we use the MeanField al- +gorithm [7, 56] to efficiently approximate the optimal so- +lution, denoted as l = MeanField(Ic, ma). We attach +the derivation and PyTorch code in the supplementary. At +last, we apply Dice Loss to leverage the refined masks l to +self-train the models as: +Lcrf = 1 − 2 � +i limi +� +i l2 +i + m2 +i +(5) +4. Experiments +We evaluate MAL on COCO dataset [1], and LVIS [57]. +The main results on COCO and LVIS are shown in Tab. 1 +and 2. The qualitative results are shown in Fig. 1 and Fig. 5. +4.1. Datasets +COCO dataset. contains 80 semantic categories. We fol- +low the standard partition, which includes train2017 (115K +images), val2017 (5K images), and test-dev (20k images). +LVIS dataset. contains 1200+ categories and 164K images. +We follow the standard partition of training and validation. +4.2. Implementation Details +We use 8 NVIDIA Tesla V100s to run the experiments. +Phase 1 (mask auto-labeling). We use AdamW [58] as +the network optimizer and set the two momentums as 0.9, +0.9. We use the cosine and annealing scheduler to adjust the +learning rate, which is set to 1.5 · 10−6 per image. The MIL +loss weight αmil, CRF loss weight αcrf, ζ, and ω in CRF +pairwise potentials are set to 4, 0.5, 0.5, 2, respectively. We +analyze the sensitivity of the loss weights and CRF hyper- +parameters in Fig. 8. We use the input resolution of 512 × +512, and a batch size of 32 (4 per GPU). For EMA, we use +a momentum of 0.996. For the task and teacher network, +we apply random flip data augmentation. On top of that, +we apply extra random color jittering, random grey-scale +conversion, and random Gaussian blur for the task network. +We train MAL for 10 epochs. It takes around 23 hours and +35 hours to train MAL with Standard ViT-Base [11] on the +COCO and LVIS datasets, respectively. +Phase 2 (Training instance segmentation models). We +select a couple of high-performance fully supervised in- +stance segmentation models, which are ConvNeXts [44] +with Cascade R-CNN [28], Swin Transformers [12] with +Mask2Former [41], ResNets [59] and ResNeXts [60] with +SOLOv2 [31]. MAL works extremely well with these ar- +chitectures, which demonstrates the great power of Mask +Auto-Labeler from the perspective of accuracy and gener- +alization. We leverage the codebase in MMDetection [61] +for phase 2. Again, we only replace the GT masks with +MAL-generated mask pseudo-labels to adjust all these fully +supervised models to box-supervised learning. +4.3. Instance segmentation results +Retention Rate. We argue that the sole mAP of instance +segmentation is not fair enough to evaluate box-supervised +instance segmentation since the performance gain can be +5 + +Method +Labeler Backbone +InstSeg Backbone +InstSeg Model +Sup +(%)Mask APval +(%)Mask APtest +(%)Ret.val +(%)Ret.test +Mask R-CNN∗ [24] +- +ResNet-101 +Mask R-CNN +Mask +38.6 +38.8 +- +- +Mask R-CNN∗ [24] +- +ResNeXt-101 +Mask R-CNN +Mask +39.5 +39.9 +- +- +CondInst [33] +- +ResNet-101 +CondInst +Mask +38.6 +39.1 +- +- +SOLOv2 [31] +- +ResNet-50 +SOLOv2 +Mask +37.5 +38.4 +- +- +SOLOv2 [31] +- +ResNet-101-DCN +SOLOv2 +Mask +41.7 +41.8 +- +- +SOLOv2 [31] +- +ResNeXt-101-DCN +SOLOv2 +Mask +42.4 +42.7 +- +- +ConvNeXt [44] +- +ConvNeXt-Small [44] +Cascade R-CNN +Mask +44.8 +45.5 +- +- +ConvNeXt [44] +- +ConvNeXt-Base [44] +Cascade R-CNN +Mask +45.4 +46.1 +- +- +Mask2Former [41] +- +Swin-Small +Mask2Former +Mask +46.1 +47.0 +- +- +BBTP† [4] +- +ResNet-101 +Mask R-CNN +Box +- +21.1 +- +59.1 +BoxInst [5] +- +ResNet-101 +CondInst +Box +33.0 +33.2 +85.5 +84.9 +BoxLevelSet [6] +- +ResNet-101-DCN +SOLOv2 +Box +35.0 +35.4 +83.9 +83.5 +DiscoBox [7] +- +ResNet-50 +SOLOv2 +Box +30.7 +32.0 +81.9 +83.3 +DiscoBox [7] +- +ResNet-101-DCN +SOLOv2 +Box +35.3 +35.8 +84.7 +85.9 +DiscoBox [7] +- +ResNeXt-101-DCN +SOLOv2 +Box +37.3 +37.9 +88.0 +88.8 +BoxTeacher [8] +- +Swin-Base +CondInst +Box +- +40.0 +- +- +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNet-50 +SOLOv2 +Box +35.0 +35.7 +93.3 +93.0 +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNet-101-DCN +SOLOv2 +Box +38.2 +38.7 +91.6 +92.6 +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNeXt-101-DCN +SOLOv2 +Box +38.9 +39.1 +91.7 +91.6 +Mask Auto-Labeler +ViT-MAE-Base [13] +ConvNeXt-Small [44] +Cascade R-CNN +Box +42.3 +43.0 +94.4 +94.5 +Mask Auto-Labeler +ViT-MAE-Base [13] +ConvNeXt-Base [44] +Cascade R-CNN +Box +42.9 +43.3 +94.5 +93.9 +Mask Auto-Labeler +ViT-MAE-Base [13] +Swin-Small [12] +Mask2Former [41] +Box +43.3 +44.1 +93.9 +93.8 +Table 1. Main results on COCO. Ret means the retention rate of box-supervised mask AP +supervised mask AP . MAL with SOLOv2/ResNeXt-101 outperforms +DiscoBox with SOLOv2/ResNeXt-101 by 1.6% on val2017 and 1.3% on test-dev. Our best model (Mask2former/Swin-Small) achieves +43.3% AP on val and 44.1% AP on test-dev. +achieved by improving box quality unrelated to segmenta- +tion quality. However, the retention rate can better reflect +the real mask quality because the fully supervised counter- +parts also get boosted by the better box results. +Results on COCO. In table 1, we show that various mod- +ern instance segmentation models can achieve up to 94.5% +performance with the pseudo-labels of the fully supervised +oracles. Our best results are 43.3% mAP on COCO test-dev +and 44.1% mAP on COCO val, achieved by using MAL +(Standard ViT-Base [11] pretrained with MAE) for phase +1, and using Mask2Former (Swin-Small) [12,41] for phase +2. There is no significant retention drop when we use the +mask pseudo-labels to train more powerful instance seg- +mentation models. On the contrary, the higher retention +rates on COCO are achieved by the heavier instance seg- +mentation models, e.g., Cascade R-CNN with ConvNeXts +and Mask2Former with Swin-Small. However, other meth- +ods have significantly lower retention rates compared with +MAL. The experiment results quantitatively imply that the +mask quality outperforms other methods by a large margin. +Results on LVIS. In table 2, we also observe that all in- +stance segmentation models work very well with the mask +pseudo-labels generated by MAL (Ret. = 93% ˜ 98%). We +visualize part of the results in figure 5. We also evaluate +the open-vocabulary ability of MAL by training MAL on +COCO dataset but generating mask pseudo-labels on LVIS, +and thus training instance segmentation models using these +mask pseudo-labels. +4.4. Image encoder variation +To support our claim that Vision Transformers are good +auto-labelers, we compare three popular networks as the im- +age encoders of MAL: Standard Vision Transformers [11, +13,16], Swin Transformer [12], ConvNeXts [44] in Tab. 4. +First, +we compare the fully supervised pretrained +weights of these three models. We choose the official fully +supervised pre-trained weights of ConvNeXts and Swin +Transformers. For Standard Vision Transformers, we adopt +a popular fully supervised approach, DeiT [16]. We ob- +serve that fully supervised Standard Vision Transformers +(DeiT) as image encoders of Mask Auto-Labeler are better +than Swin Transformers and ConvNeXts even though the +imaganet-1k performance of Swin Transformers and Con- +vNeXts is higher than that of DeiT. We argue that the suc- +cess of Standard Vision Transformers might be owed to the +self-emerging properties of Standard ViTs [9, 11] (visual- +ized in Fig. 6), and the larger-receptive field brought by +global multi-head self-attention layers. +Second, the mask pseudo-labels can be further improved +by Mask AutoEncoder (MAE) pretraining [13]. The poten- +tial reason might be that MAE pretraining enhances Stan- +dard ViTs via learning pixel-level information, which is +very important for dense-prediction tasks like segmentation. +4.5. Mask decoder variation +We compare four different modern designs of mask de- +coders: the fully connected Decoder [62], the fully convolu- +tional decoder [24,63], the attention-based decoder [30,31], +and the query-based decoder [41] in Tab. +3. We visual- +ize different designs of mask decoders in Figure 4. For the +fully connected Decoder, we use two fully connected layers +with a hidden dimension of 2048 and then output a confi- +dence map for each pixel. We reshape this output vector as +the 2D confidence map. We introduce the attention-based +decoder in Sec 3.2. For the fully convolutional Decoder, +We adopt the pixel-wise head V in the attention-based De- +6 + +Figure 5. Qualitative results of mask pseudo-labels generated by Mask Auto-Labeler on LVIS v1. +Method +Autolabeler Backbone +InstSeg Backbone +InstSeg Model +Training Data +Sup +(%)Mask APval +(%)Ret.val +Mask R-CNN [24] +- +ResNet-50-DCN +Mask R-CNN [24] +- +Mask +21.7 +- +Mask R-CNN [24] +- +ResNet-101-DCN +Mask R-CNN [24] +- +Mask +23.6 +- +Mask R-CNN [24] +- +ResNeXt-101-32x4d-FPN +Mask R-CNN [24] +- +Mask +25.5 +- +Mask R-CNN [24] +- +ResNeXt-101-64x4d-FPN +Mask R-CNN [24] +- +Mask +25.8 +- +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNet-50-DCN +Mask R-CNN [24] +LVIS v1 +Box +20.7 +95.4 +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNet-101-DCN +Mask R-CNN [24] +LVIS v1 +Box +23.0 +97.4 +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNeXt-101-32x4d-FPN +Mask R-CNN [24] +LVIS v1 +Box +23.7 +92.9 +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNeXt-101-64x4d-FPN +Mask R-CNN [24] +LVIS v1 +Box +24.5 +95.0 +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNeXt-101-32x4d-FPN +Mask R-CNN [24] +COCO +Box +23.3 +91.8 +Mask Auto-Labeler +ViT-MAE-Base [13] +ResNeXt-101-64x4d-FPN +Mask R-CNN [24] +COCO +Box +24.2 +93.8 +Table 2. Main results on LVIS v1. Training data means the dataset we use for training MAL. We also finetune it on COCO and then generate +pseudo-labels of LVIS v1. Compared with trained on LVIS v1 directly, MAL finetuned on COCO only caused around 0.35% mAP drop +on the final results, which indicates the great potential of the open-set ability of MAL. Ret means the retention rate of box-supervised mask AP +supervised mask AP . +Mask decoder +(%)Mask APval +(%)Ret.val +Fully connected decoder +35.5 +79.2 +Fully convolutional decoder +36.1 +80.5 +Attention-based decoder +42.3 +94.4 +Query-based decoder +- +- +Table 3. Ablation study of box expansion. We use Standard ViT- +MAE-Base as the image encoder of MAL in phase 1 and Cascade +RCNN with ConvNext-Small as the instance segmentation models +in phase 2. The numbers are reported in % Mask mAP. Among +different designs, the attention-based decoder performs the best. +We can not obtain reasonable results with Query-based Decoder. +coder. For the query-based decoder, we follow the design +in Mask2Former [41]. We spend much effort exploring the +query-based Decoder on MAL since it performs extremely +well on fully supervised instance segmentation. However, +the results are surprisingly unsatisfactory. We suspect the +slightly heavier layers might cause optimization issues un- +der the box-supervised losses. +Experiments show that box-supervised instance segmen- +tation favors the attention-based decoder. However, state- +of-the-art instance segmentation and object detection meth- +ods often adopt the fully convolutional decoder [15, 43] +or the query-based decoder [41]. Our proposed two-phase +framework resolves this dilemma and allows the networks +to enjoy the merits of both the attention-based Decoder and +the non-attention-based Decoders. +4.6. Clustering analysis +As the results are shown in Tab. 4, we wonder why the +Standard ViTs outperform other modern image encoders in +auto-labeling. As the comparison of classification ability +Backbone +IN-1k Acc@1 +Mask APval +Ret.val +ConvNeXt-Base [44] +83.8 +39.6 +88.4 +Swin-Base [12] +83.5 +40.2 +89.7 +ViT-DeiT-Small [64] +79.9 +40.8 +91.0 +ViT-DeiT-Base [64] +81.8 +41.1 +91.7 +ViT-MAE-Base [13] +83.6 +42.3 +94.4 +ViT-MAE-Large [13] +85.9 +42.3 +94.4 +Table 4. +Ablation study of different backbones. +All models +are pre-trained on ImageNet-1k. +ConvNeXt and Swin Trans- +former outperform DeiT on image classification, but standard ViT- +Small [16] (ViT-DeiT-Small) outperforms ConvNeXt-base and +Swin-Base on mask Auto-labeling. +Standard ViT-Base (ViT- +MAE-Base) and Standard ViT-Large (ViT-MAE-Large) pretrained +via MAE achieve the best performance on mask Auto-labeling. +does not seem to reflect the actual ability of auto-labeling, +we try to use the ability clustering to evaluate the image en- +coders because foreground(FG)/background(BG) segmen- +tation is very similar to the binary clustering problem. +Specifically, we extract the feature map output by the last +layers of Swin Transformers [12], ConvNeXts [44], Stan- +dard ViTs [11]. Then, we use the GT mask to divide the +feature vectors into the FG and BG feature sets. By evalu- +ating the average distance from the FG/BG feature vectors +to their clustering centers, we can reveal the ability of the +networks to distinguish FG and BG pixels empirically. +Formally, we define the feature vector of token i gener- +ated by backbone E as f E +i . We define the FG/BG clustering +centers f ′ +1, f ′ +0 as the mean of the FG/BG feature vectors. +Then, we use the following metric as the clustering score: +S = 1 +N +N +� +i +( f E +i +|f E +i | − +f ′ +γ(i) +|f ′ +γ(i)|)2, +(6) +7 + +50 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +6000 +100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +40050 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +600100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +400 +500 +600100 +200 +300 +400 +0 +100 +200 +300 +400 +500 +6000 +50 +100 +150 +200 +250 +300 +350 +400 +0 +100 +200 +300 +400 +500 +600Figure 6. Attention visualization of two RoI images produced by MAL. In each image group, the left-most image is the original image. +We visualize the attention map output by the 4th, 8th, 12th MHSA layers of the Standard ViTs in MAL. +(a) MAL-generated Masks are sharper and more boundary-sticky +(b) Occlusion Issues +MAL Mask +Pseudo-labels +Ground-truth +Masks +Figure 7. The lateral comparison between MAL-generated pseudo-labels (top) and GT masks (bottom) on COCO val2017. On the left, we +observe that MAL-generated pseudo-labels are sharper and more boundary-sticky than GT masks in some cases. On the right, we observe +that in highly occluded situations, human-annotated masks are still better. +25 +30 +35 +40 +0.1 0.25 0.5 0.75 1 +𝛼!"# +32 +34 +36 +1 +2 +4 +8 +16 +𝛼$%& +30 +32 +34 +36 +0.5 +1 +2 +4 +6 +𝜔 +27 +30 +33 +36 +0.1 0.25 0.5 0.75 1 +𝜁 +Figure 8. Sensitivity analysis of loss weights and CRF hyper- +parameters. We use ViT-Base [11] pretrained via MAE [13] as the +image encoder for the first phase and SOLOv2 (ResNet-50) for the +second phase. The x-axis and y-axis indicate the hyper-parameter +values and the (%)mask AP, respectively. +θ +Mask APval +Ret.val +0.6 +41.3 +92.2 +0.8 +41.7 +93.1 +1.0 +42.2 +94.2 +1.2 +42.3 +94.4 +1.4 +42.0 +93.8 +1.6 +41.8 +93.3 +Table 5. Ablation on box ex- +pansion ratio. We use Standard +ViT-Base pretrained via MAE +(ViT-MAE-Base) and Cascade +R-CNN (ConvNeXt-Small) for +phase 1 and 2. +Backbone +Score (↓) +ConvNeXt-Base [44] +0.459 +Swin-Base [12] +0.425 +ViT-DeiT-Small [64] +0.431 +ViT-DeiT-Base [64] +0.398 +ViT-MAE-Base [13] +0.324 +ViT-MAE-Large [13] +0.301 +Table 6. +Clustering scores +for different image encoders. +The smaller clustering scores +imply a better ability to dis- +tinguish foreground and back- +ground features. +where if pixel i is FG, γ(i) = 1, otherwise γ(i) = 0. +We show the clustering evaluation on the COCO val +2017 in Tab. 6. The results align our conclusion that Stan- +dard Vision Transformers are better at mask auto-labeling. +4.7. MAL masks v.s. GT masks +We show the apples to apples qualitative comparison in +Fig. 7 and make the following observations. First, MAL- +generated mask pseudo-labels are considerably sharper and +boundary-sticky than human-annotated ones since humans +have difficulties in aligning with the true boundaries. Sec- +ond, severe occlusion also presents a challenging issue. +5. Conclusion +In this work, we propose a novel two-phase frame- +work for box-supervised instance segmentation and a +novel Transformer-based architecture, Mask Auto-Labeler +(MAL), to generate high-quality mask pseudo-labels in +phase 1. We reveal that Standard Vision Transformers are +good mask auto-labelers. Moreover, we find that random +using box-expansion RoI inputs, the attention-based De- +coder, and class-agnostic training are crucial to the strong +mask auto-labeling performance. Moreover, thanks to the +two-phase framework design and MAL, we can adjust al- +most all kinds of fully supervised instance segmentation +models to box-supervised learning with little performance +drop, which shows the great generalization of MAL. +Limitations. Although great improvement has been made +by our approaches in mask auto-labeling, we still observe +many failure cases in the occlusion situation, where human +annotations are much better than MAL-generated masks. +Additionally, we meet saturation problems when scaling the +model from Standard ViT-Base to Standard ViT-Large. We +leave those problems in the future work. +Broader impacts. Our proposed Transformer-based mask +auto-labeler and the two-phase architecture serve as a stan- +dard paradigm for high-quality box-supervised instance +segmentation. 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The pseudo- +code to obtain l using mean field is attached in Alg. 1: +Algorithm 1 Mean field algorithm for CRFs. +1: procedure MEANFIELD(m, I) +2: +Ki,j ←− ω exp(− |Ii−Ij| +2ζ2 +) +3: +▷ Initialize the Gaussian kernels +4: +l ← m +▷ Initialize l using m +5: +while not converge do +▷ Iterate until convergence +6: +for i ← 1 to |l| do +7: +ˆli ← li +8: +for j ∈ N(i) do +9: +ˆli ← ˆli + Kj ∗ lj +10: +▷ Message passing +11: +end for +12: +end for +13: +l ← ϕ(ˆl) +▷ ϕ is a clamp function +14: +end while +15: +return λ(l) +▷ λ is a threshold function +16: end procedure +A.2. Additional implementation details +We use the same hyper-parameters on all benchmarks +for all image encoders (Standard ViTs [11, 13, 16], Swin +Transformers [12], and ConvNeXts [44]) and mask de- +coders (fully connected decoder, fully convolutional de- +coder, attention-based decoder, ), including batch size, opti- +mization hyper-parameters. We observe a performance drop +when we add parametric layers or multi-scale lateral/skip +connections [43, 53] between the image encoder (Standard +ViTs, Swin Transformers, ConvNeXts) and the mask de- +coder (attention-based decoder). We insert a couple of the +bi-linear interpolation layers to resize the feature map be- +tween the image encoder and the mask decoder and resize +the segmentation score map. Specifically, we resize the fea- +ture map produced by the image encoder to 1/16 (small), +1/8 (medium), 1/4 (large) size of the raw input according +to the size of the objects. We divide the objects into three +scales regarding to the area of their bound boxes. We use +the area ranges of [0, 322), [322, 962), [962, ∞) to cover +small, medium, and large objects, respectively. We resize +the mask prediction map to 512 × 512 to reach the original +resolution of the input images. +Moreover, we also try three naive ways to add classifica- +tion loss, but it does not work well with MAL. First, we add +another fully connected layer as the classification decoder, +which takes the feature map of the first fully connected layer +of the instance-aware head K. With this design, the classi- +fication causes a significant performance drop. Secondly, +we use two extra fully connected layers or the original clas- +sification decoder of standard ViTs as the classification de- +coder, which directly takes the feature map of the image +encoder. However, the classification loss does not provide +performance improvement or loss in this scenario. +A.3. Benefits for object detection +The supervised object detection models benefit from the +extra mask supervision [24], which improves detection re- +sults. +Specifically, we follow the settings in Mask R- +CNN [24]. First, we use RoI Align, the box branch, and +the box supervision without mask supervision. Second, we +add the mask branch and ground-truth mask supervision on +top of the first baseline. The second baseline is the original +Mask R-CNN. Thirdly, we replace the ground-truth masks +with the mask pseudo-labels generated by MAL on top of +the second baseline. It turns out that using MAL-generated +mask pseudo-labels for mask supervision brings in an im- +provement similar to ground-truth masks on detection. We +show the results in Tab. 7. +A.4. Additional qualitative results +We also visualize the prediction results produced by +the instance segmentation models trained with ground-truth +masks and mask pseudo-labels in Fig. 9. In most cases, we +argue that humans cannot tell which results are produced by +the models supervised by human-annotated labels. +12 + +InstSeg Backbone +Dataset +Mask Labels +(%)AP +(%)AP50 +(%)AP75 +(%)APS +(%)APM +(%)APL +ResNet-50-DCN [59] +LVIS v1 +None +22.0 +36.4 +22.9 +16.8 +29.1 +33.4 +ResNet-50-DCN [59] +LVIS v1 +GT mask +22.5 +36.9 +23.8 +16.8 +29.7 +35.0 +ResNet-50-DCN [59] +LVIS v1 +MAL mask +22.6 +37.2 +23.8 +17.3 +29.8 +34.6 +ResNet-101-DCN [59] +LVIS v1 +None +24.4 +39.5 +26.1 +17.9 +32.2 +36.7 +ResNet-101-DCN [59] +LVIS v1 +GT mask +24.6 +39.7 +26.1 +18.3 +32.1 +38.3 +ResNet-101-DCN [59] +LVIS v1 +MAL mask +25.1 +40.0 +26.7 +18.4 +32.5 +37.8 +ResNeXt-101-32x4d-FPN [53,59] +LVIS v1 +None +25.5 +41.0 +27.1 +18.8 +33.7 +38.0 +ResNeXt-101-32x4d-FPN [53,59] +LVIS v1 +GT mask +26.7 +42.1 +28.6 +19.7 +34.7 +39.4 +ResNeXt-101-32x4d-FPN [53,59] +LVIS v1 +MAL mask +26.3 +41.5 +28.3 +19.5 +34.5 +39.6 +ResNeXt-101-64x4d-FPN [53,59] +LVIS v1 +None +26.6 +42.0 +28.3 +19.8 +34.7 +39.9 +ResNeXt-101-64x4d-FPN [53,59] +LVIS v1 +GT mask +27.2 +42.8 +29.2 +20.2 +35.7 +41.0 +ResNeXt-101-64x4d-FPN [53,59] +LVIS v1 +MAL mask +27.2 +42.7 +29.1 +19.8 +35.9 +40.7 +ConvNeXt-Small [44] +COCO +None +51.5 +70.6 +56.1 +34.8 +55.2 +66.9 +ConvNeXt-Small [44] +COCO +GT mask +51.8 +70.6 +56.3 +34.5 +55.9 +66.6 +ConvNeXt-Small [44] +COCO +MAL mask +51.7 +70.5 +56.2 +35.2 +55.7 +66.8 +Table 7. Results of detection by adding different mask supervision. The models are evaluated on COCO val2017 and LVIS v1. By adding +mask supervision using ground-truth masks or mask pseudo-labels, we can get around 1% improvement on different AP metrics on LVIS +v1. On COCO val2017, the detection performance also benefits from mask pseudo-labels. Although the improvement is less than COCO’s, +the improvement is consistent over different random seeds. +Mask2former +(Swin-S) +trained with +GT Mask +Mask2former +(Swin-S) +trained with +MAL Mask +Mask2former +(Swin-S) +trained with +GT Mask +Mask2former +(Swin-S) +trained with +MAL Mask +Figure 9. The qualitative comparison between Mask2Former trained with GT mask and Mask2Former trained with MAL-generated mask +pseudo-labels. Note that we use ViT-MAE-Base as the image encoder of MAL and Swin-Small as the backbone of the Mask2Former. +13 + +Y +tv/0.99 +couchl0.30 +couchjo.99dog/0.97 +ngel08 +benchjo.47dog/0.98 +ngej0.93 +benchjo.92tvl0.97 +tv0.38 +WORKPLACE +keyboardj0.98 +mnuse|0.34 +1605 +mouse|0.89 +keyboardj0.99 +mouse0.9gtv/0.99 +tv0.54 +WORKPLACE +keyboardj0.99 +660/ +mouse|0.95 +keyboardj0.99 +mouse/0.99baseballbat/0.98 +personj0.97 +personj0.98 +oersonbaseball batj0.98 +personjo.98 +personjo.99chairl0.80 +personJo.96 +cellphone|0.72 +sife8chairl0.51 +personj0.98 +cellphone|0.89tv|0.98 +couchl0.56 +couchj0.98sonjo96 +personlo.g +chair0.92 +chairl0.7034 +nairfo.9 +chair +person0.98 +personlaano +chair0.4 +chairl0.90ha +personl0.98 +n10.99ebra +zebral0.97zebraj0.98personj0.99 +tie/0.98 +personj0.99 +personl0.99zebral0.97 +zebral0.9g +zebraj0.97zebral0.99 +zebraj0.99 +zebra/0.99personj0.98 +personl0.96 +backpacklo. +horse/0.95personj0.99 +personj0.98 +backpackjo.8 +horse|Chorse0.633 +horse0.97 \ No newline at end of file diff --git a/9dE2T4oBgHgl3EQflwec/content/tmp_files/load_file.txt b/9dE2T4oBgHgl3EQflwec/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..307b4a63883fcbcdecb58c773e12ee3b643816bb --- /dev/null +++ b/9dE2T4oBgHgl3EQflwec/content/tmp_files/load_file.txt @@ -0,0 +1,1115 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf,len=1114 +page_content='Vision Transformers Are Good Mask Auto-Labelers Shiyi Lan1 Xitong Yang2 Zhiding Yu1 Zuxuan Wu3 Jose M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Alvarez1 Anima Anandkumar1,4 1NVIDIA 2Meta AI, FAIR 3Fudan University 4Caltech https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='com/NVlabs/mask-auto-labeler Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Examples of mask pseudo-labels generated by Mask Auto-Labeler on COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Only human-annotated bounding boxes are used as supervision during training to obtain these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Abstract We propose Mask Auto-Labeler (MAL), a high-quality Transformer-based mask auto-labeling framework for in- stance segmentation using only box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' MAL takes box-cropped images as inputs and conditionally generates their mask pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='We show that Vision Transform- ers are good mask auto-labelers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our method significantly reduces the gap between auto-labeling and human annota- tion regarding mask quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Instance segmentation models trained using the MAL-generated masks can nearly match the performance of their fully-supervised counterparts, re- taining up to 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4% performance of fully supervised mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The best model achieves 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1% mAP on COCO in- stance segmentation (test-dev 2017), outperforming state- of-the-art box-supervised methods by significant margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Qualitative results indicate that masks produced by MAL are, in some cases, even better than human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Introduction Computer vision has seen significant progress over the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Tasks such as instance segmentation have made it possible to localize and segment objects with pixel-level accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, these tasks rely heavily on expan- sive human mask annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For instance, when creat- ing the COCO dataset, about 55k worker hours were spent on masks, which takes about 79% of the total annotation time [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Moreover, humans also make mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Human annotations are often misaligned with actual object bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On complicated objects, human annotation quality tends to drop significantly if there is no quality control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Due to the expensive cost and difficulty of quality control, some other large-scale detection datasets such as Open Images [2] and Objects365 [3], only contain partial or even no instance segmentation labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In light of these limitations, there is an increasing in- terest in pursuing box-supervised instance segmentation, where the goal is to predict object masks from bounding box supervision directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Recent box-supervised instance segmentation methods [4–8] have shown promising perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The emergence of these methods challenges the long-held belief that mask annotations are needed to train instance segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, there is still a non- negligible gap between state-of-the-art approaches and their fully-supervised oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our contributions: To address box-supervised instance segmentation, we introduce a two-phase framework consist- ing of a mask auto-labeling phase and an instance segmenta- tion training phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We propose a Transformer- based mask auto-labeling framework, Mask Auto-Labeler (MAL), that takes Region-of-interest (RoI) images as inputs 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='03992v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='CV] ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='600Box-supervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Cropped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Regions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Supervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Mask Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Mask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='MAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Generate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Masks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Phase 1: Mask Auto-labeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Inst Seg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Phase 2: Instance Segmentation Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Masks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' An overview of the two-phase framework of box- supervised instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the first phase, we train Mask Auto-Labeler using box supervision and conditionally gen- erate masks of the cropped regions in training images (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We then train the instance segmentation models using the generated masks (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' and conditionally generates high-quality masks (demon- strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 1) within the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our contributions can be summarized as follows: Our two-phase framework presents a versatile design compatible with any instance segmentation architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Unlike existing methods, our framework is simple and agnostic to instance segmentation module designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We show that Vision Transformers (ViTs) used as image encoders yield surprisingly strong auto-labeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also demonstrate that some specific designs in MAL, such as our attention-based decoder, multiple-instance learning with box expansion, and class-agnostic training, crucial for strong auto-labeling performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Thanks to these components, MAL sometimes even surpasses hu- mans in annotation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Using MAL-generated masks for training, instance seg- mentation models achieve up to 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4% of their fully supervised performance on COCO and LVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our re- sult significantly narrows down the gap between box- supervised and fully supervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also demonstrate the outstanding open-vocabulary general- ization of MAL by labeling novel categories not seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our method outperforms all the existing state-of-the- art box-supervised instance segmentation methods by large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' This might be attributed to good representations of ViTs and their emerging properties such as meaningful grouping [9], where we observe that the attention to objects might benefit our task significantly (demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also hypothesize that our class-agnostic training de- sign enables MAL to focus on learning general grouping instead of focusing on category information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our strong re- sults pave the way to remove the need for expensive human annotation for instance segmentation in real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Vision Transformers Transformers were initially proposed in natural language processing [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Vision Transformers [11] (ViTs) later emerged as highly competitive visual recognition models that use multi-head self-attention (MHSA) instead of con- volutions as the basic building block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' These models are re- cently marked by their competitive performance in many vi- sual recognition tasks [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We broadly categorize existing ViTs into two classes: plain ViTs, and hierarchical ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Standard Vision Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Standard ViTs [11] are the first vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Standard ViTs have the sim- plest structures, which consist of a tokenization embedding layer followed by a sequence of MHSA layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, global MHSA layers can be heavy and usually face signif- icant optimization issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' To improve their performance, many designs and training recipes are proposed to train ViTs in data-efficient manners [9,13–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Hierarchical Vision Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Hierarchical Vision Transformers [12,20–22] are pyramid-shaped architectures that aim to benefit other tasks besides image classification with their multi-scale designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On top of plain ViTs, these ViTs [20,21] separate their multi-head self-attention layers into hierarchical stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Between the stages, there are spa- tial reduction layers, such as max-pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' These ar- chitectures are usually mixed with convolutional layers [23] and often adopt efficient self-attention designs to deal with long sequence lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Instance segmentation Instance segmentation is a visual recognition task that predicts the bounding boxes and masks of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Fully supervised instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In this setting, both bounding boxes and instance-level masks are provided as the supervision signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Early works [24–27] follow a two-stage architecture that generates box proposals or seg- mentation proposals in the first stage and then produces the final segmentation and classification information in the sec- ond stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Later, instance segmentation models are broadly divided into two categories: some continue the spirit of the two-stage design and extend it to multi-stage architec- tures [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Others simplify the architecture and pro- pose one-stage instance segmentation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=', YOLACT [30], SOLO [31, 32], CondInst [33], PolarMask [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Re- cently, DETR and Deformable DETR [36, 37] show great potential of query-based approaches in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Then, methods like MaxDeepLab [38], MaskFormer [39], 2 0 100 200 300 400 0 100 200 300 400 500 600… … … MHSA FFN FFN MaxPool FC … … MHSA FFN FFN Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Bags Neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Bags Average Multiple Instance Learning Loss + + + + ++ -- -- + ++ + + - -- - Self Training EMA EMA Conditional Random Fields Loss 𝑋!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Neighbors 𝑋" Mean Field Algorithm MaxPool FC 𝐸 𝐷 𝐷!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 𝐸!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 𝑉!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 𝑉 𝐾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 𝐾 Task Network Teacher Network Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Overview of MAL architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We visualize the architecture of Mask Auto-Labeler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Mask Auto-Labeler takes cropped images as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Mask Auto-Labeler consists of two symmetric networks, Task Network and Teacher Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Each network contains the image encoder E(or Et), and the mask decoder D(or Dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use the exponential moving average (EMA) to update the weights of the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We apply multiple instance learning (MIL) loss and conditional random fields (CRFs) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The CRF loss takes the average mask predictions of the teacher network and the task network to make the training more stable and generate refined masks for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' PanopticSegFormer [40], Mask2Former [41] and Mask DINO [42] are introduced along this line and have pushed the boundary of instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On the other hand, the instance segmentation also benefits from more power- ful backbone designs, such as Swin Transformers [12, 22], ViTDet [43], and ConvNeXt [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Weakly supervised instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' There are two main styles of weakly supervised instance segmentation: learning with image-level and box-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The former uses image-level class information to perform instance seg- mentation [45–49], while the latter uses box-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' [4] leverages the tight-box priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Later, Box- Inst [5] proposes to leverage color smoothness to improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Besides that, DiscoBox [7] proposes to leverage both color smoothness and inter-image correspondence for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Other follow-ups [6,8] also leverage tight-box pri- ors and color smoothness priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Deep learning interpretation The interest in a deeper understanding of deep net- works has inspired many works to study the interpreta- tion of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For example, Class Ac- tivation Map (CAM) [50] and Grad-CAM [51] visualize the emerging localization during image classification train- ing of convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' This abil- ity has also inspired much weakly-supervised localization and shows deep connections to general weakly-supervised learning, which partly motivates our decoder design in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' DINO [9] further shows that meaning visual group- ing emerges during self-supervised learning with ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In addition, FAN [52] shows that such emerging properties in ViTs are linked to their robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Method Our work differs from previous box-supervised instance segmentation frameworks [4–8] that simultaneously learns detection and instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We leverage a two- phase framework as visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 2, which allows us to have a network focused on generating mask pseudo-labels in phase 1, and another network focused on learning in- stance segmentation [24, 28, 41, 43] in phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our pro- posed auto-labeling framework is used in phase 1 to gener- ate high-quality mask pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We propose this two-phase framework because it brings the following benefits: We can relax the learning constraints in phase 1 and focus only on mask pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Therefore, in this phase, we can take Region-of-interest (RoI) images in- stead of untrimmed images as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' This change al- lows us to use a higher resolution for small objects and a strong training technique mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1, which helps improve the mask quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We can leverage different image encoders and mask de- coders in phases 1 and 2 to achieve higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We empirically found that phases 1 and 2 favor different architectures for the image encoders and mask decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' See the ablation study in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We can use MAL-generated masks to directly train the most fully supervised instance segmentation models in phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' This makes our approach more flexible than previous architecture-specific box-supervised instance segmentation approaches [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' As phase 2 follows the previous standard pipelines, which do not need to be re-introduced here, we focus on introducing phase 1 (MAL) in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3 0 50 100 150 200 250 300 350 400 0 100 200 300 400 500 6003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' RoI input generation Most box-supervised instance segmentation ap- proaches [4–7] are trained using the entire images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, we find that using RoI images might have more benefits in box-supervised instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Moreover, we compare two intuitive sampling strategies of RoI images to obtain foreground and background pixels and explain the better strategy, box expansion, in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Benefits of using RoI inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' There are two advantages of using RoI images for inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' First, using the RoI images as inputs is naturally good for handling small objects be- cause no matter how small the objects are, the RoI images are enlarged to avoid the issues caused by low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Secondly, having RoI inputs allows MAL to focus on learn- ing segmentation and avoid being distracted from learning other complicated tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=', object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' RoI sam- pling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The sampling strategy should ensure both positive and negative pixels are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We present two straightforward sampling strategies: The first strategy is to use bounding boxes to crop the images for positive inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We crop the images using randomly generated boxes containing only background pixels for negative inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' MAL does not generate good mask pseudo-labels with cropping strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We observe that the networks tend to learn the trivial solution (all pixels are predicted as either foreground or background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The second is to expand the bounding boxes randomly and include background pixels, where negative bags are chosen from the expanded rows and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We visu- alize how we define positive/negative bags in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3 and explain the detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' This detailed design is critical to make MAL work as it prevents MAL from learning trivial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Without this design, the gen- erated masks tend to fill the entire bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Box expansion specifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Given an untrimmed image Iu ∈ RC×Hu×W u and the bounding box b = (x0, y0, x1, y1) in- dicating the x, y coordinates of the top-left corners and the bottom-right corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' To obtain background pixels, we ran- domly expand the bounding box b to b′ = (xc + βx(x0 − xc), yc +β′ x(y0 −yc), xc +βy(x1 −xc), yc +β′ y(y1 −yc)), where xc = (x0 + x1)/2, yc = (y0 + y1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' To gener- ate random values of βx, β′ x, βy, β′ y, we randomly generate θx, θy ∈ [0, θ] for x- and y-direction, where θ is the upper bound of box expansion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Next, we randomly generate βx ∈ [0, θx] and βy ∈ [0, θy].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In the end, we assign β′ x as θx−βx and β′ y as θy −βy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Finally, we use b′ to crop the im- age and obtain trimmed image It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We conduct the ablation study for θ in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' At last, We resize the trimmed image It to the size of C × Hc × W c as the input image Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Class Tokens k q v Transformer Layer (a) (b) (c) (d) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' (a) The fully connected decoder (b) The fully convolu- tional Decoder (c) The attention-based decoder (used in MAL) (d) The query-based Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' MAL architecture MAL can be divided into two symmetric networks: the task network and the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The task network consists of an image encoder denoted as E, and a mask de- coder denoted as D, demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The architec- ture of the teacher network is identical to the task network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We denote the segmentation output of the task network and the teacher network as m, mt ∈ {0, 1}N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Image encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use Standard ViTs [11] as the image encoder and drop the classification head of Standard ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We compare different image encoders in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also try feature pyramid networks on top of Standard ViTs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=', FPN [53], but it causes a performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Similar con- clusions were also found in ViTDet [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Mask decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the mask decoder D, we use a simple attention-based network inspired by YOLACT [30], which includes an instance-aware head K and a pixel-wise head V , where D(E(I)) = K(E(I)) · V (E(I)), and “ · ” repre- sents the inner-product operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the instance-aware head K, we use a max-pooling layer followed by a fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The input chan- nel dimension of K is equivalent to the output channel di- mension of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The output channel dimension of K is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the pixel-wise head V , we use four sequential convo- lutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Each is followed by a ReLU layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Between the second and the third convolutional layer, we insert a bi- linear interpolation layer to increase the feature resolution by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The input channel dimension is equivalent to the out- put channel dimension of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use 256 dimensions for hidden channels and output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also compare dif- ferent design choices of mask decoders in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Exponential moving average (EMA) teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Instead of training the teacher network directly, we leverage exponen- tial moving averages (EMA) to update the parameters in the teacher network using the parameters in the task network similar to MOCO [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The goal of using EMA Teacher is to eliminate the loss-explosion issues in training since optimizing Standard Vision Transformers is usually non- trivial [13, 14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We do not observe any significant per- formance drop or improvement on DeiT-small-based MAL after removing the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, it makes the training more stable when we use larger-scale image en- coders in MAL, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' ViT-MAE-Base [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Losses We use Multiple Instance Learning Loss Lmil and Con- ditional Random Field Loss Lcrf as the box-supervised loss: L = αmilLmil + αcrfLcrf (1) Multiple Instance Learning Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The motivation of the Multiple Instance Segmentation is to exploit the priors of tight-bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' After the student network produces the output m, we ap- ply the Multiple Instance Learning (MIL) Loss on the out- put mask m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We demonstrate the process in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We denote mi,j as the mask score at the location i, j in the image Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We define each pixel as an instance in the MIL loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Inspired by BBTP [4], we treat each row or col- umn of pixels as a bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We determine whether a bag is pos- itive or negative based on whether it passes a ground-truth box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We define the bags as B, and each bag Bi contains a row or column of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Additionally, we define the label for each bag g, and each label gi corresponds to a bag Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Therefore, we use the max pooling as the reduction func- tion and dice loss [55]: Lmil = 1 − 2 � i gi · max{Bi}2 � i max{Bi}2 + � i g2 i (2) Conditional Random Field Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The goal of CRF loss is to refine the mask prediction by imposing the smooth- ness priors via energy minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Then, we leverage this refined mask as pseudo-labels to self-train the mask predic- tion in an online-teacher manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use the average mask prediction ma = 1 2(m + mt) as the mask prediction to be refined for more stable training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Next, we define a random field X = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=', XN}, where N = Hc × W c is the size of cropped image and each Xi represents the label that corresponds to a pixel in Ic, therefore we have X ∈ {0, 1}N, meaning the back- ground or the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use l ∈ {0, 1}N to represent a labeling of X minimizing the following CRF energy: E(l|ma, Xc) = µ(X|ma, Ic) + ψ(X|Ic), (3) where µ(X|ma, Ic) represents the unary potentials, which is used to align Xi and ma i since we assume that most of the mask predictions are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Meanwhile, ψ(X|Ic) rep- resents the pairwise potential, which sharpens the refined mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Specifically, we define the pairwise potentials as: ψ(X|Ic) = � i∈{0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='.N−1}, j∈N (i) ω exp(−|Ic i − Ic j |2 2ζ2 )[Xi ̸= Xj], (4) where N(i) represents the set of 8 immediate neighbors to Xi as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Then, we use the MeanField al- gorithm [7, 56] to efficiently approximate the optimal so- lution, denoted as l = MeanField(Ic, ma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We attach the derivation and PyTorch code in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' At last, we apply Dice Loss to leverage the refined masks l to self-train the models as: Lcrf = 1 − 2 � i limi � i l2 i + m2 i (5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Experiments We evaluate MAL on COCO dataset [1], and LVIS [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The main results on COCO and LVIS are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The qualitative results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Datasets COCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' contains 80 semantic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We fol- low the standard partition, which includes train2017 (115K images), val2017 (5K images), and test-dev (20k images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' LVIS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' contains 1200+ categories and 164K images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We follow the standard partition of training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Implementation Details We use 8 NVIDIA Tesla V100s to run the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Phase 1 (mask auto-labeling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use AdamW [58] as the network optimizer and set the two momentums as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use the cosine and annealing scheduler to adjust the learning rate, which is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 · 10−6 per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The MIL loss weight αmil, CRF loss weight αcrf, ζ, and ω in CRF pairwise potentials are set to 4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5, 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We analyze the sensitivity of the loss weights and CRF hyper- parameters in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use the input resolution of 512 × 512, and a batch size of 32 (4 per GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For EMA, we use a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the task and teacher network, we apply random flip data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On top of that, we apply extra random color jittering, random grey-scale conversion, and random Gaussian blur for the task network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We train MAL for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' It takes around 23 hours and 35 hours to train MAL with Standard ViT-Base [11] on the COCO and LVIS datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Phase 2 (Training instance segmentation models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We select a couple of high-performance fully supervised in- stance segmentation models, which are ConvNeXts [44] with Cascade R-CNN [28], Swin Transformers [12] with Mask2Former [41], ResNets [59] and ResNeXts [60] with SOLOv2 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' MAL works extremely well with these ar- chitectures, which demonstrates the great power of Mask Auto-Labeler from the perspective of accuracy and gener- alization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We leverage the codebase in MMDetection [61] for phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Again, we only replace the GT masks with MAL-generated mask pseudo-labels to adjust all these fully supervised models to box-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Instance segmentation results Retention Rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We argue that the sole mAP of instance segmentation is not fair enough to evaluate box-supervised instance segmentation since the performance gain can be 5 Method Labeler Backbone InstSeg Backbone InstSeg Model Sup (%)Mask APval (%)Mask APtest (%)Ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='val (%)Ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='test Mask R-CNN∗ [24] ResNet-101 Mask R-CNN Mask 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 Mask R-CNN∗ [24] ResNeXt-101 Mask R-CNN Mask 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 CondInst [33] ResNet-101 CondInst Mask 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 SOLOv2 [31] ResNet-50 SOLOv2 Mask 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 SOLOv2 [31] ResNet-101-DCN SOLOv2 Mask 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 SOLOv2 [31] ResNeXt-101-DCN SOLOv2 Mask 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 ConvNeXt [44] ConvNeXt-Small [44] Cascade R-CNN Mask 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 ConvNeXt [44] ConvNeXt-Base [44] Cascade R-CNN Mask 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 Mask2Former [41] Swin-Small Mask2Former Mask 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 BBTP† [4] ResNet-101 Mask R-CNN Box 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 BoxInst [5] ResNet-101 CondInst Box 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 BoxLevelSet [6] ResNet-101-DCN SOLOv2 Box 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 DiscoBox [7] ResNet-50 SOLOv2 Box 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 DiscoBox [7] ResNet-101-DCN SOLOv2 Box 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 DiscoBox [7] ResNeXt-101-DCN SOLOv2 Box 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 BoxTeacher [8] Swin-Base CondInst Box 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 Mask Auto-Labeler ViT-MAE-Base [13] ResNet-50 SOLOv2 Box 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 Mask Auto-Labeler ViT-MAE-Base [13] ResNet-101-DCN SOLOv2 Box 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 Mask Auto-Labeler ViT-MAE-Base [13] ResNeXt-101-DCN SOLOv2 Box 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 Mask Auto-Labeler ViT-MAE-Base [13] ConvNeXt-Small [44] Cascade R-CNN Box 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 Mask Auto-Labeler ViT-MAE-Base [13] ConvNeXt-Base [44] Cascade R-CNN Box 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 Mask Auto-Labeler ViT-MAE-Base [13] Swin-Small [12] Mask2Former [41] Box 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Main results on COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Ret means the retention rate of box-supervised mask AP supervised mask AP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' MAL with SOLOv2/ResNeXt-101 outperforms DiscoBox with SOLOv2/ResNeXt-101 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6% on val2017 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3% on test-dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our best model (Mask2former/Swin-Small) achieves 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3% AP on val and 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1% AP on test-dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' achieved by improving box quality unrelated to segmenta- tion quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, the retention rate can better reflect the real mask quality because the fully supervised counter- parts also get boosted by the better box results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Results on COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In table 1, we show that various mod- ern instance segmentation models can achieve up to 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5% performance with the pseudo-labels of the fully supervised oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our best results are 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3% mAP on COCO test-dev and 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1% mAP on COCO val, achieved by using MAL (Standard ViT-Base [11] pretrained with MAE) for phase 1, and using Mask2Former (Swin-Small) [12,41] for phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' There is no significant retention drop when we use the mask pseudo-labels to train more powerful instance seg- mentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On the contrary, the higher retention rates on COCO are achieved by the heavier instance seg- mentation models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=', Cascade R-CNN with ConvNeXts and Mask2Former with Swin-Small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, other meth- ods have significantly lower retention rates compared with MAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The experiment results quantitatively imply that the mask quality outperforms other methods by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Results on LVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In table 2, we also observe that all in- stance segmentation models work very well with the mask pseudo-labels generated by MAL (Ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' = 93% ˜ 98%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We visualize part of the results in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also evaluate the open-vocabulary ability of MAL by training MAL on COCO dataset but generating mask pseudo-labels on LVIS, and thus training instance segmentation models using these mask pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Image encoder variation To support our claim that Vision Transformers are good auto-labelers, we compare three popular networks as the im- age encoders of MAL: Standard Vision Transformers [11, 13,16], Swin Transformer [12], ConvNeXts [44] in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' First, we compare the fully supervised pretrained weights of these three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We choose the official fully supervised pre-trained weights of ConvNeXts and Swin Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For Standard Vision Transformers, we adopt a popular fully supervised approach, DeiT [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We ob- serve that fully supervised Standard Vision Transformers (DeiT) as image encoders of Mask Auto-Labeler are better than Swin Transformers and ConvNeXts even though the imaganet-1k performance of Swin Transformers and Con- vNeXts is higher than that of DeiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We argue that the suc- cess of Standard Vision Transformers might be owed to the self-emerging properties of Standard ViTs [9, 11] (visual- ized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 6), and the larger-receptive field brought by global multi-head self-attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Second, the mask pseudo-labels can be further improved by Mask AutoEncoder (MAE) pretraining [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The poten- tial reason might be that MAE pretraining enhances Stan- dard ViTs via learning pixel-level information, which is very important for dense-prediction tasks like segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Mask decoder variation We compare four different modern designs of mask de- coders: the fully connected Decoder [62], the fully convolu- tional decoder [24,63], the attention-based decoder [30,31], and the query-based decoder [41] in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We visual- ize different designs of mask decoders in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the fully connected Decoder, we use two fully connected layers with a hidden dimension of 2048 and then output a confi- dence map for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We reshape this output vector as the 2D confidence map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We introduce the attention-based decoder in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the fully convolutional Decoder, We adopt the pixel-wise head V in the attention-based De- 6 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Qualitative results of mask pseudo-labels generated by Mask Auto-Labeler on LVIS v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Method Autolabeler Backbone InstSeg Backbone InstSeg Model Training Data Sup (%)Mask APval (%)Ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='val Mask R-CNN [24] ResNet-50-DCN Mask R-CNN [24] Mask 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 Mask R-CNN [24] ResNet-101-DCN Mask R-CNN [24] Mask 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 Mask R-CNN [24] ResNeXt-101-32x4d-FPN Mask R-CNN [24] Mask 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 Mask R-CNN [24] ResNeXt-101-64x4d-FPN Mask R-CNN [24] Mask 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 Mask Auto-Labeler ViT-MAE-Base [13] ResNet-50-DCN Mask R-CNN [24] LVIS v1 Box 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 Mask Auto-Labeler ViT-MAE-Base [13] ResNet-101-DCN Mask R-CNN [24] LVIS v1 Box 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 Mask Auto-Labeler ViT-MAE-Base [13] ResNeXt-101-32x4d-FPN Mask R-CNN [24] LVIS v1 Box 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 Mask Auto-Labeler ViT-MAE-Base [13] ResNeXt-101-64x4d-FPN Mask R-CNN [24] LVIS v1 Box 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 Mask Auto-Labeler ViT-MAE-Base [13] ResNeXt-101-32x4d-FPN Mask R-CNN [24] COCO Box 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 Mask Auto-Labeler ViT-MAE-Base [13] ResNeXt-101-64x4d-FPN Mask R-CNN [24] COCO Box 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Main results on LVIS v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Training data means the dataset we use for training MAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also finetune it on COCO and then generate pseudo-labels of LVIS v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Compared with trained on LVIS v1 directly, MAL finetuned on COCO only caused around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='35% mAP drop on the final results, which indicates the great potential of the open-set ability of MAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Ret means the retention rate of box-supervised mask AP supervised mask AP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Mask decoder (%)Mask APval (%)Ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='val Fully connected decoder 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 Fully convolutional decoder 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 Attention-based decoder 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 Query-based decoder Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Ablation study of box expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use Standard ViT- MAE-Base as the image encoder of MAL in phase 1 and Cascade RCNN with ConvNext-Small as the instance segmentation models in phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The numbers are reported in % Mask mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Among different designs, the attention-based decoder performs the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We can not obtain reasonable results with Query-based Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' For the query-based decoder, we follow the design in Mask2Former [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We spend much effort exploring the query-based Decoder on MAL since it performs extremely well on fully supervised instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, the results are surprisingly unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We suspect the slightly heavier layers might cause optimization issues un- der the box-supervised losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Experiments show that box-supervised instance segmen- tation favors the attention-based decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, state- of-the-art instance segmentation and object detection meth- ods often adopt the fully convolutional decoder [15, 43] or the query-based decoder [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our proposed two-phase framework resolves this dilemma and allows the networks to enjoy the merits of both the attention-based Decoder and the non-attention-based Decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Clustering analysis As the results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4, we wonder why the Standard ViTs outperform other modern image encoders in auto-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' As the comparison of classification ability Backbone IN-1k Acc@1 Mask APval Ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='val ConvNeXt-Base [44] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 Swin-Base [12] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 ViT-DeiT-Small [64] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 ViT-DeiT-Base [64] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 ViT-MAE-Base [13] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 ViT-MAE-Large [13] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Ablation study of different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' All models are pre-trained on ImageNet-1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' ConvNeXt and Swin Trans- former outperform DeiT on image classification, but standard ViT- Small [16] (ViT-DeiT-Small) outperforms ConvNeXt-base and Swin-Base on mask Auto-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Standard ViT-Base (ViT- MAE-Base) and Standard ViT-Large (ViT-MAE-Large) pretrained via MAE achieve the best performance on mask Auto-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' does not seem to reflect the actual ability of auto-labeling, we try to use the ability clustering to evaluate the image en- coders because foreground(FG)/background(BG) segmen- tation is very similar to the binary clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Specifically, we extract the feature map output by the last layers of Swin Transformers [12], ConvNeXts [44], Stan- dard ViTs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Then, we use the GT mask to divide the feature vectors into the FG and BG feature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' By evalu- ating the average distance from the FG/BG feature vectors to their clustering centers, we can reveal the ability of the networks to distinguish FG and BG pixels empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Formally, we define the feature vector of token i gener- ated by backbone E as f E i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We define the FG/BG clustering centers f ′ 1, f ′ 0 as the mean of the FG/BG feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' we use the following metric as the clustering score: S = 1 N N � i ( f E i |f E i | − f ′ γ(i) |f ′ γ(i)|)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' (6) 7 50 100 150 200 250 300 350 400 0 100 200 300 400 500 6000 100 200 300 400 500 600 0 100 200 300 40050 100 150 200 250 300 350 400 0 100 200 300 400 500 600100 200 300 400 500 600 0 100 200 300 400 500 600100 200 300 400 0 100 200 300 400 500 6000 50 100 150 200 250 300 350 400 0 100 200 300 400 500 600Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Attention visualization of two RoI images produced by MAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In each image group, the left-most image is the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We visualize the attention map output by the 4th, 8th, 12th MHSA layers of the Standard ViTs in MAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' (a) MAL-generated Masks are sharper and more boundary-sticky (b) Occlusion Issues MAL Mask Pseudo-labels Ground-truth Masks Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The lateral comparison between MAL-generated pseudo-labels (top) and GT masks (bottom) on COCO val2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On the left, we observe that MAL-generated pseudo-labels are sharper and more boundary-sticky than GT masks in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On the right, we observe that in highly occluded situations, human-annotated masks are still better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 25 30 35 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='75 1 𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' "# 32 34 36 1 2 4 8 16 𝛼$%& 30 32 34 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 1 2 4 6 𝜔 27 30 33 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='75 1 𝜁 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Sensitivity analysis of loss weights and CRF hyper- parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use ViT-Base [11] pretrained via MAE [13] as the image encoder for the first phase and SOLOv2 (ResNet-50) for the second phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The x-axis and y-axis indicate the hyper-parameter values and the (%)mask AP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' θ Mask APval Ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Ablation on box ex- pansion ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use Standard ViT-Base pretrained via MAE (ViT-MAE-Base) and Cascade R-CNN (ConvNeXt-Small) for phase 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Backbone Score (↓) ConvNeXt-Base [44] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='459 Swin-Base [12] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='425 ViT-DeiT-Small [64] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='431 ViT-DeiT-Base [64] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='398 ViT-MAE-Base [13] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='324 ViT-MAE-Large [13] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='301 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Clustering scores for different image encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The smaller clustering scores imply a better ability to dis- tinguish foreground and back- ground features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' where if pixel i is FG, γ(i) = 1, otherwise γ(i) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We show the clustering evaluation on the COCO val 2017 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The results align our conclusion that Stan- dard Vision Transformers are better at mask auto-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' MAL masks v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' GT masks We show the apples to apples qualitative comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 7 and make the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' First, MAL- generated mask pseudo-labels are considerably sharper and boundary-sticky than human-annotated ones since humans have difficulties in aligning with the true boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Sec- ond, severe occlusion also presents a challenging issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Conclusion In this work, we propose a novel two-phase frame- work for box-supervised instance segmentation and a novel Transformer-based architecture, Mask Auto-Labeler (MAL), to generate high-quality mask pseudo-labels in phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We reveal that Standard Vision Transformers are good mask auto-labelers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Moreover, we find that random using box-expansion RoI inputs, the attention-based De- coder, and class-agnostic training are crucial to the strong mask auto-labeling performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Moreover, thanks to the two-phase framework design and MAL, we can adjust al- most all kinds of fully supervised instance segmentation models to box-supervised learning with little performance drop, which shows the great generalization of MAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Although great improvement has been made by our approaches in mask auto-labeling, we still observe many failure cases in the occlusion situation, where human annotations are much better than MAL-generated masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Additionally, we meet saturation problems when scaling the model from Standard ViT-Base to Standard ViT-Large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We leave those problems in the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Broader impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Our proposed Transformer-based mask auto-labeler and the two-phase architecture serve as a stan- dard paradigm for high-quality box-supervised instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' If follow-up work can find and fix the issues under our proposed paradigm, there is great potential that expansive human-annotated masks are no longer needed for instance segmentation in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 8 References [1] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Doll´ar, and C Lawrence Zitnick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Microsoft coco: Common objects in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In European 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+page_content=' Instance-aware se- mantic segmentation via multi-task network cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3150–3158, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 6 [63] Jonathan Long, Evan Shelhamer, and Trevor Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Fully convolutional networks for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In Pro- ceedings of the IEEE conference on computer vision and pat- tern recognition, pages 3431–3440, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 6 [64] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Herve Jegou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Training data-efficient image transformers & distillation through at- tention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In International Conference on Machine Learning, volume 139, pages 10347–10357, July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 7, 8 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Additional details of CRF In the main paper, we define the energy terms of CRF but skip the details on how we use the Mean Field algorithm to minimize the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Here, we provide more details on how we use the Mean Field algorithm [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We define l = {l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=', lN} as the label being inferred, where N = H × W is the size of the input image and xi is the label of the i-th pixel in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We also assume that the network predicts a mask m = {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=', mN} is where mi is the unary mask score of the i-th pixel in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The pseudo- code to obtain l using mean field is attached in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 1: Algorithm 1 Mean field algorithm for CRFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 1: procedure MEANFIELD(m, I) 2: Ki,j ←− ω exp(− |Ii−Ij| 2ζ2 ) 3: ▷ Initialize the Gaussian kernels 4: l ← m ▷ Initialize l using m 5: while not converge do ▷ Iterate until convergence 6: for i ← 1 to |l| do 7: ˆli ← li 8: for j ∈ N(i) do 9: ˆli ← ˆli + Kj ∗ lj 10: ▷ Message passing 11: end for 12: end for 13: l ← ϕ(ˆl) ▷ ϕ is a clamp function 14: end while 15: return λ(l) ▷ λ is a threshold function 16: end procedure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Additional implementation details We use the same hyper-parameters on all benchmarks for all image encoders (Standard ViTs [11, 13, 16], Swin Transformers [12], and ConvNeXts [44]) and mask de- coders (fully connected decoder, fully convolutional de- coder, attention-based decoder, ), including batch size, opti- mization hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We observe a performance drop when we add parametric layers or multi-scale lateral/skip connections [43, 53] between the image encoder (Standard ViTs, Swin Transformers, ConvNeXts) and the mask de- coder (attention-based decoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We insert a couple of the bi-linear interpolation layers to resize the feature map be- tween the image encoder and the mask decoder and resize the segmentation score map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Specifically, we resize the fea- ture map produced by the image encoder to 1/16 (small), 1/8 (medium), 1/4 (large) size of the raw input according to the size of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We divide the objects into three scales regarding to the area of their bound boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We use the area ranges of [0, 322), [322, 962), [962, ∞) to cover small, medium, and large objects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We resize the mask prediction map to 512 × 512 to reach the original resolution of the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Moreover, we also try three naive ways to add classifica- tion loss, but it does not work well with MAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' First, we add another fully connected layer as the classification decoder, which takes the feature map of the first fully connected layer of the instance-aware head K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' With this design, the classi- fication causes a significant performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Secondly, we use two extra fully connected layers or the original clas- sification decoder of standard ViTs as the classification de- coder, which directly takes the feature map of the image encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' However, the classification loss does not provide performance improvement or loss in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Benefits for object detection The supervised object detection models benefit from the extra mask supervision [24], which improves detection re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Specifically, we follow the settings in Mask R- CNN [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' First, we use RoI Align, the box branch, and the box supervision without mask supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Second, we add the mask branch and ground-truth mask supervision on top of the first baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The second baseline is the original Mask R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Thirdly, we replace the ground-truth masks with the mask pseudo-labels generated by MAL on top of the second baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' It turns out that using MAL-generated mask pseudo-labels for mask supervision brings in an im- provement similar to ground-truth masks on detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' We show the results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Additional qualitative results We also visualize the prediction results produced by the instance segmentation models trained with ground-truth masks and mask pseudo-labels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' In most cases, we argue that humans cannot tell which results are produced by the models supervised by human-annotated labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' 12 InstSeg Backbone Dataset Mask Labels (%)AP (%)AP50 (%)AP75 (%)APS (%)APM (%)APL ResNet-50-DCN [59] LVIS v1 None 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 ResNet-50-DCN [59] LVIS v1 GT mask 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 ResNet-50-DCN [59] LVIS v1 MAL mask 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 ResNet-101-DCN [59] LVIS v1 None 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 ResNet-101-DCN [59] LVIS v1 GT mask 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 ResNet-101-DCN [59] LVIS v1 MAL mask 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 ResNeXt-101-32x4d-FPN [53,59] LVIS v1 None 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 ResNeXt-101-32x4d-FPN [53,59] LVIS v1 GT mask 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='4 ResNeXt-101-32x4d-FPN [53,59] LVIS v1 MAL mask 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 ResNeXt-101-64x4d-FPN [53,59] LVIS v1 None 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 ResNeXt-101-64x4d-FPN [53,59] LVIS v1 GT mask 27.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 ConvNeXt-Small [44] COCO None 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 ConvNeXt-Small [44] COCO GT mask 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='6 ConvNeXt-Small [44] COCO MAL mask 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Results of detection by adding different mask supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The models are evaluated on COCO val2017 and LVIS v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' By adding mask supervision using ground-truth masks or mask pseudo-labels, we can get around 1% improvement on different AP metrics on LVIS v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' On COCO val2017, the detection performance also benefits from mask pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Although the improvement is less than COCO’s, the improvement is consistent over different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' Mask2former (Swin-S) trained with GT Mask Mask2former (Swin-S) trained with MAL Mask Mask2former (Swin-S) trained with GT Mask Mask2former (Swin-S) trained with MAL Mask Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content=' The qualitative comparison between 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='8 horse|Chorse0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='633 horse0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} +page_content='97' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE2T4oBgHgl3EQflwec/content/2301.03992v1.pdf'} diff --git a/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/2301.05529v1.pdf.txt b/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/2301.05529v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1879a2de235ea99fc718beb7b726a381dc14567e --- /dev/null +++ b/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/2301.05529v1.pdf.txt @@ -0,0 +1,2921 @@ +UNIFORM GLOBAL STABILITY OF SWITCHED NONLINEAR +SYSTEMS IN THE KOOPMAN OPERATOR FRAMEWORK∗ +CHRISTIAN MUGISHO ZAGABE† AND ALEXANDRE MAUROY ‡ +Abstract. +In this paper, we provide a novel solution to an open problem on the global uniform stability +of switched nonlinear systems. Our results are based on the Koopman operator approach and, to +our knowledge, this is the first theoretical contribution to an open problem within that framework. +By focusing on the adjoint of the Koopman generator in the Hardy space on the polydisk, we +define equivalent linear (but infinite-dimensional) switched systems and we construct a common +Lyapunov functional for those systems, under a solvability condition of the Lie algebra generated by +the linearized vector fields. A common Lyapunov function for the original switched nonlinear systems +is derived from the Lyapunov functional by exploiting the reproducing kernel property of the Hardy +space. The Lyapunov function is shown to converge in a bounded region of the state space, which +proves global uniform stability of specific switched nonlinear systems on bounded invariant sets. +Key words. +Koopman operator, Hardy space on the polydisk, Switched systems, Uniform +stability, Common Lyapunov function. +AMS subject classifications. 47B32, 47B33, 47D06, 70K20, 93C10, 93D05. +1. Introduction. Switched systems are hybrid-type models encountered in ap- +plications where the dynamics abruptly jump from one behavior to another. They +are typically described by a family of subsystems that alternate according to a given +commutation law. Stability properties of switched systems have been the focus of +intense research effort (see e.g. [32] for a review). In this context, a natural question +is whether a switched system with an equilibrium point is uniformly stable, that is, +stable for any commutation law. It turned out that the uniform stability problem +is counter-intuitive and challenging. In the linear case, it is well-known that stable +subsystems may induce an unstable switched system. However, uniform stability is +guaranteed if the matrices associated with the subsystems are stable and commute +pairwise [24], a result which is extended in [15] to subsystems described by stable +matrices generating a solvable Lie algebra. This latter result can be explained by the +well-known equivalence between solvable Lie algebra of matrices and the existence +of a common invariant flag for those matrices, which allows to construct a common +Lyapunov function for the subsystems [34]. +In the case of switched nonlinear systems, an open problem was posed in [13] on +the relevance of Lie-algebraic conditions of vector fields for global uniform stability. +Partial solutions have been proposed in this context. It was proven in [17] that uniform +stability holds if the vector fields are individually stable and commute, in which case +a common Lyapunov function can be constructed [30, 33]. Uniform stability was also +shown for a pair of vector fields generating a third-order nilpotent Lie algebra [29] +and for particular r-order nilpotent Lie algebras [18]. However, no result has been +obtained, which solely relies on the more general solvability property of Lie algebras +of the subsystems vector fields. +In this paper, we provide a partial solution to the problem introduced in [13] by +∗Submitted to the editors +†Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni- +versity of Namur (christian.mugisho@unamur.be), +‡Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni- +versity of Namur (alexandre.mauroy@unamur.be) +1 +This manuscript is for review purposes only. +arXiv:2301.05529v1 [math.DS] 13 Jan 2023 + +2 +C. M. ZAGABE AND A. MAUROY +proving global uniform stability results for switched nonlinear systems under a gen- +eral solvability property of Lie algebras. To do so, we rely on the Koopman operator +framework [3, 21]: we depart from the classical pointwise description of dynami- +cal systems and consider instead the evolution of observable functions (here in the +Hardy space of holomorphic functions defined on the complex polydisk). Through this +approach, equivalent infinite-dimensional dynamics are generated by linear Koopman +generators, so that nonlinear systems are represented by Koopman linear systems that +are amenable to global stability analysis [19]. In particular, building on preliminary +results obtained in [34], we construct a common Lyapunov functional for switched +Koopman linear systems. A key point is to focus on the adjoint of the Koopman +generators and notice that these operators have a common invariant maximal flag if +the linear parts of the subsystems generate a solvable Lie algebra, a condition that is +milder than the original assumption proposed in [13]. Finally, we derive a common +Lyapunov function for the original switched nonlinear system and prove its conver- +gence under specific algebraic conditions on the vector field. This allows us to obtain +a bounded invariant region where the switched nonlinear system is globally uniformly +asymptotically stable. To our knowledge, this is the first time that a novel solution +to an open theoretical problem is obtained within the Koopman operator framework. +The rest of the paper is organized as follows. +In Section 2, we present some +preliminary notions on uniform stability of switched nonlinear systems and give a +general introduction to the Koopman operator framework, as well as some specific +properties in the Hardy space on the polydisk. In Section 3, we state and prove our +main result. We recast the open problem given in [13] in terms of the existence of +an invariant maximal flag and we provide a constructive proof for the existence of +a common Lyapunov function. Additional corollaries are also given, which focus on +specific classes of vector fields. Our main results are illustrated with two examples in +Section 4. Finally, concluding remarks and perspectives are given in Section 5. +Notations. We will use the following notation throughout the manuscript. For +multi-index notations α = (α1, ..., αn) ∈ Nn, we define |α| = α1 + · · · + αn and +zα = zα1 +1 · · · zαn +n . The complex conjugate and real part of a complex number a are +denoted by ¯a and ℜ(a), respectively. The transpose-conjugate of a matrix (or vector) +A is denoted by A†. The Jacobian matrix of the vector field F at x is given by JF(x). +The complex polydisk centered at 0 and of radius ρ is defined by +Dn(0, ρ) = {z ∈ Cn : |z1| < ρ, · · · , |zn| < ρ} . +In particular, Dn denotes the unit polydisk (i.e. with ρ = 1) and ∂Dn is its boundary. +Finally, the floor of a real number is denoted by ⌊x⌋. +2. Preliminaries. In this section, we introduce preliminary notions and results +on the stability theory for switched systems and on the Koopman operator framework. +2.1. Stability of switched systems. We focus on the uniform asymptotic sta- +bility property of switched systems and on the existence of a common Lyapunov +function. Some existing results that connect these two main concepts are presented +in both linear and nonlinear cases. +Definition 2.1 (Switched system). A switched system ˙x = F (σ)(x) is a (finite) +set of subsystems +(2.1) +� +˙x = F (i)(x), x ∈ X ⊂ Rn�m +i=1 +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +3 +associated with a commutation law σ : R+ → {1, · · · , m} indicating which subsystem +is activated at a given time. +In this paper, we make the following standing assumption. +Assumption 1. The commutation law σ is a piecewise constant function with a +finite number of discontinuities on every bounded time interval (see e.g. [12]). +2.1.1. Uniform stability. According to [16], stability analysis of switched sys- +tems revolves around three important problems: +• decide whether an equilibrium is stable under the action of the switched +system for any commutation law σ, in which case the equilibrium is said to +be uniformly stable, +• identify the commutation laws for which the equilibrium is stable, and +• construct the commutation law for which the equilibrium is stable. +In this paper we focus on the first problem related to uniform stability. +Definition 2.2 (Uniform stability). Assume that F (i)(xe) = 0 for all i = 1, . . . , m. +The equilibrium xe is +• uniformly asymptotically stable (UAS) if ∀ϵ > 0, ∃δ > 0 such that +∥x(0) − xe∥ ≤ δ ⇒ ∥x(t) − xe∥ ≤ ϵ, ∀t > 0, ∀σ +and +∥x(0) − xe∥ ≤ δ ⇒ lim +t→∞ x(t) = xe, ∀σ, +• globally uniformly asymptotically stable (GUAS) on D ⊆ Rn if it is UAS +and +x(0) ∈ D ⇒ lim +t→∞ x(t) = xe, ∀σ, +• globally uniformly exponentially stable (GUES) on D ⊆ Rn if ∃β, λ > 0 such +that +x(0) ∈ D ⇒ ∥x(t) − xe∥ ≤ β∥x(0) − xe∥e−λt, ∀t > 0, ∀σ. +This definition implies that the subsystems share a common equilibrium. More- +over, a necessary condition is that this equilibrium is asymptotically stable with re- +spect to the dynamics of all individual subsystems. However, this condition is not +sufficient, since the switched system might be unstable for a specific switching law. +A sufficient condition for uniform asymptotic stability is the existence of a common +Lyapunov function (CLF). +Definition 2.3 (Common Lyapunov function [12]). A positive C1- function V : +D ⊆ Rn → R is a common Lyapunov function on D ⊆ Rn for the family of subsystems +(2.1) if +∇V · F (i)(x) < 0 +∀x ∈ D \ {xe}, +∀i = 1, . . . , m. +For switched systems with a finite number of subsystems, a converse Lyapunov result +also holds ([12], [17]). +Theorem 2.4 ([17]). Suppose that D ⊆ Rn is compact and forward-invariant +with respect to the flow induced by the subsystems (2.1). The switched system (2.1) +is GUAS on D if and only if all subsystems share a CLF on D. +This manuscript is for review purposes only. + +4 +C. M. ZAGABE AND A. MAUROY +A corollary of this result provides a necessary condition for GUAS, which is based on +convex combinations of vector fields. +Corollary 2.5 ([12]). If the equilibrium of the switched system (2.1) is GUAS, +then it is a globally asymptotically stable equilibrium for the dynamics +˙x = αF (i)(x) + (1 − α)F (j)(x), +for all i, j ∈ {1, · · · , m} and for all α ∈ [0, 1]. +2.1.2. Lie-algebraic conditions in the linear case. In the case of switched +linear systems { ˙x = A(i)x, A(i) ∈ Cn×n}m +i=1, several results related to uniform stability +have been proved (see [32] for a review). We focus here on specific results based on +Lie-algebraic conditions. +Let g = span +� +A(i)� +Lie denote the Lie algebra generated by the matrices A(i), +with i = 1, · · · , m, and equipped with the Lie bracket [A(i), A(j)] = A(i)A(j)−A(j)A(i). +Definition 2.6 (Solvable Lie algebra). A Lie algebra g equipped with the Lie +bracket [., .] is said to be solvable if there exists k ∈ N such that gk = 0, where +{gj}j∈N∗ is a descendant sequence of ideals defined by +� +g1 := g +gj+1 := +� +gj, gj� +. +A general Lie-algebraic criterion for uniform exponential (asymptotic) stability of +switched linear systems is given in the following theorem. +Theorem 2.7 ([15]). +If all matrices A(i), i = 1, · · · , m, are stable (i.e. with +eigenvalues λ(i) +j +such that ℜ +� +λ(i) +j +� +< 0) and if the Lie algebra g is solvable, then the +switched linear system { ˙x = A(i)x}m +i=1 is GUES. +As shown in [23, 31], this result follows from the simultaneous triangularization of +the matrices A(i), which is a well-known property of solvable Lie algebras (see Lie’s +theorem A.5 in Appendix A). This property is in fact equivalent to the existence of a +common invariant flag for complex matrices [6]. +Definition 2.8 (Invariant flag). An invariant maximal flag of the set of matrices +{A(i)}m +i=1 is a set of subspaces {Sj}n +j=1 ⊆ Cn such that (i) A(i)Sj ⊂ Sj for all i, j, (ii) +dim(Sj) = j for all j, and (iii) Sj ⊂ Sj+1 for all j < n. +The subspaces Sj can be described through an orthonormal basis (v1, · · · , vn), so +that Sj = span {v1, · · · , vj}. Note that the vector v1 is a common eigenvector of the +matrices A(i). This basis can be used to construct a CLF. +Proposition 2.9 ([34]). +Let +(2.2) +� +˙x = A(i) x, A(i) ∈ Cn×n, x ∈ Cn�m +i=1 +be a switched linear system. Suppose that all matrices A(i) are stable and admit a +common invariant maximal flag +{0} ⊂ S1 ⊂ · · · ⊂ Sn = Cn, +Sj = span{v1, . . . , vj}. +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +5 +Then there exist ϵj > 0, j = 1, . . . , n, such that +(2.3) +V (x) = +n +� +j=1 +ϵj|v† +jx|2 +is a CLF for (2.2). +The values ϵj must satisfy the condition +(2.4) +ϵj > +max +i∈{1,...,m} +k∈{1,...,j−1} +ϵk +(n − 1)2 +4 +���v† +kA(i)vj +��� +2 +���ℜ +� +λ(i) +j +���� +���ℜ +� +λ(i) +k +���� +where λ(i) +j +are the eigenvalues of A(i). +They can be obtained iteratively from an +arbitrary value ϵ1 > 0. The geometric approach followed in [34] provides a constructive +way to obtain a CLF, a result that we will leverage in an infinite-dimensional setting +for switched nonlinear systems. +2.1.3. Lie-algebraic condition in the nonlinear case. In the context of +switched nonlinear systems, one has to consider the Lie algebra of vector fields +(2.5) +gF = span +� +F (i), i = 1, . . . , m +� +Lie +equipped with the Lie bracket +(2.6) +[F (i), F (j)](x) = JF (j)(x) F (i)(x) − JF (i)(x) F (j)(x). +It has been conjectured in [13] that Lie-algebraic conditions on (2.5) could be +used to characterize uniform stability. +This problem has been solved partially in +[29] for third-order nilpotent Lie algebras and in [18] for particular r-order nilpotent +Lie algebras. Another step toward more general Lie-algebraic conditions based on +solvability has been made in [34], a preliminary result that relies on the so-called +Koopman operator framework. However, the results obtained in [34] are restricted +to specific switched nonlinear systems that can be represented as finite-dimensional +linear ones. +In this paper, we build on this preliminary work, further exploiting +the Koopman operator framework to obtain general conditions that characterize the +GUAS property of switched nonlinear systems. +2.2. Koopman operator approach to dynamical systems. In this section, +we present the Koopman operator framework, which is key to extend the result of +Proposition 2.9 to switched nonlinear systems. We introduce the Koopman semigroup +along with its Koopman generator, cast the framework in the context of Lie groups, +and describe the finite-dimensional approximation of the operator. +2.2.1. Koopman operator. Consider a continuous-time dynamical system +(2.7) +˙x = F(x), +x ∈ X ⊂ Rn, +F ∈ C1 +which generates a flow ϕt : X → X, with t ∈ R+. The Koopman operator is defined +on a (Banach) space F and acts on observables, i.e. functions f : X → R, f ∈ F. +Definition 2.10 (Koopman semigroup [11]). The semigroup of Koopman opera- +tors (in short, Koopman semigroup) is the family of linear operators (Ut)t≥0 defined +by +Ut : F → F, +Utf = f ◦ ϕt. +This manuscript is for review purposes only. + +6 +C. M. ZAGABE AND A. MAUROY +We can also define the associated Koopman generator. +Definition 2.11 (Koopman generator [11]). The Koopman generator associated +with the vector field (2.7) is the linear operator +(2.8) +LF : D(LF ) → F, +LF f := F · ∇f +with the domain D(LF ) = {f ∈ F : F · ∇f ∈ F}. +As shown below (see Lemma 2.13), the Koopman semigroup and the Koopman gen- +erators are directly related. When the Koopman semigroup is strongly continuous +[7], i.e. +lim +t→0+ ∥Utf − f∥F = 0, the Koopman generator is the infinitesimal generator +LF f := lim +t→0+(Utf −f)/t of the Koopman semigroup. Since the Koopman operator Ut +and the generator LF are both linear, we can describe the dynamics of an observable +f on F through the linear abstract ordinary differential equation +(2.9) +˙f = LF f. +We can also briefly discuss the spectral properties of the Koopman operator. +Definition 2.12 (Koopman eigenfunction and eigenvalue [3, 21]). An eigenfunc- +tion of the Koopman operator is an observable φλ ∈ F \ {0} such that +LF φλ = λφλ. +The value λ ∈ C is the associated Koopman eigenvalue. +Under the strong continuity property, the Koopman eigenfunction also satisfies +Utφλ = eλtφλ, +∀t ≥ 0. +For a linear system ˙x = Ax, with x ∈ Rn, we denote an eigenvalue of A by ˜λj +and its associated left eigenvector by wj. Then ˜λj is a Koopman eigenvalue and the +associated Koopman eigenfunction is given by φ˜λj(x) = w† +jx [22]. For a nonlinear +system of the form (2.7) which admits a stable equilibrium xe, the eigenvalues of +JF(xe) are typically Koopman eigenvalues and the associated eigenfunctions are the +so-called principal Koopman eigenfunctions (see Remark 2.14 below). +2.2.2. Koopman operator in the Hardy space H2(Dn). From this point +on, we define the Koopman operator in the Hardy space on the polydisk (see e.g. +[25, 26, 28] for more details). This choice is well-suited to the case of analytic vector +fields that admit a stable hyperbolic equilibrium, where it allows to exploit convenient +spectral properties of the operator. +Let D be the open unit disk in C, ∂D its boundary, and Dn the unit polydisk in +Cn. The Hardy space of holomorphic functions on Dn is the space +H2(Dn) = +� +f : Dn → C, holomorphic : ∥f∥2 = lim +r→1− +� +(∂D)n |f (rω) |2dmn(ω) < ∞ +� +, +where mn is the normalized Lebesgue measure on (∂D)n. It is equipped with an inner +product defined by +⟨f, g⟩ = +� +(∂D)n f (ω) ¯g (ω) dmn(ω), +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +7 +so that the set {zα : α ∈ Nn} is the standard orthonormal basis of monomials on +H2(Dn). The monomials will be denoted by ek(z) = zα(k), where the map α : N → Nn, +k �→ α(k) refers to the lexicographic order, i.e. ek1 < ek2 if |α(k1)| < |α(k2)|, or if +|α(k1)| = |α(k2)| and αj(k1) > αj(k2) for the smallest j such that αj(k1) ̸= αj(k2). +For f and g in H2(Dn), with f = +� +k∈N +fkek and g = +� +k∈N +gkek, the isomorphism +� +k∈N +fkek �→ (fk)k≥0 +between H2(Dn) and the l2-space allows to rewrite the norm and the inner product +as +∥f∥2 = +� +k∈N +|fk|2 +and +⟨f, g⟩ = +� +k∈N +fk ¯gk. +We also note that H2(Dn) is a reproducing kernel Hilbert space (RKHS) with the +Cauchy kernel ([25, Chapter 1]) +(2.10) +k (z, ξ) = +n +� +i=1 +1 +1 − ¯ξizi +, z, ξ ∈ Dn. +It follows that one can define the evaluation functional f(z) = ⟨f, kz⟩ with kz(ω) = +k (z, ω). +If the vector field F is analytic, we can consider its analytic continuation on Dn. +Moreover, if it generates a holomorphic flow that is invariant in Dn, we can define +the Koopman semigroup on H2(Dn), which is also known as the composition operator +with symbol ϕt. The required assumptions are summarized as follows. +Assumption 2. The components Fl, l = 1, · · · , n, of the vector field F belong to +the Hardy space H2(Dn). Moreover, F generates a flow which is holomorphic and +maps Dn to Dn (forward invariance). +It is shown in [4] that the flow ϕt is holomorphic on Dn if and only if the vector field +components have a specific form (see Proposition (A.1) in the Appendix, and the works +[4, 5]). Note also that this property holds if the dynamics possess a globally stable +hyperbolic equilibrium (Assumption 3 below) in the case of non-resonant eigenvalues +(see Remark 2.14). +Now, we recall some important properties that we will use to prove our results. +Lemma 2.13. Consider a function f ∈ H2(Dn) and an evaluation functional kz, +with z ∈ Dn. Then, +1. LF zα ∈ H2(Dn) and the domain D (LF ) is dense in H2(Dn), +2. U ∗ +t kz = kϕt(z), +3. +d +dt ⟨U ∗ +t kz, f⟩ = ⟨L∗ +F U ∗ +t kz, f⟩. +Proof. +1. For all z ∈ Dn, we have +LF zα = F(z) · ∇zk = +n +� +l=1 +Fl(z) αl z(α1,...,αl−1,αl−1,αl+1,...,αn). +Since ∥fzα∥ = ∥f∥ for all f ∈ H2(Dn) and for all α ∈ Nn, it follows from +Assumption 2 that ∥LF eα∥ = +����� +n +� +l=1 +αlFl +����� ≤ +n +� +l=1 +|αl| ∥Fl∥ < ∞. Moreover +D (LF ) is dense in H2(Dn) since the monomials zα form a complete basis. +This manuscript is for review purposes only. + +8 +C. M. ZAGABE AND A. MAUROY +2. For all f ∈ H2(Dn), we have +⟨U ∗ +t kz, f⟩ = ⟨kz, Utf⟩ = (Utf) (z) +and +� +kϕt(z), f +� += f (ϕt(z)) = (Utf) (z), +so that +U ∗ +t kz = kϕt(z). +3. For all z ∈ Dn and all f ∈ D(LF ), +d +dt ⟨U ∗ +t kz, f⟩ = d +dt +� +kϕt(z), f +� += d +dtf ◦ ϕt(z) += F (ϕt(z)) .∇f (ϕt(z)) += +� +kϕt(z), LF f +� += ⟨L∗ +F U ∗ +t kz, f⟩ . +The result follows for all f since D(LF ) is dense in H2(Dn). +In the previous lemma, the second property is a well-known property of the composi- +tion operator on a RKHS. The third property is also known in the context of strongly +continuous semigroup theory (see [7]). +Finally, we make the following additional standing assumption. +Assumption 3. The vector field F admits on Dn a unique hyperbolic stable equi- +librium at 0 (without loss of generality), i.e. F(0) = 0 and the eigenvalues ˜λj of the +Jacobian matrix JF(0) satisfy ℜ{˜λj} < 0. +Remark 2.14 (Holomorphic flow and spectral properties). If Assumption 3 holds +and if the eigenvalues ˜λj are non-resonant1, then the Poincar´e linearization theorem [2] +implies that the flow ϕt is topologically conjugated to the linear flow ˜ϕt(z) = eJF (0)tz, +i.e. there exists a bi-holomorphic map h such that ϕt = h−1 ◦ ˜ϕt ◦ h. In this case, the +flow ϕt is clearly holomorphic. Moreover, the components of h are associated with +holomorphic Koopman eigenfunctions φ˜λj ∈ H2(Dn) associated with the eigenvalues +˜λj [9, 20]. These eigenfunctions are called principal eigenfunctions. Also, it can easily +be shown that, for all α ∈ Nn, +n +� +j=1 +αj˜λj is a Koopman eigenvalue associated with the +eigenfunction φα1 +˜λ1 · · · φαn +˜λn. +2.2.3. Koopman infinite matrix. Since H2(Dn) is isomorphic to l2, the Koop- +man generator can be represented by the Koopman infinite matrix +(2.11) +¯LF = +� +� +� +� +� +� +� +� +� +⟨LF e0, e0⟩ +⟨LF e0, e1⟩ +⟨LF e0, e2⟩ +⟨LF e0, e3⟩ +· · · +⟨LF e1, e0⟩ +⟨LF e1, e1⟩ +⟨LF e1, e2⟩ +⟨LF e1, e3⟩ +· · · +⟨LF e2, e0⟩ +⟨LF e2, e1⟩ +⟨LF e2, e2⟩ +⟨LF e2, e3⟩ +· · · +⟨LF e3, e0⟩ +⟨LF e3, e1⟩ +⟨LF e3, e2⟩ +⟨LF e3, e3⟩ +· · · +⟨LF e4, e0⟩ +⟨LF e4, e1⟩ +⟨LF e4, e2⟩ +⟨LF e4, e3⟩ +· · · +... +... +... +... +· · · +� +� +� +� +� +� +� +� +� +, +1The eigenvalues ˜λj are non-resonant if +n +� +j=1 +αj ˜λj = 0 with α ∈ Zn implies that α = 0. +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +9 +where the kth row contains the components of LF ek in the basis of monomials. For +f = +� +k∈N +fkek, we also have that +⟨LF f, ej⟩ = +� +k∈N +fk⟨LF ek, ej⟩. +Remark 2.15. We note that, since LF e0 = 0, the first row and column of ¯LF +contains only zero entries. By removing the first row and column, one obtains the +representation of the restriction of the Koopman generator to the subspace of functions +f that satisfy f(0) = 0. This subspace is spanned by the basis (ek)k≥1. Note that +kz − k0 belongs to this subspace, since (2.10) implies that kz(0) − k0(0) = 0. +For Fl(z) = +� +|β|≥1 +al,βzβ, the action of the Koopman operator on a basis element +is given by +LF zα = +n +� +l=1 +Fl(z) αl z(α1,...,αl−1,αl−1,αl+1,...,αn) += +n +� +l=1 +� +|β|≥1 +al,βzβ αl z(α1,...,αl−1,αl−1,αl+1,...,αn) += +n +� +l=1 +αl +� +� +� +(β1,...,βn)∈Nn +al,(β1,...,βn)z(β1+α1,...,βl+αl−1,βl+1+αl+1,...,βn+αn) +� +� . +By setting γ1 = β1 + α1, . . . , γl = βl + αl − 1, . . . , γn = βn + αn we obtain +LF zα = +n +� +l=1 +αl +� +|γ|≥|α| +al,(γ1−α1,...,γl−αl+1,...,γn−αn)z(γ1,γ2,...,γn) += +n +� +l=1 +αl +� +|γ|≥|α| +al,(γ−α)lzγ, +(2.12) +where we denote +(2.13) +(γ − α)l = (γ1 − α1, · · · , γl − αl + 1, · · · , γn − αn) . +It follows that the entries of (2.11) are given by +(2.14) +⟨LF ek, ej⟩ = +� +� +� +� +� +n +� +l=1 +αl(k) al,(α(j)−α(k))l +if |α(j)| ≥ |α(k)| +0 +if |α(j)| < |α(k)|. +Remark 2.16. For the linear part of the vector field F, where |α(j)| = 1, j = +1, · · · , n, it is clear that α(j) is the canonical basis vector of Cn, i.e. αi(j) = δij, and we +have that a(l) +α(j) = [JF(0)]lj. Also, if |α(j)| = |α(k)|, we have that (α(j)−α(k))l = α(r) +for some r ≤ n (i.e. |α(r)| = 1), with αr(j) = αr(k) + 1, αl(j) = αl(k) − 1, and +This manuscript is for review purposes only. + +10 +C. M. ZAGABE AND A. MAUROY +αi(j) = αi(k) for all i /∈ {l, r}. Then, it follows from (2.14) that +(2.15) +⟨LF ek, ej⟩ = +� +� +� +� +� +� +� +� +� +n +� +l=1 +αl(j) [JF(0)]ll +if j = k +αl(k) [JF(0)]lr +if α(j) = (α1(k), · · · , αl(k) − 1, · · · , αr(k) + 1, · · · , αn(k)), +0 +otherwise . +2.2.4. Switched Koopman systems and Lie-algebraic conditions. In the +case of a switched nonlinear system (2.1), the Koopman operator description yields +a switched linear infinite-dimensional system (in short, switched Koopman system) of +the form +(2.16) +� +˙f = LF (i)f, f ∈ D +�m +i=1 +with D = ∩m +i=1D(LF (i)). Similarly, the Lie algebra gF spanned by F (i) (see (2.5)) is +replaced by gL = span {LF (i), i = 1, . . . , m}Lie, equipped with the Lie bracket +[LF (i), LF (j)] = LF (i)LF (j) − LF (j)LF (i) . +In particular, we have the well-known relationship +(2.17) +[LF (i), LF (j)] = L[F (i),F (j)] +so that the two algebras gF and gL are isomorphic. +It follows that Lie-algebraic +conditions in gF can be recast into Lie-algebraic criteria in gL, a framework where +we can expect to obtain new results on switched systems that are reminiscent to the +linear case. In particular, since the solvability property of gF is equivalent to the +solvability property of gL, we will investigate whether this latter condition implies +the existence of a common Lyapunov functional for the switched Koopman system +(2.16). +3. Main result. This section presents our main result. We first use an illus- +trative example to show that Lie’s theorem A.5 cannot be used for nonlinear vector +fields, in contrast to the linear case (see Proposition 2.9). We then relax the algebraic +conditions suggested in [13] in order to obtain a triangular form in the Koopman +matrix representation (2.11), a property which is equivalent to the existence of an in- +variant flag for the adjoint operator L∗ +F . We finally prove uniform stability of switched +nonlinear systems under these conditions. +3.1. A first remark on the existence of the common invariant flag. The +following example shows that Lie’s theorem does not hold for infinite-dimensional +switched Koopman systems. +Example 1. Consider the two vector fields +F (1)(x1, x2) = (−αx1, −αx2) +and +F (2)(x1, x2) = (−βx1+γ +� +x2 +1 − x2 +2 +� +, −βx2+2γx1x2), +where α, β and γ are real parameters. These two vector fields generate the Lie alge- +bra g = span +� +F (1), F (2), F (3)� +Lie with F (3)(x1, x2) = (αγ(x2 +1 − x2 +2), 2αγx1x2) since +[F (1), F (2)] = F (3), [F (1), F (3)] = αF (3) and [F (2), F (3)] = βF (3). +Moreover, one +has g1 = [g, g] = span +� +F (3)� +Lie and g2 = [g1, g1] = 0, which implies that g is a +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +11 +solvable Lie algebra. However, the Koopman generators LF (1) and LF (2) associated +with the two vector fields do not share a common eigenfunction, and therefore can- +not have a common invariant flag. Indeed, the principal eigenfunctions of LF (1) are +φ˜λ(1) +1 (x1, x2) = x1 and φ˜λ(1) +2 (x1, x2) = x2, while those of LF (2) are given by +φ˜λ(2) +1 (x1, x2) = βγ +� +βx1 − γ +� +x2 +1 − x2 +2 +�� +(β − γx1)2 + γ2x2 +2 +and +φ˜λ(2) +2 (x1, x2) = +β2γx2 +(β − γx1)2 + γ2x2 +2 +. +We conclude that Lie’s theorem A.5 does not hold setting for the above example, +so that we cannot directly extend Proposition 2.9 to this case. The two Koopman +generators are not simultaneous triangularizable and do not have a common invariant +flag (see [10] for more details about simultaneous triangularization of operators and +its connection to the existence of an invariant infinite maximal flag). However, it +can be easily seen that the Koopman infinite matrices (2.11) related to the vector +fields F (1) and F (2) are both upper triangular, and therefore admit a common infinite +invariant maximal flag. In fact, this implies that the adjoint operators L∗ +F (i) have a +common invariant flag. For this reason, we will depart from the solvability condition +on vector fields (i.e. on Koopman generators), and we will deal with simultaneous +triangularization of adjoints of Koopman generators. The following result provides a +sufficient condition on the vector fields for the simultaneous triangularization of ad- +joints of Koopman generators, which appears to be less restrictive than the solvability +condition. +Lemma 3.1. Let F be an analytic vector field on Dn such that the Jacobian matrix +JF(0) is upper triangular. Then the Koopman matrix (2.11) is upper triangular, i.e. +⟨LF ek, ej⟩ = 0 for all k > j. Moreover, the adjoint L∗ +F of the Koopman generator +admits an infinite invariant maximal flag generated by the monomials ek, i.e. Sk = +span{e1, . . . , ek}. +Proof. It follows from (2.14) that ⟨LF ek, ej⟩ = 0 if |α(k)| > |α(j)| (i.e. the Koop- +man matrix (2.11) is always upper triangular by matrix blocks related to monomials of +the same total degree). In the case |α(k)| = |α(j)| with k > j, the lexicographic order +implies that one can have α(j) = (α1(k), · · · , αl(k)−1, · · · , αr(k)+1, · · · , αn(k)) only +with r < l. Since [JF(0)]lr = 0 for all l > r, it follows from (2.15) that ⟨LF ek, ej⟩ = 0 +when k > j. Finally, it is clear that L∗ +F ej ∈ span{e1, . . . , ej} since ⟨ek, L∗ +F ej⟩ = 0 for +all k > j. +Remark 3.2. When the Jacobian matrix is upper triangular, it is well-known that +[JF(0)]jj = ˜λj. In this case, it follows from (2.15) that the diagonal entries of the +(upper triangular) Koopman matrix are given by +(3.1) +⟨LF ej, ej⟩ = +n +� +l=1 +αl(j)˜λl. +Since these values are the Koopman eigenvalues in the case of non-resonant eigenvalues +˜λj (see Remark 2.14), we will denote λj = ⟨LF ej, ej⟩ by a slight abuse of notation. +Corollary 3.3. Let +� +F (i)�m +i=1 be a switched nonlinear system on Dn and sup- +pose that the Lie algebra of matrices span +� +JF (i)(0) +� +Lie is solvable. Then there ex- +ists a change of variables z �→ �z = P −1z on Cn such that the adjoint operators +L∗ +� +F (i) of the Koopman generators (with �F (i)(�z) = P −1F (i)(P �z)) admit a common in- +This manuscript is for review purposes only. + +12 +C. M. ZAGABE AND A. MAUROY +finite invariant maximal flag. Moreover, +� +�F (i)�m +i=1 is a switched nonlinear system on +Dn � +0, ∥P −1∥∞ +� +. +Proof. Since span +� +JF (i)(0) +� +Lie is solvable, Lie’s theorem A.5 implies that the +matrices JF (i)(0) are simultaneously triangularizable, i.e. there exists a matrix P such +that JF (i)(0) = PT (i)P −1 for all i, where T (i) is upper triangular. Let set F (i)(z) = +JF (i)(0)z + ˜F (i)(z) to separate the linear and the nonlinear parts of the dynamics. In +the new coordinates �z = P −1z, we obtain the dynamics �F (i)(�z) = P −1JF (i)(0)P �z + +P −1 ˜Fi(P �z) = T (i)�z + �˜F i(�z). It follows from Lemma 3.1 that monomials �ek, with +�ek(�z) = (�z)α(k), generate a common invariant maximal flag for L∗ +� +F (i). In addition, for +all z ∈ Dn and all j = 1, · · · , n, we have +|�zj| ≤ ∥�z∥∞ = +��P −1z +�� +∞ ≤ +��P −1�� +∞ ∥z∥∞ < ∥P −1∥∞. +Remark 3.4. It is clear that the change of coordinates z �→ �z = P −1z is defined +up to a multiplicative constant. Without loss of generality, we will consider in the +sequel that ∥P −1∥∞ = 1, so that +� +�F (i)�m +i=1 is a switched nonlinear system on the +unit polydisk Dn. +Instead of a nilpotency or solvability condition on the vector fields F (i), we only +require a milder solvability condition on the Jacobian matrices JF (i)(xe) to guarantee +the triangular form of the Koopman matrix (2.11). It is noticeable that this local +condition is much less restrictive than the global solvability condition mentioned in +the original open problem [13]. Also, it was shown in [1] that the triangular form of the +vector fields (and therefore of the Jacobian matrices) is not sufficient to guarantee the +GUAS property of a switched nonlinear system on Rn. In the next section, however, +we use the solvability condition on the Jacobian matrices to prove the GUAS property +in a bounded invariant region of the state space. This result is consistent with the +local stability result derived in [14]. +3.2. A common Lyapunov function for switched nonlinear systems. We +now aim to show that, for some positive sequence (ϵk)∞ +k=1, the series +(3.2) +V(f) = +∞ +� +k=1 +ϵk |⟨f, ek⟩|2 +is a Lyapunov functional for the switched Koopman system (2.16). Before starting +our main result, we need a few lemmas. +Lemma 3.5. Let ˙z = F(z) be a vector field on the polydisk Dn which generates a +flow ϕt. Suppose that there exist a sequence of positive numbers (ϵk)k≥1 and ρ ∈]0, 1] +such that Dn(0, ρ) is forward invariant with respect to ϕt and such that the series +� +k≥1 +|α(k)|ϵkρ2|α(k)| +is convergent. Then, the series +(3.3) +V(kz − k0) = +∞ +� +k=1 +ϵk |⟨kz, ek⟩|2 +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +13 +and +(3.4) +∞ +� +k=1 +ϵk +d +dt |⟨U ∗ +t (kz − k0), ek⟩|2 +are absolutely and uniformly convergent on Dn(0, ρ) for all t > 0. +Proof. For the first series, we have +ϵk |⟨kz − k0, ek⟩|2 = ϵk |⟨kz, ek⟩|2 = ϵk +���zα(k)��� +2 +< ϵkρ2|α(k)| ≤ |α(k)|ϵkρ2|α(k)| +for all z ∈ Dn(0, ρ) and all k ≥ 1. For the second series, we have +ϵk +���� +d +dt |⟨U ∗ +t (kz − k0), ek⟩|2 +���� = ϵk +���� +d +dt +���(ϕt(z))α(k) − (ϕt(0))α(k)��� +2���� += 2ϵk +����ℜ +� d +dt (ϕt(z))α(k) · +� +ϕt(z) +�α(k)����� +≤ 2ϵk +���� +d +dt (ϕt(z))α(k) +���� · +���� +� +ϕt(z) +�α(k)���� +< 2ϵkρ|α(k)| +���� +d +dt (ϕt(z))α(k) +���� +≤ 2ϵkρ|α(k)| +n +� +s=1 +αs(k) +���F (s) (ϕt(z)) +��� +��(ϕt(z))s +��αs(k)−1 +n +� +l=1,l̸=s +��(ϕt(z))l +��αl(k) +< 2ϵkρ2|α(k)|−1 +n +� +s=1 +αs(k) +���F (s) (ϕt(z)) +��� +for all z ∈ Dn(0, ρ), t > 0 and k ≥ 1. By using the maximum modulus principle for +bounded domains A.2 with the holomorphic function F (s) ◦ ϕt, we can denote +M = +max +z∈∂Dn,s=1,··· ,n |(F (s) ◦ ϕt)(z)| +and we obtain +ϵk +���� +d +dt |⟨U ∗ +t (kz − k0), ek⟩|2 +���� < 2M|α(k)|ϵkρ2|α(k)|−1. +Finally, absolute and uniform convergence of both series follow from the Weierstrass +test (A.4). +Lemma 3.6. Let ˙z = F(z) be a nonlinear system on Dn with an upper triangular +Jacobian matrix JF(0) and let ˙f = LF f be its corresponding Koopman system on +D (LF ) ⊂ H2(Dn). +If the series (3.3) and (3.4) are absolutely and uniformly convergent in Dn for all +t > 0, then, for all double sequences of positive real numbers (bjk)j≥1,k≥1 such that +(3.5) +∞ +� +k=1 +bjk ≤ 1, +This manuscript is for review purposes only. + +14 +C. M. ZAGABE AND A. MAUROY +one has +d +dtV (U ∗ +t (kz − k0)) ≤2 +∞ +� +j=1 +bjjϵj |cj|2 ℜ (λj) ++ 2 +∞ +� +j=2 +j−1 +� +k=1 +� +bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩) +� +, +(3.6) +with cj = ⟨U ∗ +t (kz − k0), ej⟩ and λj = ⟨LF ej, ej⟩. +Proof. Suppose that z ∈ Dn is such that the series (3.3) and (3.4) are absolutely +and uniformly convergent. Then, by using Lemma 2.13 (3), we obtain +d +dtϵk |⟨U ∗ +t (kz − k0), ek⟩|2 = 2ϵkℜ +�� d +dtU ∗ +t (kz − k0), ek +� +⟨U ∗ +t (kz − k0), ek⟩ +� += 2ϵkℜ +� +⟨L∗ +F U ∗ +t (kz − k0), ek⟩ ⟨U ∗ +t (kz − k0), ek⟩ +� += 2ϵk +∞ +� +j=1 +ℜ (cj¯ck ⟨ej, LF ek⟩) +where we used the decomposition U ∗ +t (kz − k0) = +∞ +� +j=1 +cjej. Since (3.4) is absolutely +and uniformly convergent, term by term derivation yields +d +dtV (U ∗ +t (kz − k0)) = +∞ +� +k=1 +d +dtϵk |⟨U ∗ +t (kz − k0), ek⟩|2 += 2 +∞ +� +k=1 +∞ +� +j=1 +ϵkℜ (cj¯ck ⟨ej, LF ek⟩) += 2 +∞ +� +j=1 +ϵj |cj|2 ℜ (λj) + 2 +∞ +� +j=2 +j−1 +� +k=1 +ϵkℜ (cj¯ck ⟨ej, LF ek⟩) +where we used the triangular form of ¯LF (which follows from Lemma 3.1 since JF(0) +is triangular) and λj = ⟨LF ej, ej⟩. +Using (3.5), we have +d +dtV (U ∗ +t (kz − k0)) ≤ 2 +∞ +� +j=1 +� +� +j +� +k=1 +bjk + +∞ +� +k=j+1 +bjk +� +� ϵj |cj|2 ℜ (λj) + 2 +∞ +� +j=2 +j−1 +� +k=1 +ϵkℜ (cj¯ck ⟨ej, LF ek⟩) += 2 +∞ +� +j=1 +bjjϵj |cj|2 ℜ (λj) + 2 +∞ +� +j=2 +j−1 +� +k=1 +bjkϵj |cj|2 ℜ (λj) ++ 2 +∞ +� +j=1 +∞ +� +k=j+1 +bjkϵj |cj|2 ℜ (λj) + 2 +∞ +� +j=2 +j−1 +� +k=1 +ϵkℜ (cj¯ck ⟨ej, LF ek⟩) += 2 +∞ +� +j=1 +bjjϵj |cj|2 ℜ (λj) ++ 2 +∞ +� +j=2 +j−1 +� +k=1 +� +bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩) +� +. +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +15 +Under Assumption 3, it follows from (3.1) that ℜ{λj} < 0 for all j. Therefore, the +time derivative (3.6) of the Lyapunov functional is negative if negative terms related +to the diagonal entries ⟨LF ej, ej⟩ = λj and ⟨LF ek, ek⟩ = λk compensate (possibly +positive) cross-terms related to ⟨ej, LF ek⟩. We note that a term associated with a +diagonal entry will be used to compensate an infinity of cross-terms (associated with +entries in the corresponding row and column of the Koopman matrix), and the values +bjk play the role of weights in the compensation process. +We are now in position to state our main result. +Theorem 3.7. Let +(3.7) +� +˙z = F (i)(z) +�m +i=1 +be a switched nonlinear system on Dn and assume that +• all subsystems of (3.7) have a common hyperbolic equilibrium ze = 0 that is +globally asymptotically stable on the polydisk Dn, +• the Lie algebra span +� +JF (i)(0) +� +Lie is solvable (and therefore there exists a +matrix P such that P −1JF (i)(0)P are upper triangular), +• there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to +the flows ϕt +i of F (i). +Consider double sequences of positive real numbers +� +b(i) +jk +� +j≥1,k≥1, with i = 1, . . . , m, +such that b(i) +jk b(i) +kj > 0 if ⟨L � +F (i)�ek, �ej⟩ ̸= 0 (where �ej(�z) = �zα(j) are monomials in the +new coordinates �z = P −1z) and such that +∞ +� +k=1 +b(i) +jk ≤ 1, and define the double sequence +(3.8) +� +� +�Q(i) +jk +def += +� +� +� +� +� +��� +L � +F (i)�ek, �ej +���2 +4 +��ℜ +�� +L � +F (i)�ej, �ej +���� ��ℜ +�� +L � +F (i)�ek, �ek +���� +1 +b(i) +jk b(i) +kj +if +� +L � +F (i)�ek, �ej +� +̸= 0 +0 +otherwise +� +� +� +j≥2,1≤k≤j−1 +. +If the series +(3.9) ++∞ +� +k=1 +|α(k)| ϵk ρ2|α(k)| +is convergent with +(3.10) +ϵj > +max +i=1,··· ,m +k=1,...,j−1 +ϵk Q(i) +jk , +then the switched system (3.7) is GUAS on Dn(0, ρ). Moreover the series +V (z) = +∞ +� +k=1 +ϵk +��� +� +P −1z +�α(k)��� +2 +is a common global Lyapunov function on Dn(0, ρ). +This manuscript is for review purposes only. + +16 +C. M. ZAGABE AND A. MAUROY +Proof. Consider the switched system +(3.11) +� +˙�z = �F (i)(�z) +�m +i=1 +defined on Dn. By Corollary 3.3, the monomials �ek(�z) = (�z)α(k) generate a common +infinite invariant maximal flag for ¯L � +F (i). We first show that the candidate Lyapunov +functional �V(f) = +∞ +� +k=1 +ϵk |⟨f, �ek⟩|2 satisfies +d +dt +�V +� +(�U (i) +t )∗ (k�z − k0) +� +< 0 +for all i = 1, · · · , m, where �U (i) +t +denotes the Koopman semigroup associated with the +subsystem ˙�z = �F (i)(�z). Lemma 3.5 with (3.9) implies that the series (3.3) and (3.4) +are absolutely convergent on Dn (0, ρ). Then, it follows from Lemma 3.6 that +d +dt +�V +� +(�U (i) +t )∗ (k�z − k0) +� +≤2 +∞ +� +j=1 +b(i) +jj ϵj +���c(i) +j +��� +2 +ℜ +� +λ(i) +j +� ++ 2 +∞ +� +j=2 +j−1 +� +k=1 +b(i) +jk ϵj +���c(i) +j +��� +2 +ℜ +� +λ(i) +j +� ++ b(i) +kj ϵk +���c(i) +k +��� +2 +ℜ +� +λ(i) +k +� ++ ϵkℜ +� +c(i) +j ¯c(i) +k +� +�ej, L � +F (i)�ek +�� +where c(i) +j += +� +(�U (i) +t )∗ (k�z − k0) , �ej +� +and λ(i) +j += +� +L � +F (i)�ej, �ej +� +. Since ℜ{λ(i) +j } < 0 (see +(3.1)), one has to find a sequence of positive numbers (ϵj)j≥1 such that +b(i) +jk ϵj +���c(i) +j +��� +2 ���ℜ +� +λ(i) +j +���� + b(i) +kj ϵk +���c(i) +k +��� +2 ���ℜ +� +λ(i) +k +���� > ϵk +���ℜ +� +c(i) +j ¯c(i) +k +� +�ej, L � +F (i)�ek +����� +for all i = 1, · · · , m and for all j, k with j > k such that +(3.12) +� +�ej, L � +F (i)�ek +� +̸= 0. +By using the inequality +���ℜ +� +c(i) +j ¯c(i) +k +� +�ej, L � +F (i)�ek +����� ≤ +���c(i) +j +��� +���c(i) +k +��� +��� +�ej, L � +F (i)�ek +��� , +one has to satisfy +b(i) +jk ϵj +���c(i) +j +��� +2 ���ℜ +� +λ(i) +j +���� + b(i) +kj ϵk +���c(i) +k +��� +2 ���ℜ +� +λ(i) +k +���� > ϵk +���c(i) +j +��� +���c(i) +k +��� +��� +L � +F (i)�ek, �ej +��� +or equivalently +(3.13) +ϵj > ϵk +� +�− +b(i) +kj +b(i) +jk +���ℜ +� +λ(i) +k +���� +���ℜ +� +λ(i) +j +���� +����� +c(i) +k +c(i) +j +����� +2 ++ +��� +L � +F (i)�ek, �ej +��� +b(i) +jk +���ℜ +� +λ(i) +j +���� +����� +c(i) +k +c(i) +j +����� +� +� def += ϵk h +������ +c(i) +k +c(i) +j +����� +� +. +It is easy to see that the real quadratic function h has the maximal value +Q(i) +jk = +��� +L � +F (i)�ek, �ej +���2 +4 +���ℜ +� +λ(i) +j +���� +���ℜ +� +λ(i) +k +���� +1 +b(i) +jk b(i) +kj +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +17 +so that (3.13) is satisfied if we choose iteratively ϵj according to (3.10). It follows that +we have +d +dt +�V +� +(�U (i) +t )∗ (k�z − k0) +� +< 2 +∞ +� +j=1 +b(i) +jj ϵj +���c(i) +j +��� +2 +ℜ +� +λ(i) +j +� +< − min +j +� +b(i) +jj +���ℜ +� +λ(i) +j +���� +� ∞ +� +j=1 +ϵj +���c(i) +j +��� +2 += − min +j +� +b(i) +jj +���ℜ +� +λ(i) +j +���� +� +�V +� +(�U (i) +t )∗ (k�z − k0) +� +. +With the evaluation functional k�z, we can define +�V : Dn (0, ρ) → R+, +�V (�z) = �V (k�z − k0) +and, using Lemma 2.13, we verify that +�V +� +�ϕ(i) +t (�z) +� += �V +� +k�ϕ(i) +t +(�z) − k0 +� += �V +� +k�ϕ(i) +t +(�z) − k�ϕ(i) +t +(0) +� += �V +� +(�U (i) +t )∗ (k�z − k0) +� +< �V (k�z − k0) = �V (�z). +In addition, if we define V = �V ◦ P −1 : Dn (0, ρ) → R+, we have +V +� +ϕ(i) +t (z) +� += �V +� +P −1ϕ(i) +t (P �z) +� += �V +� +�ϕ(i) +t (�z +� +< �V (�z) = V (P �z) = V (z). +Therefore, we have the CLF +(3.14) +V (z) = +∞ +� +k=1 +ϵk |⟨k�z − k0, �ek⟩|2 = +∞ +� +k=1 +ϵk |⟨k�z, �ek⟩|2 = +∞ +� +k=1 +ϵk +��� +� +P −1z +�α(k)��� +2 +for the switched nonlinear system (3.7). Finally, since Dn (0, ρ) is forward invariant +with respect to ϕ(i) +t , the switched system (3.7) is GUAS on Dn(0, ρ). +Note that, if the assumptions of Theorem 3.7 are satisfied but the polydisk +Dn(0, ρ) is not forward invariant with respect to the flow generated by the subsys- +tems, then the switched system is GUAS in the largest sublevel set of the Lyapunov +function that is contained in Dn(0, ρ). +The condition on the boundedness of the double sequence (3.8) could be inter- +preted as the dominance of diagonal entries of the matrix ¯L � +F (i) (i.e., the Koopman +eigenvalues (3.1)) with respect to the other entries. Moreover, the number of non- +zero cross-terms (3.12) to be compensated affects the way we define the sequence of +weights b(i) +jk and therefore the sequence ϵj in (3.10). If the double sequence (3.8) has +an upper bound Q < 1, one can set ϵk = Q for all k. However, such case rarely +appears. Instead, if Q > 1, one might have ϵk = O(Qk) and it is clear that (3.9) +diverges for all ρ since k − 2|α(k)| → ∞ as k → ∞ (except in the case n = 1 where +|α(k)| = k, see also Remark 3.10 below). In the following, we will consider specific +vector fields such that the series (3.9) converges for a proper choice of sequence b(i) +jk , +so that Theorem 3.7 can be used. +For polynomial vector fields of the form F (i) +l +(z) = +r +� +k=1 +a(i) +l,kzα(k), we denote by K(i) +the number of nonzero terms (without counting the monomial zl in F (i) +l +), i.e. +(3.15) +K(i) = +m +� +l=1 +# +� +k ̸= l : a(i) +l,k ̸= 0 +� +This manuscript is for review purposes only. + +18 +C. M. ZAGABE AND A. MAUROY +where # is the cardinal of a set. In this case, we have the following result. +Corollary 3.8. Let +(3.16) +� +˙z = F (i)(z) +�m +i=1 +be a switched nonlinear system on Dn, where F (i) are polynomial vector fields. Assume +that +• all subsystems of (3.16) have a common hyperbolic equilibrium ze = 0 that is +globally asymptotically stable on Dn, +• the Lie algebra span +� +JF (i)(0) +� +Lie is solvable (and therefore there exists a +matrix P such that PJF (i)(0)P −1 are upper triangular), +• the unit polydisk Dn is forward invariant with respect to the flows ϕ(i) +t +gener- +ated by F (i). +If +(3.17) +max +i=1,··· ,m lim sup +j∈N +max +k=1,...,j−1 +( �K(i))2 ��� +L � +F (i)�ek, �ej +���2 +��ℜ +�� +L � +F (i)�ej, �ej +���� ��ℜ +�� +L � +F (i)�ek, �ek +���� < 1, +where �K(i) is the number of nonzero terms of �F (i)(�z) = P −1F (i)(P �z) (see (3.15)) and +where �ej(�z) = �zα(j) are the monomials in the new coordinates �z = P −1z, then (3.16) +is GUAS on Dn. +Proof. The result follows from Theorem 3.7 with the sequence +(3.18) +� +� +� +� +� +� +� +b(i) +jj = (1 − ξ) +b(i) +jk = +ξ +2K(i) +if j ̸= k with +� +L � +F (i)�ek, �ej +� +̸= 0 or +� +L � +F (i)�ej, �ek +� +̸= 0 +b(i) +jk = 0, +if j ̸= k with +� +L � +F (i)�ek, �ej +� += 0 or +� +L � +F (i)�ej, �ek +� += 0, +with ξ ∈]0, 1[. +It is clear from (2.14) that, for a fixed j and for all k ∈ N \ {j}, +there are at most K(i) nonzero values ⟨L � +F (i)�ek, �ej⟩ and at most K(i) nonzero values +⟨L � +F (i)�ej, �ek⟩, so that the sequence (3.18) satisfies +∞ +� +k=1 +b(i) +jk ≤ 1. The elements Q(i) +jk of +the double sequence (3.8) are given by +(3.19) +Q(i) +jk = +( �K(i))2 ��� +L � +F (i)�ek, �ej +���2 +ξ2 ��ℜ +�� +L � +F (i)�ej, �ej +���� ��ℜ +�� +L � +F (i)�ek, �ek +����. +The condition (3.17) implies that +max +i=1,··· ,m lim sup +j∈N +max +k=1,...,j−1 Q(i) +jk +def += Q < 1 for some +ξ ∈]0, 1[, so that (3.10) is satisfied with +(3.20) +ϵj ∼ max +k∈Kj {ϵk Q} +for j ≫ 1, with Kj = {k ∈ {1, . . . , j − 1} : +� +L � +F (i)�ek, �ej +� +̸= 0 for some i ∈ {1, . . . , m}}. +The sequence (3.20) yields ϵj = O(Qj) for j ≫ 1. It follows that (3.9) is convergent +for any ρ ≤ 1 and Theorem 3.7 implies that the switched system (3.16) is GUAS on +Dn. +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +19 +Another result is obtained when a diagonal dominance property is assumed for +the Jacobian matrices JF (i)(0). +Corollary 3.9. Let +(3.21) +� +˙z = F (i)(z) +�m +i=1 +be a switched nonlinear system on Dn, with F (i) +l +(z) = ++∞ +� +k=1 +a(i) +l,kzα(k) and ++∞ +� +k=1 +|a(i) +l,k| < ∞ +for all i = {1, . . . , m} and l ∈ {1, . . . , n}. Assume that +• all subsystems of (3.21) have a common hyperbolic equilibrium ze = 0 that is +globally asymptotically stable on Dn, +• the Lie algebra span +� +JF (i)(0) +� +Lie is solvable (and therefore there exists a +matrix P such that PJF (i)(0)P −1 are upper triangular), +• there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to +the flows ϕ(i) +t +generated by F (i). +If there exist ξ ∈]0, 1[ and κ ∈]0, 1[ with ξ + κ < 1 such that, for all q, r ∈ +{1, · · · , n} with q < r (when n > 1), +(3.22) +���J �F (i)(0)]qr +��� +2 +< +� +2ξ +n2 − n +�2 ���ℜ([J �F (i)(0)]rr) +��� +���ℜ([J �F (i)(0)]qq) +��� , +(3.23) +���[J �F (i)(0)]qr +��� < +2ξ +n2 − n +���ℜ([J �F (i)(0)]qq) +��� +and +(3.24) +max +i=1,··· ,m lim sup +j∈N +max +k=1,...,j−1 +⟨L � +F (i) �ek,�ej⟩̸=0 +�∞ +l=1 +��� +L � +F (i)�el, �ej +��� �∞ +l=1 +��� +L � +F (i)�ek, �el +��� +κ2 ��ℜ +�� +L � +F (i)�ej, �ej +���� ��ℜ +�� +L � +F (i)�ek, �ek +���� < 1 +ρ2 , +where �ej(�z) = �zα(j) are monomials in the new coordinates �z = P −1z, then (3.21) is +GUAS on Dn(0, ρ). +Proof. We will denote by D +def += (n2 − n)/2 the number of upper off-diagonal +entries of the Jacobian matrices J �F (i)(0). The result follows from Theorem 3.7 with +the sequence +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +b(i) +jj = (1 − ξ − κ) +b(i) +jk = +ξ +2D +if j ̸= k with |α(j)| = |α(k)|, and if +� +L � +F (i)�ek, �ej +� +̸= 0 or +� +L � +F (i)�ej, �ek +� +̸= 0 +b(i) +jk = 0 +if |α(j)| = |α(k)|, +� +L � +F (i)�ek, �ej +� += 0 and +� +L � +F (i)�ej, �ek +� += 0 +b(i) +jk = κ +2 +��� +L � +F (i)�ek, �ej +��� +�∞ +l=1 +��� +L � +F (i)�el, �ej +��� +if |α(k)| < |α(j)| +b(i) +jk = κ +2 +��� +L � +F (i)�ej, �ek +��� +�∞ +l=1 +��� +L � +F (i)�ej, �el +��� +if |α(k)| > |α(j)| +with ξ ∈]0, 1[ and κ ∈]0, 1[. It follows from (2.15) in Remark 2.16 and the fact that +the Jacobian matrices J �F (i)(0) are upper triangular that, for a fixed j and all k ̸= j +This manuscript is for review purposes only. + +20 +C. M. ZAGABE AND A. MAUROY +with |α(k)| = |α(j)|, there are at most D nonzero values +� +L � +F (i)�ek, �ej +� +and at most D +nonzero values +� +L � +F (i)�ej, �ek +� +. Therefore, the sequence b(i) +jk satisfies +∞ +� +k=1 +b(i) +jk < (1 − ξ − κ) + ξ + κ +2 +�j +k=1 +��� +L � +F (i)�ek, �ej +��� +�∞ +l=1 +��� +L � +F (i)�el, �ej +��� + κ +2 +�∞ +k=j+1 +��� +L � +F (i)�ej, �ek +��� +�∞ +l=1 +��� +L � +F (i)�ej, �el +��� +< 1. +The elements Q(i) +jk of the double sequence (3.8) are given by +(3.25) +Q(i) +jk = +� +� +� +� +� +� +� +� +� +� +� +� +� +D2 ��� +L � +F (i)�ek, �ej +���2 +ξ2 ��ℜ +�� +L � +F (i)�ej, �ej +���� ��ℜ +�� +L � +F (i)�ek, �ek +���� +if |α(j)| = |α(k)| +�∞ +l=1 +��� +L � +F (i)�el, �ej +��� �∞ +l=1 +��� +L � +F (i)�ek, �el +��� +κ2 ��ℜ +�� +L � +F (i)�ej, �ej +���� ��ℜ +�� +L � +F (i)�ek, �ek +���� +if |α(k)| ̸= |α(j)| and +� +L � +F (i)�ek, �ej +� +̸= 0 +0 +otherwise. +We note that +∞ +� +l=1 +��� +L � +F (i)�el, �ej +��� and +∞ +� +l=1 +��� +L � +F (i)�ek, �el +��� are finite according to the as- +sumption. +Next, we show that the conditions (3.22) and (3.23) imply that Q(i) +jk < 1 if |α(j)| = +|α(k)|. Indeed, it follows from (2.15) and (3.25) that this latter inequality is equivalent +to +α2 +q(k)|[J �F (i)(0)]qr|2 < ξ2 +D2 +����� +n +� +l=1 +αl(j)ℜ([J �F (i)(0)]ll) +����� +����� +n +� +l=1 +αl(k)ℜ([J �F (i)(0)]ll) +����� +for all j > k such that α(j) = (α1(k), · · · , αq(k)−1, · · · , αr(k)+1, · · · , αn(k)) for some +q < r. Since the diagonal entries of the (upper-triangular) Jacobian matrices J �F (i)(0) +are the eigenvalues and therefore have negative real parts, the most restrictive case is +obtained with αl(k) = 0 for all l ̸= q, which yields +α2 +q(k)|[J �F (i)(0)]qr|2 < ξ2 +D2 +���(αq(k) − 1)ℜ([J �F (i)(0)]qq) + ℜ([J �F (i)(0)]rr) +��� +���αq(k)ℜ([J �F (i)(0)]qq) +��� . +When αq(k) = 1, this inequality is equivalent to (3.22). When αq(k) > 1, we can +rewrite +(αq(k) − 1)|[J �F (i)(0)]qr|2 + |[J �F (i)(0)]qr|2 +< ξ2 +D2 +� +(αq(k) − 1) +���ℜ([J �F (i)(0)]qq) +��� +2 ++ +���ℜ([J �F (i)(0)]rr) +��� +���ℜ([J �F (i)(0)]qq) +��� +� +. +Using (3.22), we have that the above inequality is satisfied if +(αq(k) − 1)|[J �F (i)(0)]qr|2 < ξ2 +D2 (αq(k) − 1) +���ℜ([J �F (i)(0)]qq) +��� +2 +, +which is equivalent to (3.23). +While Q(i) +jk < 1 for |α(j)| = |α(k)|, it is easy to see that Q(i) +jk > 1 for |α(j)| > |α(k)|. +The condition (3.24) therefore implies that +max +i=1,··· ,m lim sup +j∈N +max +k=1,...,j−1 Q(i) +jk +def += Q < 1/ρ2 +and (3.10) is satisfied with +(3.26) +ϵj ∼ max +k∈Kj {ϵk Q} +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +21 +for j ≫ 1, with +Kj = {k ∈ 1, . . . , j − 1 : +� +L � +F (i)�ek, �ej +� +̸= 0 for some i ∈ {1, . . . , m} and |α(k)| < |α(j)|}. +Hence, the sequence (3.26) yields ϵj = O(Q|α(j)|). It follows that (3.9) is convergent +and Theorem 3.7 implies that the switched system (3.21) is GUAS on Dn(0, ρ). +For the particular case where the Jacobian matrices JF (i)(0) are simultaneously +diagonalizable (i.e. they are diagonalizable and they commute), the diagonal dom- +inance conditions (3.22) and (3.23) are trivially satisfied. We should mention that +the Lie-algebraic property of commutation is only needed for the Jacobian matrices +JF (i)(0), an assumption which contrasts with the commutation property imposed on +vector fields in [17], [30] and [33]. +Remark 3.10. In the case n = 1, we recover the trivial GUAS property of switched +systems from Corollary 3.9. Indeed, consider the vector fields F (i)(z) = +∞ +� +k=1 +a(i) +k zk +on D, with +∞ +� +k=1 +|a(i) +k | < ∞, and assume that the subsystems have a globally stable +equilibrium at the origin. The Lie-algebra generated by the scalars JF (i)(0) is trivially +solvable and D(0, ρ) is forward invariant for all ρ. Moreover, the conditions (3.22) and +(3.23) are trivially satisfied. Then Corollary 3.9 implies that the switched system is +GUAS on D(0, ρ) for ρ ∈]0, 1[ which satisfies (3.24). It follows from (2.14) that +∞ +� +l=1 +|⟨LF (i)el, ej⟩| = +j +� +l=1 +l|a(i) +j−l+1| = +j +� +l=1 +(j − l + 1)|a(i) +l |, +∞ +� +l=1 +|⟨LF (i)ek, el⟩| = k +∞ +� +l=1 +|a(i) +l |, +and |ℜ (⟨ej, LF (i)ej⟩)| = j +���ℜ(a(i) +1 ) +���. With κ arbitrarily close to 1 (since ξ can be taken +arbitrarily small in (3.22) and (3.23)), condition (3.24) is rewritten as +max +i=1,··· ,m lim sup +j∈N +�j +l=1(j − l + 1)|a(i) +l | �∞ +l=1 |a(i) +l | +j +���ℜ(a(i) +1 ) +��� +2 +< 1 +ρ2 +and, using (j − l + 1)|a(i) +l | ≤ j|a(i) +l | for all l, we obtain +ρ < +min +i=1,··· ,m +�∞ +l=1 |a(i) +l | +���ℜ(a(i) +1 ) +��� +. +4. Examples. This section presents two examples that illustrate our results. We +will focus on specific cases that satisfy the assumptions of Corollaries 3.8 and 3.9 and, +without loss of generality, we will directly consider Jacobian matrices in triangular +form. +4.1. Example 1: polynomial vector fields. Similarly to Example 1, we con- +sider the vector fields on the bidisk D2 +(4.1) +F (1)(z1, z2) = +� +−az1 +−az2 +and F (2)(z1, z2) = +� +� +� +−az1 + b +� +z2 +1 − z1z2 +2 +� +−az2 + b +2z1z2, +This manuscript is for review purposes only. + +22 +C. M. ZAGABE AND A. MAUROY +where b > 0 and a > 3b. For all ρ < 1, the bidisk D2(0, ρ) is invariant with respect +to the flows of F (i). Indeed, for all z ∈ ∂D2(0, ρ) (i.e. |zl| = ρ for some l), one has to +verify that ℜ +� +F (i) +l +(z) ¯zl +� +< 0. We have +• |zl| = ρ ⇒ ℜ +� +F (1) +l +(z)¯zl +� += −aρ2 < 0, +• |z1| = ρ ⇒ ℜ +� +F (2) +1 +(z)¯z1 +� += −aρ2 + bρ2ℜ +� +z1 − z2 +2 +� +< 0, +• |z2| = ρ ⇒ ℜ +� +F (2) +2 +(z)¯z2 +� += ρ2 +� +−a + b +2ℜ (z1) +� +< 0. +It is clear that vector field F (1) generates a holomorphic flow on D2. +The same +property holds for F (2) since the conditions of Proposition A.1 are satisfied with +h1 (z′ +1) = h2 (z′ +2) = 0 (i.e. ℜ{G1(z)} = ℜ{−a + b +� +z1 − z2 +2 +� +} < 0 and ℜ{G2(z)} = +ℜ{−a + b +2z1} < 0). The unique global stable equilibrium of the subsystems in the +bidisk D2 is the origin. According to (2.14), the entries of the Koopman matrices +¯LF (1) and ¯LF (2) are given by +⟨LF (1)ek, ej⟩ = +� +−a |α(j)| +if k = j +0 +otherwise +and +⟨LF (2)ek, ej⟩ = +� +� +� +� +� +� +� +� +� +� +� +� +� +−a |α(j)| +if k = j +b +� +α1(j) + α2(j) − 2 +2 +� +if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j) +−b α1(j) +if α1(k) = α1(j) and α2(k) = α2(j) − 2 ≥ 0 +0 +otherwise . +Since ⟨LF (1)ek, ej⟩=0 for all k ̸= j, we have Q(1) +jk = 0 for all k, j in (3.17). Moreover, +K(2) = 3 and the condition (3.17) can be rewritten as +Q = lim sup +j∈N +|α(j)|>1 +max +� +� +� +� +� +9b2 +a2 +� +α1(j) + α2(j)−2 +2 +�2 +|α(j)| (|α(j)| − 1) , 9b2 +a2 +(α1(j))2 +|α(j)| (|α(j)| − 2) +� +� +� +� +� += 9b2 +a2 < 1 +and is satisfied since a > 3b. Hence, it follows from Corollary 3.8 that the switched +system (4.1) is GUAS in D2. Note that, in this case, a CLF is given by +V (z) = +∞ +� +k=1 +Q−k ���zα(k)��� +2 +. +4.2. Example 2: analytic vector fields. The following example is taken from +[1] and [12]. Consider the switched system defined by the vector fields +F (1)(x1, x2) = +� +−x1 + 1 +µ sin2(x1)x2 +1x2, −x2 +� +F (2)(x1, x2) = +� +−x1 + 1 +µ cos2(x1)x2 +1x2, −x2 +� +, +where µ ≥ 12/5. Both subsystems are globally asymptotically stable, but the switched +system is not GUAS in R2, as shown in [1] by using the fact that the convex com- +bination F = +� +F (1) + F (2)� +/2 of the two subsystems is not globally asymptotically +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +23 +stable in R2 (see Corollary 2.5). Yet, our result allows to infer the GUAS property +in a specific region of the invariant real square ] − 1, 1[2. To do so, we complexify +the dynamics and define the vector fields on the bidisk D2. For all ρ < 1, the bidisk +D2(0, ρ) is invariant with respect to the flows of F (i). Indeed, we have +• |z1| = ρ ⇒ ℜ +� +F (1) +1 +(z)¯z1 +� += ρ2 +� +−1 + 1 +µℜ +� +z1z2 sin2(z1) +�� +< 0 since +1 +µ +��ℜ +� +z1z2 sin2(z1) +��� ≤ ρ2 ��sin2(z1) +�� +µ +< 1 +where we used max +z1∈D +��sin2(z1) +�� < 12/5 ≤ µ, +• |z2| = ρ ⇒ ℜ +� +F (1) +2 +(z)¯z2 +� += −ρ2 < 0. +The same result follows for F (2) (with max +z1∈D +��cos2(z1) +�� < 12/5 ≤ µ). The vector field +F (1) generates a holomorphic flow on D2 since the conditions of Proposition A.1 are +satisfied with h1 (z′ +1) = h2 (z′ +2) = 0 (i.e. ℜ{G1(z)} = ℜ{−1 + 1 +µ sin2(z1)z1z2} < 0 and +ℜ{G2(z)} = −1 < 0. The same result holds for F (2). The Taylor expansion of the +vector fields yields +F (1)(z) = +� +−z1 + 1 +µ +∞ +� +p=1 +(−1)p+122p−1 +(2p)! +z2p+2 +1 +z2, −z2 +� +F (2)(z) = +� +−z1 + 1 +µz2 +1z2 + 1 +µ +∞ +� +p=1 +(−1)p22p−1 +(2p)! +z2p+2 +1 +z2, −z2 +� +. +According to (2.14), the entries of the Koopman matrices ¯LF (1) and ¯LF (2) are given +by +⟨LF (1)ek, ej⟩ = +� +� +� +� +� +� +� +� +� +− |α(k)| +if k = j +α1(k) +µ +(−1)p+122p−1 +(2p)! +if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 +0 +otherwise +and +⟨LF (2)ek, ej⟩ = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− |α(k)| +if k = j +α1(k) +µ +if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 +α1(k) +µ +(−1)p22p−1 +(2p)! +if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 +0 +otherwise +This manuscript is for review purposes only. + +24 +C. M. ZAGABE AND A. MAUROY +where p = 1, · · · , +�α1(j) − 1 +2 +� +. This implies that we have +(4.2) +∞ +� +l=1 +|⟨LF (1)el, ej⟩| = |α(j)| + 1 +µ +⌊ α1(j)−1 +2 +⌋ +� +l=1 +(α1(j) − 1 − 2l)22l−1 +(2l)! +∞ +� +l=1 +|⟨LF (2)el, ej⟩| = |α(j)| + α1(j) − 1 +µ ++ 1 +µ +⌊ α1(j)−1 +2 +⌋ +� +l=1 +(α1(j) − 1 − 2l)22l−1 +(2l)! +∞ +� +l=1 +|⟨LF (1)ek, el⟩| = |α(k)| + α1(k) +µ +∞ +� +p=1 +22p−1 +(2p)! = |α(k)| + α1(k) +2µ +(cosh(2) − 1) +∞ +� +l=1 +|⟨LF (2)ek, el⟩| = |α(k)| + α1(k) +µ +� +1 + +∞ +� +p=1 +22p−1 +(2p)! +� += |α(k)| + α1(k) +2µ +(cosh(2) + 1) . +Since the Jacobian matrices JF (i)(0) are diagonal, the conditions (3.22) and (3.23) +are trivially satisfied (with ξ arbitrarily small). Moreover, we observe from (4.2) that +∞ +� +l=1 +|⟨LF (1)el, ej⟩| ≤ +∞ +� +l=1 +|⟨LF (2)el, ej⟩| and +∞ +� +l=1 +|⟨LF (1)ek, el⟩| ≤ +∞ +� +l=1 +|⟨LF (2)ek, el⟩| for all +k, j. It follows that, with κ arbitrarily close to 1, condition (3.24) can be rewritten as +lim sup +j∈N +max +k=1,...,j−1 +�∞ +l=1 |⟨LF (2)el, ej⟩| �∞ +l=1 |⟨LF (2)ek, el⟩| +|ℜ (⟨LF (2)ej, ej⟩)| |ℜ (⟨LF (2)ek, ek⟩)| +< 1 +ρ2 , +which is verified for ρ = +� +1 + 1 +2µ (cosh(2) + 1) +�−1 +. Indeed, from (4.2), we have +∞ +� +l=1 +|⟨LF (2)el, ej⟩| < |α(j)| + α1(j) +µ +� +1 + +∞ +� +l=1 +22l−1 +(2l)! +� += |α(j)| + α1(j) +2µ +(cosh(2) + 1) +≤ |α(j)| +� +1 + 1 +2µ (cosh(2) + 1) +� +. +and +∞ +� +l=1 +|⟨LF (2)ek, el⟩| ≤ |α(k)| +� +1 + 1 +2µ (cosh(2) + 1) +� +. +It follows from Corollary 3.9 that the switched system is GUAS on D2(0, ρ). See +Figure 1 for the different values of ρ depending on µ. Note that a CLF is given by +V (z) = +∞ +� +k=1 +Q−2|α(k)| ���zα(k)��� +2 +where Q = 1/ρ. +5. Conclusion and perspectives. This paper provides new advances on the +uniform stability problem for switched nonlinear systems satisfying Lie-algebraic solv- +ability conditions. First, we have shown that the solvability condition on nonlinear +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +25 +Fig. 1. The switched system is shown to be GUAS on a polydisk of radius ρ that depends on +the parameter µ. +vector fields does not guarantee the existence of a common invariant flag and, instead, +we have imposed the solvability condition only on the linear part of the vector fields. +Then we have constructed a common Lyapunov functional for an equivalent infinite- +dimensional switched linear system obtained with the adjoint of the Koopman genera- +tor on the Hardy space of the polydisk. Finally we have derived a common Lyapunov +function via evaluation functionals to prove that specific switched nonlinear systems +are uniformly globally asymptotically stable on invariant sets. Our results heavily +rely on the Koopman operator framework, which appears to be a valid tool to tackle +theoretical questions from a novel angle. +We envision several perspectives for future research. Our results apply to specific +types of switched nonlinear systems within the frame of Lie-algebraic solvability con- +ditions. They could be extended to more general dynamics, including dynamics that +possess a limit cycle or a general attractor. In the same line, the Koopman operator- +based techniques developed in this paper could be applied to other types of stability +than uniform stability. More importantly, the obtained stability results are limited +to bounded invariant sets, mainly due to the convergence properties of the Lyapunov +functions and the very definition of the Hardy space on the polydisk. We envision +that these results could possibly be adapted to infer global stability in Rn. Finally, +our results are not restricted to switched systems and have direct implications in the +global stability properties of nonlinear dynamical systems, which will be investigated +in a future publication. +Appendix A. General theorems. +We recall here some general results that are used in the proofs of our results. +Proposition A.1 ([4]). +Let F : Dn → Cn be holomorphic. +Then F is an +infinitesimal generator on Dn if and only if, for all l = 1, · · · , n and for all z ∈ Dn, +Fl(z) = Gl(z) (zl − hl (z′ +l)) +where z′ +l = (z1, · · · , zl−1, zl+1, · · · zn), hl : Dn−1 → D is holomorphic, Gl : Dn → C is +holomorphic, and ℜ ((1 − hl (z′ +l) ¯zl) Gl(z)) ≤ 0. +Theorem A.2 (Maximum Modulus Principle for bounded domains [27]). +Let +Dn ⊂ Cn be a bounded domain and f : Dn → C be a continuous function, whose +restriction to Dn is holomorphic. Then |f| attains a maximum on the boundary ∂Dn. +Theorem A.3 (Abel’s multidimensional lemma [27] p.36). +Let +� +α∈Nn +aαzα be +This manuscript is for review purposes only. + +1 +0.9 +0.8 +0.7 +0.6 +0.5 +0 +10 +20 +30 +40 +50 +μ26 +C. M. ZAGABE AND A. MAUROY +a power series. If there exist r ∈ Cn such that sup +α∈Nn |aαrα| < ∞, then the series +� +α∈Nn +aαzα is normally convergent for all z ∈ Cn such that |z1| < |r1|, · · · , |zn| < |rn|. +Theorem A.4 (Weierstrass’s M-test). +Let ++∞ +� +k=1 +fn(z) be a series of functions on +a domain Dn of Cn. If there exists a sequence of real numbers Mk such that +• Mk > 0 for all k, +• the numerical series ++∞ +� +k=1 +Mk is convergent and +• ∀k, ∀z ∈ Dn, |fk(z)| ≤ Mk. +Then the series ++∞ +� +k=1 +fn(z) is absolutely and uniformly convergent on Dn. +Theorem A.5 (Lie’s theorem [8] p.49). +Let X be a nonzero n-complex vector +space, and g be a solvable Lie subalgebra of the Lie algebra of n × n complex matrices. +Then X has a basis (v1, . . . , vn) with respect to which every element of g has an upper +triangular form. +REFERENCES +[1] D. Angeli and D. Liberzon, A note on uniform global asymptotic stability of nonlinear +switched systems in triangular form, in Proc. 14th Int. Symp. on Mathematical Theory of +Networks and Systems (MTNS), 2000. +[2] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, vol. 250, +Springer Science & Business Media, 2012. +[3] M. Budiˇsi´c, R. Mohr, and I. 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Morse, Basic problems in stability and design of switched systems, +IEEE control systems magazine, 19 (1999), pp. 59–70. +[17] J. L. Mancilla-Aguilar, A condition for the stability of switched nonlinear systems, IEEE +Transactions on Automatic Control, 45 (2000), pp. 2077–2079. +[18] M. Margaliot and D. Liberzon, Lie-algebraic stability conditions for nonlinear switched +systems and differential inclusions, Systems & control letters, 55 (2006), pp. 8–16. +This manuscript is for review purposes only. + +UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS +27 +[19] A. Mauroy and I. Mezi´c, Global stability analysis using the eigenfunctions of the Koopman +operator, IEEE Transactions on Automatic Control, 61 (2016), pp. 3356–3369. +[20] A. Mauroy, I. Mezi´c, and J. Moehlis, Isostables, isochrons, and Koopman spectrum for the +action–angle representation of stable fixed point dynamics, Physica D: Nonlinear Phenom- +ena, 261 (2013), pp. 19–30. +[21] A. Mauroy, Y. Susuki, and I. 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Scheidemann, Introduction to complex analysis in several variables, Springer, 2005. +[28] J. H. Shapiro, Composition operators: and classical function theory, Springer Science & Busi- +ness Media, 2012. +[29] Y. Sharon and M. Margaliot, Third-order nilpotency, finite switchings and asymptotic sta- +bility, in Proceedings of the 44th IEEE Conference on Decision and Control, IEEE, 2005, +pp. 5415–5420. +[30] H. Shim, D. Noh, and J. H. Seo, Common Lyapunov function for exponentially stable non- +linear systems, 2001. +[31] R. Shorten and K. Narendra, On the stability and existence of common Lyapunov functions +for stable linear switching systems, in Proceedings of the 37th IEEE Conference on Decision +and Control (Cat. No. 98CH36171), vol. 4, IEEE, 1998, pp. 3723–3724. +[32] R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, Stability criteria for switched +and hybrid systems, SIAM review, 49 (2007), pp. 545–592. +[33] L. Vu and D. Liberzon, Common Lyapunov functions for families of commuting nonlinear +systems, Systems & control letters, 54 (2005), pp. 405–416. +[34] C. M. Zagabe and A. Mauroy, Switched nonlinear systems in the Koopman operator frame- +work: Toward a Lie-algebraic condition for uniform stability, in 2021 European Control +Conference (ECC), IEEE, 2021, pp. 281–286. +This manuscript is for review purposes only. + diff --git a/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/load_file.txt b/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54b8ad8538b89c644ff188842a890184391c12d2 --- /dev/null +++ b/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/load_file.txt @@ -0,0 +1,1083 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf,len=1082 +page_content='UNIFORM GLOBAL STABILITY OF SWITCHED NONLINEAR SYSTEMS IN THE KOOPMAN OPERATOR FRAMEWORK∗ CHRISTIAN MUGISHO ZAGABE† AND ALEXANDRE MAUROY ‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we provide a novel solution to an open problem on the global uniform stability of switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our results are based on the Koopman operator approach and, to our knowledge, this is the first theoretical contribution to an open problem within that framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By focusing on the adjoint of the Koopman generator in the Hardy space on the polydisk, we define equivalent linear (but infinite-dimensional) switched systems and we construct a common Lyapunov functional for those systems, under a solvability condition of the Lie algebra generated by the linearized vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A common Lyapunov function for the original switched nonlinear systems is derived from the Lyapunov functional by exploiting the reproducing kernel property of the Hardy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Lyapunov function is shown to converge in a bounded region of the state space, which proves global uniform stability of specific switched nonlinear systems on bounded invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator, Hardy space on the polydisk, Switched systems, Uniform stability, Common Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 47B32, 47B33, 47D06, 70K20, 93C10, 93D05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Switched systems are hybrid-type models encountered in ap- plications where the dynamics abruptly jump from one behavior to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' They are typically described by a family of subsystems that alternate according to a given commutation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Stability properties of switched systems have been the focus of intense research effort (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' [32] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this context, a natural question is whether a switched system with an equilibrium point is uniformly stable, that is, stable for any commutation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It turned out that the uniform stability problem is counter-intuitive and challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the linear case, it is well-known that stable subsystems may induce an unstable switched system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, uniform stability is guaranteed if the matrices associated with the subsystems are stable and commute pairwise [24], a result which is extended in [15] to subsystems described by stable matrices generating a solvable Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This latter result can be explained by the well-known equivalence between solvable Lie algebra of matrices and the existence of a common invariant flag for those matrices, which allows to construct a common Lyapunov function for the subsystems [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case of switched nonlinear systems, an open problem was posed in [13] on the relevance of Lie-algebraic conditions of vector fields for global uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Partial solutions have been proposed in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It was proven in [17] that uniform stability holds if the vector fields are individually stable and commute, in which case a common Lyapunov function can be constructed [30, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Uniform stability was also shown for a pair of vector fields generating a third-order nilpotent Lie algebra [29] and for particular r-order nilpotent Lie algebras [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, no result has been obtained, which solely relies on the more general solvability property of Lie algebras of the subsystems vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we provide a partial solution to the problem introduced in [13] by ∗Submitted to the editors †Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni- versity of Namur (christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='mugisho@unamur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='be), ‡Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni- versity of Namur (alexandre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='mauroy@unamur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='be) 1 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='05529v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='DS] 13 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY proving global uniform stability results for switched nonlinear systems under a gen- eral solvability property of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' To do so, we rely on the Koopman operator framework [3, 21]: we depart from the classical pointwise description of dynami- cal systems and consider instead the evolution of observable functions (here in the Hardy space of holomorphic functions defined on the complex polydisk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Through this approach, equivalent infinite-dimensional dynamics are generated by linear Koopman generators, so that nonlinear systems are represented by Koopman linear systems that are amenable to global stability analysis [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, building on preliminary results obtained in [34], we construct a common Lyapunov functional for switched Koopman linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A key point is to focus on the adjoint of the Koopman generators and notice that these operators have a common invariant maximal flag if the linear parts of the subsystems generate a solvable Lie algebra, a condition that is milder than the original assumption proposed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, we derive a common Lyapunov function for the original switched nonlinear system and prove its conver- gence under specific algebraic conditions on the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This allows us to obtain a bounded invariant region where the switched nonlinear system is globally uniformly asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' To our knowledge, this is the first time that a novel solution to an open theoretical problem is obtained within the Koopman operator framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In Section 2, we present some preliminary notions on uniform stability of switched nonlinear systems and give a general introduction to the Koopman operator framework, as well as some specific properties in the Hardy space on the polydisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In Section 3, we state and prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We recast the open problem given in [13] in terms of the existence of an invariant maximal flag and we provide a constructive proof for the existence of a common Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Additional corollaries are also given, which focus on specific classes of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our main results are illustrated with two examples in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, concluding remarks and perspectives are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We will use the following notation throughout the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For multi-index notations α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=', αn) ∈ Nn, we define |α| = α1 + · · · + αn and zα = zα1 1 · · · zαn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The complex conjugate and real part of a complex number a are denoted by ¯a and ℜ(a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The transpose-conjugate of a matrix (or vector) A is denoted by A†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Jacobian matrix of the vector field F at x is given by JF(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The complex polydisk centered at 0 and of radius ρ is defined by Dn(0, ρ) = {z ∈ Cn : |z1| < ρ, · · · , |zn| < ρ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, Dn denotes the unit polydisk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' with ρ = 1) and ∂Dn is its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, the floor of a real number is denoted by ⌊x⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this section, we introduce preliminary notions and results on the stability theory for switched systems and on the Koopman operator framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Stability of switched systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We focus on the uniform asymptotic sta- bility property of switched systems and on the existence of a common Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Some existing results that connect these two main concepts are presented in both linear and nonlinear cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 (Switched system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A switched system ˙x = F (σ)(x) is a (finite) set of subsystems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) � ˙x = F (i)(x), x ∈ X ⊂ Rn�m i=1 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 3 associated with a commutation law σ : R+ → {1, · · · , m} indicating which subsystem is activated at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we make the following standing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The commutation law σ is a piecewise constant function with a finite number of discontinuities on every bounded time interval (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' According to [16], stability analysis of switched sys- tems revolves around three important problems: decide whether an equilibrium is stable under the action of the switched system for any commutation law σ, in which case the equilibrium is said to be uniformly stable, identify the commutation laws for which the equilibrium is stable, and construct the commutation law for which the equilibrium is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper we focus on the first problem related to uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2 (Uniform stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assume that F (i)(xe) = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The equilibrium xe is uniformly asymptotically stable (UAS) if ∀ϵ > 0, ∃δ > 0 such that ∥x(0) − xe∥ ≤ δ ⇒ ∥x(t) − xe∥ ≤ ϵ, ∀t > 0, ∀σ and ∥x(0) − xe∥ ≤ δ ⇒ lim t→∞ x(t) = xe, ∀σ, globally uniformly asymptotically stable (GUAS) on D ⊆ Rn if it is UAS and x(0) ∈ D ⇒ lim t→∞ x(t) = xe, ∀σ, globally uniformly exponentially stable (GUES) on D ⊆ Rn if ∃β, λ > 0 such that x(0) ∈ D ⇒ ∥x(t) − xe∥ ≤ β∥x(0) − xe∥e−λt, ∀t > 0, ∀σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This definition implies that the subsystems share a common equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' More- over, a necessary condition is that this equilibrium is asymptotically stable with re- spect to the dynamics of all individual subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, this condition is not sufficient, since the switched system might be unstable for a specific switching law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A sufficient condition for uniform asymptotic stability is the existence of a common Lyapunov function (CLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3 (Common Lyapunov function [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A positive C1- function V : D ⊆ Rn → R is a common Lyapunov function on D ⊆ Rn for the family of subsystems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) if ∇V · F (i)(x) < 0 ∀x ∈ D \\ {xe}, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For switched systems with a finite number of subsystems, a converse Lyapunov result also holds ([12], [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4 ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that D ⊆ Rn is compact and forward-invariant with respect to the flow induced by the subsystems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The switched system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) is GUAS on D if and only if all subsystems share a CLF on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY A corollary of this result provides a necessary condition for GUAS, which is based on convex combinations of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the equilibrium of the switched system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) is GUAS, then it is a globally asymptotically stable equilibrium for the dynamics ˙x = αF (i)(x) + (1 − α)F (j)(x), for all i, j ∈ {1, · · · , m} and for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lie-algebraic conditions in the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case of switched linear systems { ˙x = A(i)x, A(i) ∈ Cn×n}m i=1, several results related to uniform stability have been proved (see [32] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We focus here on specific results based on Lie-algebraic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let g = span � A(i)� Lie denote the Lie algebra generated by the matrices A(i), with i = 1, · · · , m, and equipped with the Lie bracket [A(i), A(j)] = A(i)A(j)−A(j)A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6 (Solvable Lie algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A Lie algebra g equipped with the Lie bracket [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='] is said to be solvable if there exists k ∈ N such that gk = 0, where {gj}j∈N∗ is a descendant sequence of ideals defined by � g1 := g gj+1 := � gj, gj� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A general Lie-algebraic criterion for uniform exponential (asymptotic) stability of switched linear systems is given in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If all matrices A(i), i = 1, · · · , m, are stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' with eigenvalues λ(i) j such that ℜ � λ(i) j � < 0) and if the Lie algebra g is solvable, then the switched linear system { ˙x = A(i)x}m i=1 is GUES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' As shown in [23, 31], this result follows from the simultaneous triangularization of the matrices A(i), which is a well-known property of solvable Lie algebras (see Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This property is in fact equivalent to the existence of a common invariant flag for complex matrices [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 (Invariant flag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' An invariant maximal flag of the set of matrices {A(i)}m i=1 is a set of subspaces {Sj}n j=1 ⊆ Cn such that (i) A(i)Sj ⊂ Sj for all i, j, (ii) dim(Sj) = j for all j, and (iii) Sj ⊂ Sj+1 for all j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The subspaces Sj can be described through an orthonormal basis (v1, · · · , vn), so that Sj = span {v1, · · · , vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that the vector v1 is a common eigenvector of the matrices A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This basis can be used to construct a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 ([34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) � ˙x = A(i) x, A(i) ∈ Cn×n, x ∈ Cn�m i=1 be a switched linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that all matrices A(i) are stable and admit a common invariant maximal flag {0} ⊂ S1 ⊂ · · · ⊂ Sn = Cn, Sj = span{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 5 Then there exist ϵj > 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , n, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) V (x) = n � j=1 ϵj|v† jx|2 is a CLF for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The values ϵj must satisfy the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) ϵj > max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',m} k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1} ϵk (n − 1)2 4 ���v† kA(i)vj ��� 2 ���ℜ � λ(i) j ���� ���ℜ � λ(i) k ���� where λ(i) j are the eigenvalues of A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' They can be obtained iteratively from an arbitrary value ϵ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The geometric approach followed in [34] provides a constructive way to obtain a CLF, a result that we will leverage in an infinite-dimensional setting for switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lie-algebraic condition in the nonlinear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the context of switched nonlinear systems, one has to consider the Lie algebra of vector fields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5) gF = span � F (i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m � Lie equipped with the Lie bracket (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6) [F (i), F (j)](x) = JF (j)(x) F (i)(x) − JF (i)(x) F (j)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It has been conjectured in [13] that Lie-algebraic conditions on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5) could be used to characterize uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This problem has been solved partially in [29] for third-order nilpotent Lie algebras and in [18] for particular r-order nilpotent Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Another step toward more general Lie-algebraic conditions based on solvability has been made in [34], a preliminary result that relies on the so-called Koopman operator framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, the results obtained in [34] are restricted to specific switched nonlinear systems that can be represented as finite-dimensional linear ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we build on this preliminary work, further exploiting the Koopman operator framework to obtain general conditions that characterize the GUAS property of switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator approach to dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this section, we present the Koopman operator framework, which is key to extend the result of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 to switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We introduce the Koopman semigroup along with its Koopman generator, cast the framework in the context of Lie groups, and describe the finite-dimensional approximation of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider a continuous-time dynamical system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) ˙x = F(x), x ∈ X ⊂ Rn, F ∈ C1 which generates a flow ϕt : X → X, with t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Koopman operator is defined on a (Banach) space F and acts on observables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' functions f : X → R, f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10 (Koopman semigroup [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The semigroup of Koopman opera- tors (in short, Koopman semigroup) is the family of linear operators (Ut)t≥0 defined by Ut : F → F, Utf = f ◦ ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY We can also define the associated Koopman generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11 (Koopman generator [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Koopman generator associated with the vector field (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) is the linear operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) LF : D(LF ) → F, LF f := F · ∇f with the domain D(LF ) = {f ∈ F : F · ∇f ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' As shown below (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13), the Koopman semigroup and the Koopman gen- erators are directly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When the Koopman semigroup is strongly continuous [7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' lim t→0+ ∥Utf − f∥F = 0, the Koopman generator is the infinitesimal generator LF f := lim t→0+(Utf −f)/t of the Koopman semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since the Koopman operator Ut and the generator LF are both linear, we can describe the dynamics of an observable f on F through the linear abstract ordinary differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) ˙f = LF f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We can also briefly discuss the spectral properties of the Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12 (Koopman eigenfunction and eigenvalue [3, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' An eigenfunc- tion of the Koopman operator is an observable φλ ∈ F \\ {0} such that LF φλ = λφλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The value λ ∈ C is the associated Koopman eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Under the strong continuity property, the Koopman eigenfunction also satisfies Utφλ = eλtφλ, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For a linear system ˙x = Ax, with x ∈ Rn, we denote an eigenvalue of A by ˜λj and its associated left eigenvector by wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then ˜λj is a Koopman eigenvalue and the associated Koopman eigenfunction is given by φ˜λj(x) = w† jx [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For a nonlinear system of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) which admits a stable equilibrium xe, the eigenvalues of JF(xe) are typically Koopman eigenvalues and the associated eigenfunctions are the so-called principal Koopman eigenfunctions (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator in the Hardy space H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' From this point on, we define the Koopman operator in the Hardy space on the polydisk (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' [25, 26, 28] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This choice is well-suited to the case of analytic vector fields that admit a stable hyperbolic equilibrium, where it allows to exploit convenient spectral properties of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let D be the open unit disk in C, ∂D its boundary, and Dn the unit polydisk in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Hardy space of holomorphic functions on Dn is the space H2(Dn) = � f : Dn → C, holomorphic : ∥f∥2 = lim r→1− � (∂D)n |f (rω) |2dmn(ω) < ∞ � , where mn is the normalized Lebesgue measure on (∂D)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is equipped with an inner product defined by ⟨f, g⟩ = � (∂D)n f (ω) ¯g (ω) dmn(ω), This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 7 so that the set {zα : α ∈ Nn} is the standard orthonormal basis of monomials on H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The monomials will be denoted by ek(z) = zα(k), where the map α : N → Nn, k �→ α(k) refers to the lexicographic order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ek1 < ek2 if |α(k1)| < |α(k2)|, or if |α(k1)| = |α(k2)| and αj(k1) > αj(k2) for the smallest j such that αj(k1) ̸= αj(k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For f and g in H2(Dn), with f = � k∈N fkek and g = � k∈N gkek, the isomorphism � k∈N fkek �→ (fk)k≥0 between H2(Dn) and the l2-space allows to rewrite the norm and the inner product as ∥f∥2 = � k∈N |fk|2 and ⟨f, g⟩ = � k∈N fk ¯gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We also note that H2(Dn) is a reproducing kernel Hilbert space (RKHS) with the Cauchy kernel ([25, Chapter 1]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) k (z, ξ) = n � i=1 1 1 − ¯ξizi , z, ξ ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that one can define the evaluation functional f(z) = ⟨f, kz⟩ with kz(ω) = k (z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the vector field F is analytic, we can consider its analytic continuation on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, if it generates a holomorphic flow that is invariant in Dn, we can define the Koopman semigroup on H2(Dn), which is also known as the composition operator with symbol ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The required assumptions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The components Fl, l = 1, · · · , n, of the vector field F belong to the Hardy space H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, F generates a flow which is holomorphic and maps Dn to Dn (forward invariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is shown in [4] that the flow ϕt is holomorphic on Dn if and only if the vector field components have a specific form (see Proposition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) in the Appendix, and the works [4, 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note also that this property holds if the dynamics possess a globally stable hyperbolic equilibrium (Assumption 3 below) in the case of non-resonant eigenvalues (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Now, we recall some important properties that we will use to prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider a function f ∈ H2(Dn) and an evaluation functional kz, with z ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' LF zα ∈ H2(Dn) and the domain D (LF ) is dense in H2(Dn), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' U ∗ t kz = kϕt(z), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' d dt ⟨U ∗ t kz, f⟩ = ⟨L∗ F U ∗ t kz, f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all z ∈ Dn, we have LF zα = F(z) · ∇zk = n � l=1 Fl(z) αl z(α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αl−1,αl−1,αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since ∥fzα∥ = ∥f∥ for all f ∈ H2(Dn) and for all α ∈ Nn, it follows from Assumption 2 that ∥LF eα∥ = ����� n � l=1 αlFl ����� ≤ n � l=1 |αl| ∥Fl∥ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover D (LF ) is dense in H2(Dn) since the monomials zα form a complete basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all f ∈ H2(Dn), we have ⟨U ∗ t kz, f⟩ = ⟨kz, Utf⟩ = (Utf) (z) and � kϕt(z), f � = f (ϕt(z)) = (Utf) (z), so that U ∗ t kz = kϕt(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all z ∈ Dn and all f ∈ D(LF ), d dt ⟨U ∗ t kz, f⟩ = d dt � kϕt(z), f � = d dtf ◦ ϕt(z) = F (ϕt(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='∇f (ϕt(z)) = � kϕt(z), LF f � = ⟨L∗ F U ∗ t kz, f⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The result follows for all f since D(LF ) is dense in H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the previous lemma, the second property is a well-known property of the composi- tion operator on a RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The third property is also known in the context of strongly continuous semigroup theory (see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, we make the following additional standing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The vector field F admits on Dn a unique hyperbolic stable equi- librium at 0 (without loss of generality), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' F(0) = 0 and the eigenvalues ˜λj of the Jacobian matrix JF(0) satisfy ℜ{˜λj} < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14 (Holomorphic flow and spectral properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If Assumption 3 holds and if the eigenvalues ˜λj are non-resonant1, then the Poincar´e linearization theorem [2] implies that the flow ϕt is topologically conjugated to the linear flow ˜ϕt(z) = eJF (0)tz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' there exists a bi-holomorphic map h such that ϕt = h−1 ◦ ˜ϕt ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this case, the flow ϕt is clearly holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the components of h are associated with holomorphic Koopman eigenfunctions φ˜λj ∈ H2(Dn) associated with the eigenvalues ˜λj [9, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' These eigenfunctions are called principal eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Also, it can easily be shown that, for all α ∈ Nn, n � j=1 αj˜λj is a Koopman eigenvalue associated with the eigenfunction φα1 ˜λ1 · · · φαn ˜λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman infinite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since H2(Dn) is isomorphic to l2, the Koop- man generator can be represented by the Koopman infinite matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) ¯LF = � � � � � � � � � ⟨LF e0, e0⟩ ⟨LF e0, e1⟩ ⟨LF e0, e2⟩ ⟨LF e0, e3⟩ · · ⟨LF e1, e0⟩ ⟨LF e1, e1⟩ ⟨LF e1, e2⟩ ⟨LF e1, e3⟩ · · ⟨LF e2, e0⟩ ⟨LF e2, e1⟩ ⟨LF e2, e2⟩ ⟨LF e2, e3⟩ · · ⟨LF e3, e0⟩ ⟨LF e3, e1⟩ ⟨LF e3, e2⟩ ⟨LF e3, e3⟩ · · ⟨LF e4, e0⟩ ⟨LF e4, e1⟩ ⟨LF e4, e2⟩ ⟨LF e4, e3⟩ · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' · · � � � � � � � � � , 1The eigenvalues ˜λj are non-resonant if n � j=1 αj ˜λj = 0 with α ∈ Zn implies that α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 9 where the kth row contains the components of LF ek in the basis of monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For f = � k∈N fkek, we also have that ⟨LF f, ej⟩ = � k∈N fk⟨LF ek, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We note that, since LF e0 = 0, the first row and column of ¯LF contains only zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By removing the first row and column, one obtains the representation of the restriction of the Koopman generator to the subspace of functions f that satisfy f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This subspace is spanned by the basis (ek)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that kz − k0 belongs to this subspace, since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) implies that kz(0) − k0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For Fl(z) = � |β|≥1 al,βzβ, the action of the Koopman operator on a basis element is given by LF zα = n � l=1 Fl(z) αl z(α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αl−1,αl−1,αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αn) = n � l=1 � |β|≥1 al,βzβ αl z(α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αl−1,αl−1,αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αn) = n � l=1 αl � � � (β1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βn)∈Nn al,(β1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βn)z(β1+α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βl+αl−1,βl+1+αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βn+αn) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By setting γ1 = β1 + α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , γl = βl + αl − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , γn = βn + αn we obtain LF zα = n � l=1 αl � |γ|≥|α| al,(γ1−α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',γl−αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',γn−αn)z(γ1,γ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',γn) = n � l=1 αl � |γ|≥|α| al,(γ−α)lzγ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12) where we denote (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13) (γ − α)l = (γ1 − α1, · · · , γl − αl + 1, · · · , γn − αn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that the entries of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) ⟨LF ek, ej⟩ = � � � � � n � l=1 αl(k) al,(α(j)−α(k))l if |α(j)| ≥ |α(k)| 0 if |α(j)| < |α(k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the linear part of the vector field F, where |α(j)| = 1, j = 1, · · · , n, it is clear that α(j) is the canonical basis vector of Cn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' αi(j) = δij, and we have that a(l) α(j) = [JF(0)]lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Also, if |α(j)| = |α(k)|, we have that (α(j)−α(k))l = α(r) for some r ≤ n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' |α(r)| = 1), with αr(j) = αr(k) + 1, αl(j) = αl(k) − 1, and This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY αi(j) = αi(k) for all i /∈ {l, r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) ⟨LF ek, ej⟩ = � � � � � � � � � n � l=1 αl(j) [JF(0)]ll if j = k αl(k) [JF(0)]lr if α(j) = (α1(k), · · · , αl(k) − 1, · · · , αr(k) + 1, · · · , αn(k)), 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Switched Koopman systems and Lie-algebraic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case of a switched nonlinear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1), the Koopman operator description yields a switched linear infinite-dimensional system (in short, switched Koopman system) of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) � ˙f = LF (i)f, f ∈ D �m i=1 with D = ∩m i=1D(LF (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Similarly, the Lie algebra gF spanned by F (i) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5)) is replaced by gL = span {LF (i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m}Lie, equipped with the Lie bracket [LF (i), LF (j)] = LF (i)LF (j) − LF (j)LF (i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, we have the well-known relationship (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) [LF (i), LF (j)] = L[F (i),F (j)] so that the two algebras gF and gL are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that Lie-algebraic conditions in gF can be recast into Lie-algebraic criteria in gL, a framework where we can expect to obtain new results on switched systems that are reminiscent to the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, since the solvability property of gF is equivalent to the solvability property of gL, we will investigate whether this latter condition implies the existence of a common Lyapunov functional for the switched Koopman system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This section presents our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We first use an illus- trative example to show that Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 cannot be used for nonlinear vector fields, in contrast to the linear case (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We then relax the algebraic conditions suggested in [13] in order to obtain a triangular form in the Koopman matrix representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11), a property which is equivalent to the existence of an in- variant flag for the adjoint operator L∗ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We finally prove uniform stability of switched nonlinear systems under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A first remark on the existence of the common invariant flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The following example shows that Lie’s theorem does not hold for infinite-dimensional switched Koopman systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider the two vector fields F (1)(x1, x2) = (−αx1, −αx2) and F (2)(x1, x2) = (−βx1+γ � x2 1 − x2 2 � , −βx2+2γx1x2), where α, β and γ are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' These two vector fields generate the Lie alge- bra g = span � F (1), F (2), F (3)� Lie with F (3)(x1, x2) = (αγ(x2 1 − x2 2), 2αγx1x2) since [F (1), F (2)] = F (3), [F (1), F (3)] = αF (3) and [F (2), F (3)] = βF (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, one has g1 = [g, g] = span � F (3)� Lie and g2 = [g1, g1] = 0, which implies that g is a This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 11 solvable Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, the Koopman generators LF (1) and LF (2) associated with the two vector fields do not share a common eigenfunction, and therefore can- not have a common invariant flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, the principal eigenfunctions of LF (1) are φ˜λ(1) 1 (x1, x2) = x1 and φ˜λ(1) 2 (x1, x2) = x2, while those of LF (2) are given by φ˜λ(2) 1 (x1, x2) = βγ � βx1 − γ � x2 1 − x2 2 �� (β − γx1)2 + γ2x2 2 and φ˜λ(2) 2 (x1, x2) = β2γx2 (β − γx1)2 + γ2x2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We conclude that Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 does not hold setting for the above example, so that we cannot directly extend Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The two Koopman generators are not simultaneous triangularizable and do not have a common invariant flag (see [10] for more details about simultaneous triangularization of operators and its connection to the existence of an invariant infinite maximal flag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, it can be easily seen that the Koopman infinite matrices (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) related to the vector fields F (1) and F (2) are both upper triangular, and therefore admit a common infinite invariant maximal flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In fact, this implies that the adjoint operators L∗ F (i) have a common invariant flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For this reason, we will depart from the solvability condition on vector fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' on Koopman generators), and we will deal with simultaneous triangularization of adjoints of Koopman generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The following result provides a sufficient condition on the vector fields for the simultaneous triangularization of ad- joints of Koopman generators, which appears to be less restrictive than the solvability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let F be an analytic vector field on Dn such that the Jacobian matrix JF(0) is upper triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then the Koopman matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) is upper triangular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ⟨LF ek, ej⟩ = 0 for all k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the adjoint L∗ F of the Koopman generator admits an infinite invariant maximal flag generated by the monomials ek, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Sk = span{e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , ek}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that ⟨LF ek, ej⟩ = 0 if |α(k)| > |α(j)| (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' the Koop- man matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) is always upper triangular by matrix blocks related to monomials of the same total degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case |α(k)| = |α(j)| with k > j, the lexicographic order implies that one can have α(j) = (α1(k), · · · , αl(k)−1, · · · , αr(k)+1, · · · , αn(k)) only with r < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since [JF(0)]lr = 0 for all l > r, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) that ⟨LF ek, ej⟩ = 0 when k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, it is clear that L∗ F ej ∈ span{e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , ej} since ⟨ek, L∗ F ej⟩ = 0 for all k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When the Jacobian matrix is upper triangular, it is well-known that [JF(0)]jj = ˜λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this case, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) that the diagonal entries of the (upper triangular) Koopman matrix are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) ⟨LF ej, ej⟩ = n � l=1 αl(j)˜λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since these values are the Koopman eigenvalues in the case of non-resonant eigenvalues ˜λj (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14), we will denote λj = ⟨LF ej, ej⟩ by a slight abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let � F (i)�m i=1 be a switched nonlinear system on Dn and sup- pose that the Lie algebra of matrices span � JF (i)(0) � Lie is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then there ex- ists a change of variables z �→ �z = P −1z on Cn such that the adjoint operators L∗ � F (i) of the Koopman generators (with �F (i)(�z) = P −1F (i)(P �z)) admit a common in- This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY finite invariant maximal flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, � �F (i)�m i=1 is a switched nonlinear system on Dn � 0, ∥P −1∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since span � JF (i)(0) � Lie is solvable, Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 implies that the matrices JF (i)(0) are simultaneously triangularizable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' there exists a matrix P such that JF (i)(0) = PT (i)P −1 for all i, where T (i) is upper triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let set F (i)(z) = JF (i)(0)z + ˜F (i)(z) to separate the linear and the nonlinear parts of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the new coordinates �z = P −1z, we obtain the dynamics �F (i)(�z) = P −1JF (i)(0)P �z + P −1 ˜Fi(P �z) = T (i)�z + �˜F i(�z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 that monomials �ek, with �ek(�z) = (�z)α(k), generate a common invariant maximal flag for L∗ � F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In addition, for all z ∈ Dn and all j = 1, · · · , n, we have |�zj| ≤ ∥�z∥∞ = ��P −1z �� ∞ ≤ ��P −1�� ∞ ∥z∥∞ < ∥P −1∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is clear that the change of coordinates z �→ �z = P −1z is defined up to a multiplicative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Without loss of generality, we will consider in the sequel that ∥P −1∥∞ = 1, so that � �F (i)�m i=1 is a switched nonlinear system on the unit polydisk Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Instead of a nilpotency or solvability condition on the vector fields F (i), we only require a milder solvability condition on the Jacobian matrices JF (i)(xe) to guarantee the triangular form of the Koopman matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is noticeable that this local condition is much less restrictive than the global solvability condition mentioned in the original open problem [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Also, it was shown in [1] that the triangular form of the vector fields (and therefore of the Jacobian matrices) is not sufficient to guarantee the GUAS property of a switched nonlinear system on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the next section, however, we use the solvability condition on the Jacobian matrices to prove the GUAS property in a bounded invariant region of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This result is consistent with the local stability result derived in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A common Lyapunov function for switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We now aim to show that, for some positive sequence (ϵk)∞ k=1, the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) V(f) = ∞ � k=1 ϵk |⟨f, ek⟩|2 is a Lyapunov functional for the switched Koopman system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Before starting our main result, we need a few lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let ˙z = F(z) be a vector field on the polydisk Dn which generates a flow ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that there exist a sequence of positive numbers (ϵk)k≥1 and ρ ∈]0, 1] such that Dn(0, ρ) is forward invariant with respect to ϕt and such that the series � k≥1 |α(k)|ϵkρ2|α(k)| is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) V(kz − k0) = ∞ � k=1 ϵk |⟨kz, ek⟩|2 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 13 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) ∞ � k=1 ϵk d dt |⟨U ∗ t (kz − k0), ek⟩|2 are absolutely and uniformly convergent on Dn(0, ρ) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the first series, we have ϵk |⟨kz − k0, ek⟩|2 = ϵk |⟨kz, ek⟩|2 = ϵk ���zα(k)��� 2 < ϵkρ2|α(k)| ≤ |α(k)|ϵkρ2|α(k)| for all z ∈ Dn(0, ρ) and all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the second series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' we have ϵk ���� d dt |⟨U ∗ t (kz − k0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ek⟩|2 ���� = ϵk ���� d dt ���(ϕt(z))α(k) − (ϕt(0))α(k)��� 2���� = 2ϵk ����ℜ � d dt (ϕt(z))α(k) · � ϕt(z) �α(k)����� ≤ 2ϵk ���� d dt (ϕt(z))α(k) ���� · ���� � ϕt(z) �α(k)���� < 2ϵkρ|α(k)| ���� d dt (ϕt(z))α(k) ���� ≤ 2ϵkρ|α(k)| n � s=1 αs(k) ���F (s) (ϕt(z)) ��� ��(ϕt(z))s ��αs(k)−1 n � l=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='l̸=s ��(ϕt(z))l ��αl(k) < 2ϵkρ2|α(k)|−1 n � s=1 αs(k) ���F (s) (ϕt(z)) ��� for all z ∈ Dn(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' t > 0 and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By using the maximum modulus principle for bounded domains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2 with the holomorphic function F (s) ◦ ϕt, we can denote M = max z∈∂Dn,s=1,··· ,n |(F (s) ◦ ϕt)(z)| and we obtain ϵk ���� d dt |⟨U ∗ t (kz − k0), ek⟩|2 ���� < 2M|α(k)|ϵkρ2|α(k)|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, absolute and uniform convergence of both series follow from the Weierstrass test (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let ˙z = F(z) be a nonlinear system on Dn with an upper triangular Jacobian matrix JF(0) and let ˙f = LF f be its corresponding Koopman system on D (LF ) ⊂ H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) are absolutely and uniformly convergent in Dn for all t > 0, then, for all double sequences of positive real numbers (bjk)j≥1,k≥1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5) ∞ � k=1 bjk ≤ 1, This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY one has d dtV (U ∗ t (kz − k0)) ≤2 ∞ � j=1 bjjϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 � bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6) with cj = ⟨U ∗ t (kz − k0), ej⟩ and λj = ⟨LF ej, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that z ∈ Dn is such that the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) are absolutely and uniformly convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13 (3), we obtain d dtϵk |⟨U ∗ t (kz − k0), ek⟩|2 = 2ϵkℜ �� d dtU ∗ t (kz − k0), ek � ⟨U ∗ t (kz − k0), ek⟩ � = 2ϵkℜ � ⟨L∗ F U ∗ t (kz − k0), ek⟩ ⟨U ∗ t (kz − k0), ek⟩ � = 2ϵk ∞ � j=1 ℜ (cj¯ck ⟨ej, LF ek⟩) where we used the decomposition U ∗ t (kz − k0) = ∞ � j=1 cjej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) is absolutely and uniformly convergent, term by term derivation yields d dtV (U ∗ t (kz − k0)) = ∞ � k=1 d dtϵk |⟨U ∗ t (kz − k0), ek⟩|2 = 2 ∞ � k=1 ∞ � j=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) = 2 ∞ � j=1 ϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) where we used the triangular form of ¯LF (which follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 since JF(0) is triangular) and λj = ⟨LF ej, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5), we have d dtV (U ∗ t (kz − k0)) ≤ 2 ∞ � j=1 � � j � k=1 bjk + ∞ � k=j+1 bjk � � ϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) = 2 ∞ � j=1 bjjϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 bjkϵj |cj|2 ℜ (λj) + 2 ∞ � j=1 ∞ � k=j+1 bjkϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) = 2 ∞ � j=1 bjjϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 � bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 15 Under Assumption 3, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) that ℜ{λj} < 0 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Therefore, the time derivative (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6) of the Lyapunov functional is negative if negative terms related to the diagonal entries ⟨LF ej, ej⟩ = λj and ⟨LF ek, ek⟩ = λk compensate (possibly positive) cross-terms related to ⟨ej, LF ek⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We note that a term associated with a diagonal entry will be used to compensate an infinity of cross-terms (associated with entries in the corresponding row and column of the Koopman matrix), and the values bjk play the role of weights in the compensation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We are now in position to state our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) � ˙z = F (i)(z) �m i=1 be a switched nonlinear system on Dn and assume that all subsystems of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) have a common hyperbolic equilibrium ze = 0 that is globally asymptotically stable on the polydisk Dn, the Lie algebra span � JF (i)(0) � Lie is solvable (and therefore there exists a matrix P such that P −1JF (i)(0)P are upper triangular), there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to the flows ϕt i of F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider double sequences of positive real numbers � b(i) jk � j≥1,k≥1, with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m, such that b(i) jk b(i) kj > 0 if ⟨L � F (i)�ek, �ej⟩ ̸= 0 (where �ej(�z) = �zα(j) are monomials in the new coordinates �z = P −1z) and such that ∞ � k=1 b(i) jk ≤ 1, and define the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) � � �Q(i) jk def = � � � � � ��� L � F (i)�ek, �ej ���2 4 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� 1 b(i) jk b(i) kj if � L � F (i)�ek, �ej � ̸= 0 0 otherwise � � � j≥2,1≤k≤j−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) +∞ � k=1 |α(k)| ϵk ρ2|α(k)| is convergent with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) ϵj > max i=1,··· ,m k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 ϵk Q(i) jk , then the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover the series V (z) = ∞ � k=1 ϵk ��� � P −1z �α(k)��� 2 is a common global Lyapunov function on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) � ˙�z = �F (i)(�z) �m i=1 defined on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3, the monomials �ek(�z) = (�z)α(k) generate a common infinite invariant maximal flag for ¯L � F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We first show that the candidate Lyapunov functional �V(f) = ∞ � k=1 ϵk |⟨f, �ek⟩|2 satisfies d dt �V � (�U (i) t )∗ (k�z − k0) � < 0 for all i = 1, · · · , m, where �U (i) t denotes the Koopman semigroup associated with the subsystem ˙�z = �F (i)(�z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) implies that the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) are absolutely convergent on Dn (0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6 that d dt �V � (�U (i) t )∗ (k�z − k0) � ≤2 ∞ � j=1 b(i) jj ϵj ���c(i) j ��� 2 ℜ � λ(i) j � + 2 ∞ � j=2 j−1 � k=1 b(i) jk ϵj ���c(i) j ��� 2 ℜ � λ(i) j � + b(i) kj ϵk ���c(i) k ��� 2 ℜ � λ(i) k � + ϵkℜ � c(i) j ¯c(i) k � �ej, L � F (i)�ek �� where c(i) j = � (�U (i) t )∗ (k�z − k0) , �ej � and λ(i) j = � L � F (i)�ej, �ej � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since ℜ{λ(i) j } < 0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1)), one has to find a sequence of positive numbers (ϵj)j≥1 such that b(i) jk ϵj ���c(i) j ��� 2 ���ℜ � λ(i) j ���� + b(i) kj ϵk ���c(i) k ��� 2 ���ℜ � λ(i) k ���� > ϵk ���ℜ � c(i) j ¯c(i) k � �ej, L � F (i)�ek ����� for all i = 1, · · · , m and for all j, k with j > k such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12) � �ej, L � F (i)�ek � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By using the inequality ���ℜ � c(i) j ¯c(i) k � �ej, L � F (i)�ek ����� ≤ ���c(i) j ��� ���c(i) k ��� ��� �ej, L � F (i)�ek ��� , one has to satisfy b(i) jk ϵj ���c(i) j ��� 2 ���ℜ � λ(i) j ���� + b(i) kj ϵk ���c(i) k ��� 2 ���ℜ � λ(i) k ���� > ϵk ���c(i) j ��� ���c(i) k ��� ��� L � F (i)�ek, �ej ��� or equivalently (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13) ϵj > ϵk � �− b(i) kj b(i) jk ���ℜ � λ(i) k ���� ���ℜ � λ(i) j ���� ����� c(i) k c(i) j ����� 2 + ��� L � F (i)�ek, �ej ��� b(i) jk ���ℜ � λ(i) j ���� ����� c(i) k c(i) j ����� � � def = ϵk h ������ c(i) k c(i) j ����� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is easy to see that the real quadratic function h has the maximal value Q(i) jk = ��� L � F (i)�ek, �ej ���2 4 ���ℜ � λ(i) j ���� ���ℜ � λ(i) k ���� 1 b(i) jk b(i) kj This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 17 so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13) is satisfied if we choose iteratively ϵj according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that we have d dt �V � (�U (i) t )∗ (k�z − k0) � < 2 ∞ � j=1 b(i) jj ϵj ���c(i) j ��� 2 ℜ � λ(i) j � < − min j � b(i) jj ���ℜ � λ(i) j ���� � ∞ � j=1 ϵj ���c(i) j ��� 2 = − min j � b(i) jj ���ℜ � λ(i) j ���� � �V � (�U (i) t )∗ (k�z − k0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' With the evaluation functional k�z, we can define �V : Dn (0, ρ) → R+, �V (�z) = �V (k�z − k0) and, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13, we verify that �V � �ϕ(i) t (�z) � = �V � k�ϕ(i) t (�z) − k0 � = �V � k�ϕ(i) t (�z) − k�ϕ(i) t (0) � = �V � (�U (i) t )∗ (k�z − k0) � < �V (k�z − k0) = �V (�z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In addition, if we define V = �V ◦ P −1 : Dn (0, ρ) → R+, we have V � ϕ(i) t (z) � = �V � P −1ϕ(i) t (P �z) � = �V � �ϕ(i) t (�z � < �V (�z) = V (P �z) = V (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Therefore, we have the CLF (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) V (z) = ∞ � k=1 ϵk |⟨k�z − k0, �ek⟩|2 = ∞ � k=1 ϵk |⟨k�z, �ek⟩|2 = ∞ � k=1 ϵk ��� � P −1z �α(k)��� 2 for the switched nonlinear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, since Dn (0, ρ) is forward invariant with respect to ϕ(i) t , the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that, if the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 are satisfied but the polydisk Dn(0, ρ) is not forward invariant with respect to the flow generated by the subsys- tems, then the switched system is GUAS in the largest sublevel set of the Lyapunov function that is contained in Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The condition on the boundedness of the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) could be inter- preted as the dominance of diagonal entries of the matrix ¯L � F (i) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=', the Koopman eigenvalues (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1)) with respect to the other entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the number of non- zero cross-terms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12) to be compensated affects the way we define the sequence of weights b(i) jk and therefore the sequence ϵj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) has an upper bound Q < 1, one can set ϵk = Q for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, such case rarely appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Instead, if Q > 1, one might have ϵk = O(Qk) and it is clear that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) diverges for all ρ since k − 2|α(k)| → ∞ as k → ∞ (except in the case n = 1 where |α(k)| = k, see also Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the following, we will consider specific vector fields such that the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) converges for a proper choice of sequence b(i) jk , so that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For polynomial vector fields of the form F (i) l (z) = r � k=1 a(i) l,kzα(k), we denote by K(i) the number of nonzero terms (without counting the monomial zl in F (i) l ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) K(i) = m � l=1 # � k ̸= l : a(i) l,k ̸= 0 � This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY where # is the cardinal of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this case, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) � ˙z = F (i)(z) �m i=1 be a switched nonlinear system on Dn, where F (i) are polynomial vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assume that all subsystems of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) have a common hyperbolic equilibrium ze = 0 that is globally asymptotically stable on Dn, the Lie algebra span � JF (i)(0) � Lie is solvable (and therefore there exists a matrix P such that PJF (i)(0)P −1 are upper triangular), the unit polydisk Dn is forward invariant with respect to the flows ϕ(i) t gener- ated by F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 ( �K(i))2 ��� L � F (i)�ek, �ej ���2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� < 1, where �K(i) is the number of nonzero terms of �F (i)(�z) = P −1F (i)(P �z) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15)) and where �ej(�z) = �zα(j) are the monomials in the new coordinates �z = P −1z, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) is GUAS on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The result follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 with the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='18) � � � � � � � b(i) jj = (1 − ξ) b(i) jk = ξ 2K(i) if j ̸= k with � L � F (i)�ek, �ej � ̸= 0 or � L � F (i)�ej, �ek � ̸= 0 b(i) jk = 0, if j ̸= k with � L � F (i)�ek, �ej � = 0 or � L � F (i)�ej, �ek � = 0, with ξ ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is clear from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that, for a fixed j and for all k ∈ N \\ {j}, there are at most K(i) nonzero values ⟨L � F (i)�ek, �ej⟩ and at most K(i) nonzero values ⟨L � F (i)�ej, �ek⟩, so that the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='18) satisfies ∞ � k=1 b(i) jk ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The elements Q(i) jk of the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='19) Q(i) jk = ( �K(i))2 ��� L � F (i)�ek, �ej ���2 ξ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) implies that max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 Q(i) jk def = Q < 1 for some ξ ∈]0, 1[, so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) is satisfied with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='20) ϵj ∼ max k∈Kj {ϵk Q} for j ≫ 1, with Kj = {k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , j − 1} : � L � F (i)�ek, �ej � ̸= 0 for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='20) yields ϵj = O(Qj) for j ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) is convergent for any ρ ≤ 1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 implies that the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) is GUAS on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 19 Another result is obtained when a diagonal dominance property is assumed for the Jacobian matrices JF (i)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) � ˙z = F (i)(z) �m i=1 be a switched nonlinear system on Dn, with F (i) l (z) = +∞ � k=1 a(i) l,kzα(k) and +∞ � k=1 |a(i) l,k| < ∞ for all i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m} and l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assume that all subsystems of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) have a common hyperbolic equilibrium ze = 0 that is globally asymptotically stable on Dn, the Lie algebra span � JF (i)(0) � Lie is solvable (and therefore there exists a matrix P such that PJF (i)(0)P −1 are upper triangular), there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to the flows ϕ(i) t generated by F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If there exist ξ ∈]0, 1[ and κ ∈]0, 1[ with ξ + κ < 1 such that, for all q, r ∈ {1, · · · , n} with q < r (when n > 1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) ���J �F (i)(0)]qr ��� 2 < � 2ξ n2 − n �2 ���ℜ([J �F (i)(0)]rr) ��� ���ℜ([J �F (i)(0)]qq) ��� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) ���[J �F (i)(0)]qr ��� < 2ξ n2 − n ���ℜ([J �F (i)(0)]qq) ��� and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 ⟨L � F (i) �ek,�ej⟩̸=0 �∞ l=1 ��� L � F (i)�el, �ej ��� �∞ l=1 ��� L � F (i)�ek, �el ��� κ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� < 1 ρ2 , where �ej(�z) = �zα(j) are monomials in the new coordinates �z = P −1z, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We will denote by D def = (n2 − n)/2 the number of upper off-diagonal entries of the Jacobian matrices J �F (i)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The result follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 with the sequence � � � � � � � � � � � � � � � � � � � � � � � � � b(i) jj = (1 − ξ − κ) b(i) jk = ξ 2D if j ̸= k with |α(j)| = |α(k)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' and if � L � F (i)�ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej � ̸= 0 or � L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ek � ̸= 0 b(i) jk = 0 if |α(j)| = |α(k)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' � L � F (i)�ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej � = 0 and � L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ek � = 0 b(i) jk = κ 2 ��� L � F (i)�ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej ��� �∞ l=1 ��� L � F (i)�el,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej ��� if |α(k)| < |α(j)| b(i) jk = κ 2 ��� L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ek ��� �∞ l=1 ��� L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �el ��� if |α(k)| > |α(j)| with ξ ∈]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1[ and κ ∈]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16 and the fact that the Jacobian matrices J �F (i)(0) are upper triangular that, for a fixed j and all k ̸= j This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY with |α(k)| = |α(j)|, there are at most D nonzero values � L � F (i)�ek, �ej � and at most D nonzero values � L � F (i)�ej, �ek � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Therefore, the sequence b(i) jk satisfies ∞ � k=1 b(i) jk < (1 − ξ − κ) + ξ + κ 2 �j k=1 ��� L � F (i)�ek, �ej ��� �∞ l=1 ��� L � F (i)�el, �ej ��� + κ 2 �∞ k=j+1 ��� L � F (i)�ej, �ek ��� �∞ l=1 ��� L � F (i)�ej, �el ��� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The elements Q(i) jk of the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='25) Q(i) jk = � � � � � � � � � � � � � D2 ��� L � F (i)�ek, �ej ���2 ξ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� if |α(j)| = |α(k)| �∞ l=1 ��� L � F (i)�el, �ej ��� �∞ l=1 ��� L � F (i)�ek, �el ��� κ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� if |α(k)| ̸= |α(j)| and � L � F (i)�ek, �ej � ̸= 0 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We note that ∞ � l=1 ��� L � F (i)�el, �ej ��� and ∞ � l=1 ��� L � F (i)�ek, �el ��� are finite according to the as- sumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Next, we show that the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) imply that Q(i) jk < 1 if |α(j)| = |α(k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='25) that this latter inequality is equivalent to α2 q(k)|[J �F (i)(0)]qr|2 < ξ2 D2 ����� n � l=1 αl(j)ℜ([J �F (i)(0)]ll) ����� ����� n � l=1 αl(k)ℜ([J �F (i)(0)]ll) ����� for all j > k such that α(j) = (α1(k), · · · , αq(k)−1, · · · , αr(k)+1, · · · , αn(k)) for some q < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since the diagonal entries of the (upper-triangular) Jacobian matrices J �F (i)(0) are the eigenvalues and therefore have negative real parts, the most restrictive case is obtained with αl(k) = 0 for all l ̸= q, which yields α2 q(k)|[J �F (i)(0)]qr|2 < ξ2 D2 ���(αq(k) − 1)ℜ([J �F (i)(0)]qq) + ℜ([J �F (i)(0)]rr) ��� ���αq(k)ℜ([J �F (i)(0)]qq) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When αq(k) = 1, this inequality is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When αq(k) > 1, we can rewrite (αq(k) − 1)|[J �F (i)(0)]qr|2 + |[J �F (i)(0)]qr|2 < ξ2 D2 � (αq(k) − 1) ���ℜ([J �F (i)(0)]qq) ��� 2 + ���ℜ([J �F (i)(0)]rr) ��� ���ℜ([J �F (i)(0)]qq) ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22), we have that the above inequality is satisfied if (αq(k) − 1)|[J �F (i)(0)]qr|2 < ξ2 D2 (αq(k) − 1) ���ℜ([J �F (i)(0)]qq) ��� 2 , which is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' While Q(i) jk < 1 for |α(j)| = |α(k)|, it is easy to see that Q(i) jk > 1 for |α(j)| > |α(k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) therefore implies that max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 Q(i) jk def = Q < 1/ρ2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) is satisfied with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='26) ϵj ∼ max k∈Kj {ϵk Q} This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 21 for j ≫ 1, with Kj = {k ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , j − 1 : � L � F (i)�ek, �ej � ̸= 0 for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m} and |α(k)| < |α(j)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Hence, the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='26) yields ϵj = O(Q|α(j)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) is convergent and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 implies that the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the particular case where the Jacobian matrices JF (i)(0) are simultaneously diagonalizable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' they are diagonalizable and they commute), the diagonal dom- inance conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) are trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We should mention that the Lie-algebraic property of commutation is only needed for the Jacobian matrices JF (i)(0), an assumption which contrasts with the commutation property imposed on vector fields in [17], [30] and [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case n = 1, we recover the trivial GUAS property of switched systems from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, consider the vector fields F (i)(z) = ∞ � k=1 a(i) k zk on D, with ∞ � k=1 |a(i) k | < ∞, and assume that the subsystems have a globally stable equilibrium at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Lie-algebra generated by the scalars JF (i)(0) is trivially solvable and D(0, ρ) is forward invariant for all ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) are trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 implies that the switched system is GUAS on D(0, ρ) for ρ ∈]0, 1[ which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that ∞ � l=1 |⟨LF (i)el, ej⟩| = j � l=1 l|a(i) j−l+1| = j � l=1 (j − l + 1)|a(i) l |, ∞ � l=1 |⟨LF (i)ek, el⟩| = k ∞ � l=1 |a(i) l |, and |ℜ (⟨ej, LF (i)ej⟩)| = j ���ℜ(a(i) 1 ) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' With κ arbitrarily close to 1 (since ξ can be taken arbitrarily small in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23)), condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) is rewritten as max i=1,··· ,m lim sup j∈N �j l=1(j − l + 1)|a(i) l | �∞ l=1 |a(i) l | j ���ℜ(a(i) 1 ) ��� 2 < 1 ρ2 and, using (j − l + 1)|a(i) l | ≤ j|a(i) l | for all l, we obtain ρ < min i=1,··· ,m �∞ l=1 |a(i) l | ���ℜ(a(i) 1 ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This section presents two examples that illustrate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We will focus on specific cases that satisfy the assumptions of Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 and, without loss of generality, we will directly consider Jacobian matrices in triangular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Example 1: polynomial vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Similarly to Example 1, we con- sider the vector fields on the bidisk D2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) F (1)(z1, z2) = � −az1 −az2 and F (2)(z1, z2) = � � � −az1 + b � z2 1 − z1z2 2 � −az2 + b 2z1z2, This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY where b > 0 and a > 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all ρ < 1, the bidisk D2(0, ρ) is invariant with respect to the flows of F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, for all z ∈ ∂D2(0, ρ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' |zl| = ρ for some l), one has to verify that ℜ � F (i) l (z) ¯zl � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We have |zl| = ρ ⇒ ℜ � F (1) l (z)¯zl � = −aρ2 < 0, |z1| = ρ ⇒ ℜ � F (2) 1 (z)¯z1 � = −aρ2 + bρ2ℜ � z1 − z2 2 � < 0, |z2| = ρ ⇒ ℜ � F (2) 2 (z)¯z2 � = ρ2 � −a + b 2ℜ (z1) � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is clear that vector field F (1) generates a holomorphic flow on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The same property holds for F (2) since the conditions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 are satisfied with h1 (z′ 1) = h2 (z′ 2) = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ℜ{G1(z)} = ℜ{−a + b � z1 − z2 2 � } < 0 and ℜ{G2(z)} = ℜ{−a + b 2z1} < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The unique global stable equilibrium of the subsystems in the bidisk D2 is the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14), the entries of the Koopman matrices ¯LF (1) and ¯LF (2) are given by ⟨LF (1)ek, ej⟩ = � −a |α(j)| if k = j 0 otherwise and ⟨LF (2)ek, ej⟩ = � � � � � � � � � � � � � −a |α(j)| if k = j b � α1(j) + α2(j) − 2 2 � if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j) −b α1(j) if α1(k) = α1(j) and α2(k) = α2(j) − 2 ≥ 0 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since ⟨LF (1)ek, ej⟩=0 for all k ̸= j, we have Q(1) jk = 0 for all k, j in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, K(2) = 3 and the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) can be rewritten as Q = lim sup j∈N |α(j)|>1 max � � � � � 9b2 a2 � α1(j) + α2(j)−2 2 �2 |α(j)| (|α(j)| − 1) , 9b2 a2 (α1(j))2 |α(j)| (|α(j)| − 2) � � � � � = 9b2 a2 < 1 and is satisfied since a > 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Hence, it follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 that the switched system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) is GUAS in D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that, in this case, a CLF is given by V (z) = ∞ � k=1 Q−k ���zα(k)��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Example 2: analytic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The following example is taken from [1] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider the switched system defined by the vector fields F (1)(x1, x2) = � −x1 + 1 µ sin2(x1)x2 1x2, −x2 � F (2)(x1, x2) = � −x1 + 1 µ cos2(x1)x2 1x2, −x2 � , where µ ≥ 12/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Both subsystems are globally asymptotically stable, but the switched system is not GUAS in R2, as shown in [1] by using the fact that the convex com- bination F = � F (1) + F (2)� /2 of the two subsystems is not globally asymptotically This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 23 stable in R2 (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Yet, our result allows to infer the GUAS property in a specific region of the invariant real square ] − 1, 1[2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' To do so, we complexify the dynamics and define the vector fields on the bidisk D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all ρ < 1, the bidisk D2(0, ρ) is invariant with respect to the flows of F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, we have |z1| = ρ ⇒ ℜ � F (1) 1 (z)¯z1 � = ρ2 � −1 + 1 µℜ � z1z2 sin2(z1) �� < 0 since 1 µ ��ℜ � z1z2 sin2(z1) ��� ≤ ρ2 ��sin2(z1) �� µ < 1 where we used max z1∈D ��sin2(z1) �� < 12/5 ≤ µ, |z2| = ρ ⇒ ℜ � F (1) 2 (z)¯z2 � = −ρ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The same result follows for F (2) (with max z1∈D ��cos2(z1) �� < 12/5 ≤ µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The vector field F (1) generates a holomorphic flow on D2 since the conditions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 are satisfied with h1 (z′ 1) = h2 (z′ 2) = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ℜ{G1(z)} = ℜ{−1 + 1 µ sin2(z1)z1z2} < 0 and ℜ{G2(z)} = −1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The same result holds for F (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Taylor expansion of the vector fields yields F (1)(z) = � −z1 + 1 µ ∞ � p=1 (−1)p+122p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' z2p+2 1 z2, −z2 � F (2)(z) = � −z1 + 1 µz2 1z2 + 1 µ ∞ � p=1 (−1)p22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' z2p+2 1 z2, −z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14), the entries of the Koopman matrices ¯LF (1) and ¯LF (2) are given by ⟨LF (1)ek, ej⟩ = � � � � � � � � � − |α(k)| if k = j α1(k) µ (−1)p+122p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 0 otherwise and ⟨LF (2)ek, ej⟩ = � � � � � � � � � � � � � � � − |α(k)| if k = j α1(k) µ if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 α1(k) µ (−1)p22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 0 otherwise This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY where p = 1, · · · , �α1(j) − 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This implies that we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) ∞ � l=1 |⟨LF (1)el, ej⟩| = |α(j)| + 1 µ ⌊ α1(j)−1 2 ⌋ � l=1 (α1(j) − 1 − 2l)22l−1 (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ∞ � l=1 |⟨LF (2)el, ej⟩| = |α(j)| + α1(j) − 1 µ + 1 µ ⌊ α1(j)−1 2 ⌋ � l=1 (α1(j) − 1 − 2l)22l−1 (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ∞ � l=1 |⟨LF (1)ek, el⟩| = |α(k)| + α1(k) µ ∞ � p=1 22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' = |α(k)| + α1(k) 2µ (cosh(2) − 1) ∞ � l=1 |⟨LF (2)ek, el⟩| = |α(k)| + α1(k) µ � 1 + ∞ � p=1 22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' � = |α(k)| + α1(k) 2µ (cosh(2) + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since the Jacobian matrices JF (i)(0) are diagonal, the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) are trivially satisfied (with ξ arbitrarily small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, we observe from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) that ∞ � l=1 |⟨LF (1)el, ej⟩| ≤ ∞ � l=1 |⟨LF (2)el, ej⟩| and ∞ � l=1 |⟨LF (1)ek, el⟩| ≤ ∞ � l=1 |⟨LF (2)ek, el⟩| for all k, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that, with κ arbitrarily close to 1, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) can be rewritten as lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 �∞ l=1 |⟨LF (2)el, ej⟩| �∞ l=1 |⟨LF (2)ek, el⟩| |ℜ (⟨LF (2)ej, ej⟩)| |ℜ (⟨LF (2)ek, ek⟩)| < 1 ρ2 , which is verified for ρ = � 1 + 1 2µ (cosh(2) + 1) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2), we have ∞ � l=1 |⟨LF (2)el, ej⟩| < |α(j)| + α1(j) µ � 1 + ∞ � l=1 22l−1 (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' � = |α(j)| + α1(j) 2µ (cosh(2) + 1) ≤ |α(j)| � 1 + 1 2µ (cosh(2) + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' and ∞ � l=1 |⟨LF (2)ek, el⟩| ≤ |α(k)| � 1 + 1 2µ (cosh(2) + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 that the switched system is GUAS on D2(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' See Figure 1 for the different values of ρ depending on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that a CLF is given by V (z) = ∞ � k=1 Q−2|α(k)| ���zα(k)��� 2 where Q = 1/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Conclusion and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This paper provides new advances on the uniform stability problem for switched nonlinear systems satisfying Lie-algebraic solv- ability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' First, we have shown that the solvability condition on nonlinear This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The switched system is shown to be GUAS on a polydisk of radius ρ that depends on the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' vector fields does not guarantee the existence of a common invariant flag and, instead, we have imposed the solvability condition only on the linear part of the vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then we have constructed a common Lyapunov functional for an equivalent infinite- dimensional switched linear system obtained with the adjoint of the Koopman genera- tor on the Hardy space of the polydisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally we have derived a common Lyapunov function via evaluation functionals to prove that specific switched nonlinear systems are uniformly globally asymptotically stable on invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our results heavily rely on the Koopman operator framework, which appears to be a valid tool to tackle theoretical questions from a novel angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We envision several perspectives for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our results apply to specific types of switched nonlinear systems within the frame of Lie-algebraic solvability con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' They could be extended to more general dynamics, including dynamics that possess a limit cycle or a general attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the same line, the Koopman operator- based techniques developed in this paper could be applied to other types of stability than uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' More importantly, the obtained stability results are limited to bounded invariant sets, mainly due to the convergence properties of the Lyapunov functions and the very definition of the Hardy space on the polydisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We envision that these results could possibly be adapted to infer global stability in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, our results are not restricted to switched systems and have direct implications in the global stability properties of nonlinear dynamical systems, which will be investigated in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' General theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We recall here some general results that are used in the proofs of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let F : Dn → Cn be holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then F is an infinitesimal generator on Dn if and only if, for all l = 1, · · · , n and for all z ∈ Dn, Fl(z) = Gl(z) (zl − hl (z′ l)) where z′ l = (z1, · · · , zl−1, zl+1, · · · zn), hl : Dn−1 → D is holomorphic, Gl : Dn → C is holomorphic, and ℜ ((1 − hl (z′ l) ¯zl) Gl(z)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2 (Maximum Modulus Principle for bounded domains [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let Dn ⊂ Cn be a bounded domain and f : Dn → C be a continuous function, whose restriction to Dn is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then |f| attains a maximum on the boundary ∂Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3 (Abel’s multidimensional lemma [27] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let � α∈Nn aαzα be This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 0 10 20 30 40 50 μ26 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY a power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If there exist r ∈ Cn such that sup α∈Nn |aαrα| < ∞, then the series � α∈Nn aαzα is normally convergent for all z ∈ Cn such that |z1| < |r1|, · · · , |zn| < |rn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4 (Weierstrass’s M-test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let +∞ � k=1 fn(z) be a series of functions on a domain Dn of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If there exists a sequence of real numbers Mk such that Mk > 0 for all k, the numerical series +∞ � k=1 Mk is convergent and ∀k, ∀z ∈ Dn, |fk(z)| ≤ Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then the series +∞ � k=1 fn(z) is absolutely and uniformly convergent on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 (Lie’s theorem [8] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let X be a nonzero n-complex vector space, and g be a solvable Lie subalgebra of the Lie algebra of n × n complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then X has a basis (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , vn) with respect to which every element of g has an upper triangular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Angeli and D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Mauroy, Switched nonlinear systems in the Koopman operator frame- work: Toward a Lie-algebraic condition for uniform stability, in 2021 European Control Conference (ECC), IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 281–286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} diff --git a/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf b/AtFQT4oBgHgl3EQf9DeI/content/2301.13449v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..560fc25dff4cd92cf5b5ee608502308c633faaeb --- /dev/null +++ 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In data-driven mod- +eling, machine learning methods such as linear regression, neural +networks or decision-tree based methods are being applied. +While these methods do not require domain knowledge, they are +sensitive to data quality. Therefore, improving data quality in a +dataset is beneficial for creating machine learning-based models. +The improvement of data quality can be implemented through +preprocessing methods. A selected type of preprocessing is feature +engineering, which focuses on evaluating and improving the +quality of certain features inside the dataset. Feature engineering +methods include methods such as feature creation, feature ex- +pansion, or feature selection. In this work, a Python framework +containing different feature engineering methods is presented. +This framework contains different methods for feature creation, +expansion and selection; in addition, methods for transforming +or filtering data are implemented. The implementation of the +framework is based on the Python library scikit-learn. The +framework is demonstrated on a case study of a use case +from energy demand prediction. A data-driven model is created +including selected feature engineering methods. The results show +an improvement in prediction accuracy through the engineered +features. +Keywords: Energy Systems Modeling, Data-driven Modeling, +Feature Engineering, Python, Frameworks +I. INTRODUCTION +Modeling and simulation is an crucial step in the design and +optimization of energy systems. While traditional modeling +methods rely on system parameters, a recent approach focuses +on creating data-driven models based on measurement data +from an underlying system. In data-driven modeling, models +are not created based on system parameters, but on existing +measurement data. These models are based on machine learn- +ing (ML) methods [1]. While the area of machine learning +includes a wide range of methods such as clustering algorithms +or classifiers, the focus in data-driven modeling is set to +regression analysis for prediction and forecasting [2]. In re- +gression analysis, methods such as linear regression, decision- +tree based regression, or neural networks are being applied +[3]. While some of these methods, such as linear regression, +can be classified as white-box ML methods, others, such as +neural networks, are classified as black-box ML methods due +to their lack of comprehensibility [4]. While white-box ML +methods give more insight about their internal structure than +black box ML methods, their architecture is simpler, making +it more difficult to model complex dependencies, for instance +non-linearities [5]. To capture such dependencies using white- +box ML models, information about the dependencies can be +passed to the model through the dataset. This step is called +feature engineering. The main purpose of feature engineering +is to augment the existing dataset [6]. This can be done through +adding new information, or expanding or reducing the existing +feature set. In addition, the quality of a single feature can be +improved, for instance through transformation or filtering [7]. +The area of feature engineering covers a wide number of +methods, such as feature expansion [8] or feature selection +[9]. The term feature creation covers the creation of features +to add new information. Methods of feature creation include +encodings of time-based features, such as cyclic features [10], +or categorical encoding [11]. Similarly, feature expansion +is the method of creating new features based on existing +features. Feature expansion covers classical methods such as +polynomial expansion [8] or spline interpolation [12]. +In contrast to feature creation and expansion, feature selection +aims to reduce the size of the feature set. While large feature +sets may contain more information than smaller feature sets, +there may be redundancy in the data, as well as sparsity [13] or +multicollinearity [14]. To reduce the sparsity or multicollinear- +ity, as well as to remove redundant features, feature selection +mechanisms are applied. While methods such as Principal +Component Analysis (PCA) [15] aim to reduce the feature +set through transformation, feature selection methods discard +features based on certain criteria [16]. Feature selection can +be implemented for instance through sequential methods, such +as forward or backward selection [17], or through correlation +criteria [9]. Correlation criteria include measures based on +the Pearson Correlation Coefficient, as well as entropy-based +criteria [16]. The feature selection is then implemented through +a threshold-based selection. Threshold-based feature selection +analyzes features based on the selected criterion, and discards +features below a certain threshold. +Mainly, the methods of feature engineering are applied during +the first steps of creating a data-driven model, creating an en- +gineered dataset. This engineered dataset is then used to train +the model [18]. However, feature engineering methods can +also be used in combination with model selection procedures, +such as grid search [19]. Feature engineering methods are +widely used in applications from the energy domain, such as +in prediction for building energy demand [20] or photovoltaic +power prediction [18]. +arXiv:2301.01720v1 [cs.LG] 4 Jan 2023 + +A. Main Contribution +In the creation of data-driven models, a significant factor is +the quality of the underlying dataset. To improve the dataset +quality, feature engineering methods can be applied. +The main contribution of this work is a Python framework for +feature engineering that can be used for data-driven model +creation. The framework implements different methods for +feature creation, feature expansion, feature selection or trans- +formation. The feature engineering framework is implemented +in Python based on the scikit-learn framework and can be +imported as a Python package. The functionality of the frame- +work is demonstrated on a case study of an energy demand +prediction use case. The results of the case study show an +improvement prediction accuracy through the applied feature +engineering steps. +II. METHOD +The presented framework implements various feature engi- +neering methods in Python based on the research in [21] and +on the interfaces defined by scikit-learn. The methods are im- +plemented using either scikit-learn’s TransformerMixin or +SelectorMixin interface. The framework implements meth- +ods for feature expansion, feature creation, feature selection, +as well as transformation and filtering operations. +A. Feature Creation and Expansion +In the framework, different methods for feature creation and +expansion are implemented. These methods create new fea- +tures from time values or from expansion of existing features. +To create new features, the implemented framework supports +categorical encoding and cyclic encoding of time-based values. +Cyclic Features Cyclic features can be used to model time +values through cyclic functions [10]. Cyclic features were +implemented in [21], as well as in [22] and [23]. In the +implementation of the framework, sinusoidal signals xsin, xcos +with a selected frequency f can be created based on a sample +series n: +xsin[n] = sin(2πfn) +(1) +xcos[n] = cos(2πfn) +(2) +The implementation offers the creation of features with a zero- +order hold function for a certain time period, for instance TS = +1 day for a signal with a time period of T = 1 week. +Categorical Features Categorical encoding creates a repre- +sentation of discrete numerical values through a number of +features with boolean values [11], [21]. In this implementation, +for a number of categorical features x0,....,N for a feature x +with discrete possible values v0,....,N, a single feature xi is +defined as: +xi = +� +1 +x = vi +0 +else +(3) +The framework offers categorical encoding for time-based +values. In addition, a division factor is implemented to create +an encoding of a downsampled version of the time values. +Feature Expansion For feature expansion, the framework im- +plements wrappers for scikit-learn’s PolynomialFeatures and +SplineTransformer classes. The method of polynomial expan- +sion was applied in [21]. The parameters for the expansion +methods are passed through the wrapper. +Time-based Features The framework implements a method of +dynamic timeseries unrolling to create features xn−1, xn−2, +... xn−N from an existing feature x. The method of dynamic +timeseries unrolling is based on the research in [24], [25], and +[22]. While [25] and [24] use dynamic timeseries unrolling for +both input and target features of a model, allowing the creation +of auto-recursive models, this implementation only supports +dynamic timeseries unrolling for the input features, similar +to the method used in [22]. In this implementation, dynamic +timeseries unrolling is implemented through filter operations +from the scipy.signal library. The dynamic features are created +through the convolution of the signal x with a Kronecker delta +for i = 1...N: +xdyn,i[n] = x[n] ∗ δ[n − i] +(4) +This operation creates delayed signals xdyn,1, ..., xdyn,N. In +our implementation, for the samples in the delayed signals, +for which no values are available, zero values are used. +B. Feature Selection +In the framework, several threshold-based feature selection +methods are implemented. These methods analyze the input +and target features based on a certain criterion, and then +discard features with a low value of the criterion. A widely +used criterion is the Pearson Correlation Coefficient, which +is used to detect linear correlations between features [18]. +The Pearson Correlation Coefficient calculates the correlation +between two features for samples x0,....,N, y0,...,N with mean +values ¯x and ¯y: +rx,y = +�N +i=0(xi − ¯x)(yi − ¯y) +��N +i=0(xi − ¯x)2 �N +i=0(yi − ¯y)2 +(5) +While the Pearson correlation identifies linear correlations, +non-linear dependencies are not detected. To detect non-linear +dependencies, criteria such as Maximum Information Coeffi- +cient (MIC) [26], ennemi [27], dCor [28] or the Randomized +Dependence Coefficient (RDC) [29] can be used. +The framework provides classes for the criteria Pearson Corre- +lation Coefficient, F-statistic based on the Pearson Correlation +Coefficient, as well as thresholds based on the MIC, ennemi +and RDC. +C. Transformation and Filtering Operations +To transform features, the framework implements the Box- +cox transformation as well as the square root and inverse +transformation. In addition, the framework provides filtering +operations, which were applied in timeseries prediction for +instance in [7]. Discrete-time based filters can be implemented +in Python through the functions implemented in scipy.signal. +The scipy.signal library offers functions for calculating the +coefficients for different types of digital filters. A digital filter + +of order N can be defined through the transfer function H(z) +in a direct form: +H(z) = +�N +i=0 bizi +�N +i=0 aizi +(6) +The filter coefficients ai and bi define the behavior of the +filter. The scipy.signal library offers functions to compute the +filter coefficients for filter types such as the Butterworth or +Chebyshev filter [30]. While scipy.signal offers the compu- +tation of analog and digital filter coefficients, the framework +implementation focuses on digital filter implementations. The +framework implements the Butterworth and Chebyshev fil- +ter as scikit-learn TransformerMixin classes. In addition, an +envelope detection filter was implemented for demodulation +of modulated signals. This filter was implemented using the +pandas rolling average function. For all filters, offset com- +pensation before and after applying the filter operation and a +mask for handling NaN values were implemented. The direct +form filter classes of the framework offer a simple option for +extension. Different architectures can be implemented by re- +defining the implemented method for coefficient calculation. +This allows to create filters with different Finite Impulse +Response (FIR) or Infinite Impulse Response (IIR) structures. +D. Composite Transformers +In feature engineering, it is often the case that only a se- +lected subset of features should be transformed. To offer the +possibility to transform only selected features, a composite +transformer wrapper was implemented. This wrapper offers to +either automatically replace features through their transformed +versions, or add transformed features separately to the dataset. +E. Implementation +The framework offers compatibility with the sklearn.Pipeline +implementation, +making +it +possible +to +use +objects +as +part of a ML pipeline. The parameters of each objects +can be adapted through grid search, for instance using +sklearn.model_selection.GridSearchCV. In addition, every cre- +ated object can be stored to and loaded from a Pickle file using +the save_pkl or load_pkl method. +While the filtering, feature expansion and feature cre- +ation methods support operations on a numpy.ndarray or +pd.Dataframe or pd.Series object, the feature creation methods +require a pd.Dataframe or pd.Series object with a DateTimeIn- +dex or TimedeltaIndex to create samples based on a certain +date. +III. CASE STUDY +The framework is demonstrated on a use case from prediction +for energy systems modeling. For this purpose, a mixed office- +campus building is selected. A prediction model should be +trained based on existing measurement data. The data-driven +model is created using a workflow based on the implemented +methods. +A. Application +In this case study, the energy demand of a mixed office-campus +building should be evaluated. The data was provided from the +research in [24]. The energy demand of a building is subject +to various factors. Main factors that influence building energy +demand are thermal characteristics and Heating, Ventilation, +Air Conditioning and Cooling (HVAC) system behavior [31]. +Additionally, building energy demand may be dependent on +occupancy [3] or subject to seasonal trends [10]. Many of these +factors show non-linear behavior, which makes it difficult to +address them through a purely linear model. Therefore, feature +engineering was used to model additional factors. +B. Data-driven Model +For the selected application, a data-driven model of the build- +ing energy demand should be created. To demonstrate the +effect of feature engineering, two models were trained based +on the existing measurement data: a basic regression model +and a regression model with engineered features. +Measurement Data The energy demand was measured during +a period from 05/2019 to 03/2020, with a sampling time of +1h [24]. The measurement data includes features based on +weather data, such as temperature, as well as occupancy data, +such as registrations. The rest of the features are time-based, +such as daytime or weekday. +TABLE I +FEATURE SET FOR ENERGY CONSUMPTION PREDICTION +Feature Name +Unit +Description +temperature +°C +Outdoor Temperature +daytime +h +Daytime +weekday +d +Weekday from 0 to 6 +holiday +Public holiday +daylight +day or night +registrations +registrations for lectures +Consumption +kWh +Energy Consumption +Model Architecture For the energy demand, a linear regression +model should be trained. The linear regression architecture was +selected due to its simplicity and comprehensibility as a white- +box ML model. Non-linear behavior of the underlying system +should be incorporated through feature engineering. +Feature Engineering To model the non-linear behavior of +the energy demand, categorical features and cyclical features +were used in combination with Butterworth Filtering, dynamic +timeseries unrolling and feature selection through the Pearson +Correlation Coefficient. An overview of the implemented +workflow is depicted in Figure 1. +Training Parameters For the model training, a train-test split +of 0.8 was selected together with a 5-fold cross-validation. +For the model with engineered features, the parameters for +the steps timeseries unrolling and feature selection were deter- +mined through a grid search based on the metrics Coefficient +of Determination (R2), mean squared error (MSE) and Mean +Absolute Percentage Error (MAPE). + +Basic Feature +Set +Extended +Feature Set +Engineered +Feature Set +Feature Selection – +Pearson Correlation +Fig. 1. Implemented Workflow. +C. Experimental Results +The two models were trained on the measurement data and +compared in terms of performance metrics. Additionally, anal- +yses of the predicted values through timeseries analysis and +prediction error plots were performed. +Performance Metrics To evaluate the performance of the +model, the metrics R2, Coefficient of Variation of the Root +Mean Square Error (CV-RMSE) and MAPE were used [21]. +Table II gives an overview of the metrics. +TABLE II +PERFORMANCE METRICS +Model +R2 +CV-RMSE +MAPE +Basic Regression +0.548 +0.267 +22.764 % +Engineered Features +0.638 +0.201 +17.493% +From the performance metrics, an improvement in prediction +accuracy for the linear regression model through the engi- +neered features could be observed. +Timeseries Analysis The improvement in prediction accuracy +could also be observed from the timeseries analysis depicted +in Figure 2. +2020-01-05 +2020-01-06 +2020-01-07 +2020-01-08 +2020-01-09 +2020-01-10 +2020-01-11 +2020-01-12 +2020-01-13 +2020-01-14 +2020-01-15 +2020-01-16 +2020-01-17 +2020-01-18 +2020-01-19 +2020-01-20 +2020-01-21 +2020-01-22 +2020-01-23 +2020-01-24 +2020-01-25 +20 +40 +60 +80 +Time [Days] +Consumption [kWh] +Measurement value +Basic Regression +Engineered Features +Fig. 2. Timeseries Analysis for period of 25 days from test set. +The timeseries analysis showed that the cyclic behavior of +the day-night changes in the energy demand could be more +accurately replicated by the model with engineered features. +Additionally, the prediction using engineered features shows a +higher accuracy in replicating low energy demand values than +the basic regression. This effect can be observed in Figure 3. +2020-01-15 +2020-01-16 +2020-01-17 +2020-01-18 +2020-01-19 +2020-01-20 +20 +40 +60 +Time [Days] +Consumption [kWh] +Measurement value +Basic Regression +Engineered Features +Fig. 3. Timeseries Analysis for period of five days from test set. +For both models, the residual error was analyzed through +prediction error plots (Figure 4). The prediction error plots +show that the residual error is decreased for the model with +engineered features. In addition, the homogenity of the error +distribution is improved through the applied feature engineer- +ing methods. +20 +30 +40 +50 +60 +70 +80 +20 +30 +40 +50 +60 +70 +80 +True Value [kWh] +Predicted Value [kWh] +Basic Regression +Optimal Prediction +20 +25 +30 +35 +40 +45 +50 +55 +20 +25 +30 +35 +40 +45 +50 +55 +True Value [kWh] +Predicted Value [kWh] +Engineered Features +Optimal Prediction +Fig. 4. Prediction Error Plots for Energy Consumption +Since performance metrics, timeseries analysis and prediction +error plots show an improvement in accuracy, the feature engi- +neering steps are suggested to be beneficial for the prediction +model. +IV. RELATED WORK +In the creation of data-driven models in Python, many frame- +works have been implemented. One of the most well-known +Python ML frameworks is the scikit-learn framework, which +provides methods such as data preprocessing, feature engineer- +ing, clustering, and implementations of various ML models. +The scikit-learn framework offers interfaces which can be used +to implement additional methods. Due to the popularity of +scikit-learn, various frameworks extending scikit-learn have +been implemented. For instance, the imblearn framework [32] +focuses on extending scikit-learn’s functionality to processing +imbalanced datasets. In addition, the imblearn framework +offers different resampling methods. The mlxtend framework +[33] offers feature extraction methods such as PCA, or fea- +ture selection methods such as sequential feature selection. +Additionally, different evaluation and utility functions are +implemented. In contrast, libraries such as statsmodels [34] +provide their own interface for their regression models. The + +statsmodels framework provides models based on stochastic +and statistical methods, such as the Weighted Least Squares +(WLS). In the area of feature engineering, different Python +packages have been created. The feature-engine [35] library +contains a large collection of feature engineering methods, +which are implemented based on scikit-learn. The featuretools +framework [36] allows the synthesis of features from relational +databases. offers functionality for feature encoding, as well as +different transformations or aggregate functions. Additionally, +this framework offers transformations, feature encoding, ag- +gregate functions, as well as coordinate transformations. +V. CONCLUSION +This paper presents a Python framework for feature engineer- +ing that provides different methods through a standardized +interface. The framework is based on the scikit-learn package +and offers different methods. The framework offers classic +feature engineering methods such feature expansion, as well +as as feature creation, feature selection or transformation and +filter operations. The framework is implemented as a Python +package and can be included in different projects. Through +the specifically defined interfaces of the framework, additional +methods can be added with low effort. Finally, we demonstrate +the framework on a case study of energy demand prediction, +using a workflow created from a subset of the implemented +methods for data-driven model creation. +A. Future Work +The current version of the framework gives many options +for extensions. For instance, additional feature engineering +methods can be added using the provided interfaces of the +framework. In addition, combinations of the implemented +feature engineering methods can be used for prediction in +different use cases. +REFERENCES +[1] A. Mosavi, M. Salimi, S. F. Ardabili, T. Rabczuk, S. Shamshirband, +and A. 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Veeramachaneni, “Deep feature synthesis: Towards +automating data science endeavors,” in 2015 IEEE International Con- +ference on Data Science and Advanced Analytics, DSAA 2015, Paris, +France, October 19-21, 2015. +IEEE, 2015, pp. 1–10. + diff --git a/CNAzT4oBgHgl3EQfwP6m/content/tmp_files/load_file.txt b/CNAzT4oBgHgl3EQfwP6m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8bbfad1fc5a2e3ef2323af93625f11dbc5420b6 --- /dev/null +++ b/CNAzT4oBgHgl3EQfwP6m/content/tmp_files/load_file.txt @@ -0,0 +1,501 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf,len=500 +page_content='Augmenting data-driven models for energy systems through feature engineering: A Python framework for feature engineering Sandra Wilfling Abstract—Data-driven modeling is an approach in energy systems modeling that has been gaining popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In data-driven mod- eling, machine learning methods such as linear regression, neural networks or decision-tree based methods are being applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While these methods do not require domain knowledge, they are sensitive to data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Therefore, improving data quality in a dataset is beneficial for creating machine learning-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The improvement of data quality can be implemented through preprocessing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A selected type of preprocessing is feature engineering, which focuses on evaluating and improving the quality of certain features inside the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature engineering methods include methods such as feature creation, feature ex- pansion, or feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In this work, a Python framework containing different feature engineering methods is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This framework contains different methods for feature creation, expansion and selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' in addition, methods for transforming or filtering data are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The implementation of the framework is based on the Python library scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework is demonstrated on a case study of a use case from energy demand prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A data-driven model is created including selected feature engineering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The results show an improvement in prediction accuracy through the engineered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Keywords: Energy Systems Modeling, Data-driven Modeling, Feature Engineering, Python, Frameworks I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' INTRODUCTION Modeling and simulation is an crucial step in the design and optimization of energy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While traditional modeling methods rely on system parameters, a recent approach focuses on creating data-driven models based on measurement data from an underlying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In data-driven modeling, models are not created based on system parameters, but on existing measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' These models are based on machine learn- ing (ML) methods [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While the area of machine learning includes a wide range of methods such as clustering algorithms or classifiers, the focus in data-driven modeling is set to regression analysis for prediction and forecasting [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In re- gression analysis, methods such as linear regression, decision- tree based regression, or neural networks are being applied [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While some of these methods, such as linear regression, can be classified as white-box ML methods, others, such as neural networks, are classified as black-box ML methods due to their lack of comprehensibility [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While white-box ML methods give more insight about their internal structure than black box ML methods, their architecture is simpler, making it more difficult to model complex dependencies, for instance non-linearities [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' To capture such dependencies using white- box ML models, information about the dependencies can be passed to the model through the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This step is called feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The main purpose of feature engineering is to augment the existing dataset [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This can be done through adding new information, or expanding or reducing the existing feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In addition, the quality of a single feature can be improved, for instance through transformation or filtering [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The area of feature engineering covers a wide number of methods, such as feature expansion [8] or feature selection [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The term feature creation covers the creation of features to add new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Methods of feature creation include encodings of time-based features, such as cyclic features [10], or categorical encoding [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Similarly, feature expansion is the method of creating new features based on existing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature expansion covers classical methods such as polynomial expansion [8] or spline interpolation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In contrast to feature creation and expansion, feature selection aims to reduce the size of the feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While large feature sets may contain more information than smaller feature sets, there may be redundancy in the data, as well as sparsity [13] or multicollinearity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' To reduce the sparsity or multicollinear- ity, as well as to remove redundant features, feature selection mechanisms are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While methods such as Principal Component Analysis (PCA) [15] aim to reduce the feature set through transformation, feature selection methods discard features based on certain criteria [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature selection can be implemented for instance through sequential methods, such as forward or backward selection [17], or through correlation criteria [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Correlation criteria include measures based on the Pearson Correlation Coefficient, as well as entropy-based criteria [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The feature selection is then implemented through a threshold-based selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Threshold-based feature selection analyzes features based on the selected criterion, and discards features below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Mainly, the methods of feature engineering are applied during the first steps of creating a data-driven model, creating an en- gineered dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This engineered dataset is then used to train the model [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' However, feature engineering methods can also be used in combination with model selection procedures, such as grid search [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature engineering methods are widely used in applications from the energy domain, such as in prediction for building energy demand [20] or photovoltaic power prediction [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='01720v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='LG] 4 Jan 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Main Contribution In the creation of data-driven models, a significant factor is the quality of the underlying dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' To improve the dataset quality, feature engineering methods can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The main contribution of this work is a Python framework for feature engineering that can be used for data-driven model creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework implements different methods for feature creation, feature expansion, feature selection or trans- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The feature engineering framework is implemented in Python based on the scikit-learn framework and can be imported as a Python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The functionality of the frame- work is demonstrated on a case study of an energy demand prediction use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The results of the case study show an improvement prediction accuracy through the applied feature engineering steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' METHOD The presented framework implements various feature engi- neering methods in Python based on the research in [21] and on the interfaces defined by scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The methods are im- plemented using either scikit-learn’s TransformerMixin or SelectorMixin interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework implements meth- ods for feature expansion, feature creation, feature selection, as well as transformation and filtering operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature Creation and Expansion In the framework, different methods for feature creation and expansion are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' These methods create new fea- tures from time values or from expansion of existing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' To create new features, the implemented framework supports categorical encoding and cyclic encoding of time-based values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Cyclic Features Cyclic features can be used to model time values through cyclic functions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Cyclic features were implemented in [21], as well as in [22] and [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In the implementation of the framework, sinusoidal signals xsin, xcos with a selected frequency f can be created based on a sample series n: xsin[n] = sin(2πfn) (1) xcos[n] = cos(2πfn) (2) The implementation offers the creation of features with a zero- order hold function for a certain time period, for instance TS = 1 day for a signal with a time period of T = 1 week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Categorical Features Categorical encoding creates a repre- sentation of discrete numerical values through a number of features with boolean values [11], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In this implementation, for a number of categorical features x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='.,N for a feature x with discrete possible values v0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='.,N, a single feature xi is defined as: xi = � 1 x = vi 0 else (3) The framework offers categorical encoding for time-based values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In addition, a division factor is implemented to create an encoding of a downsampled version of the time values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature Expansion For feature expansion, the framework im- plements wrappers for scikit-learn’s PolynomialFeatures and SplineTransformer classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The method of polynomial expan- sion was applied in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The parameters for the expansion methods are passed through the wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Time-based Features The framework implements a method of dynamic timeseries unrolling to create features xn−1, xn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' xn−N from an existing feature x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The method of dynamic timeseries unrolling is based on the research in [24], [25], and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While [25] and [24] use dynamic timeseries unrolling for both input and target features of a model, allowing the creation of auto-recursive models, this implementation only supports dynamic timeseries unrolling for the input features, similar to the method used in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In this implementation, dynamic timeseries unrolling is implemented through filter operations from the scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='signal library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The dynamic features are created through the convolution of the signal x with a Kronecker delta for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='N: xdyn,i[n] = x[n] ∗ δ[n − i] (4) This operation creates delayed signals xdyn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=', xdyn,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In our implementation, for the samples in the delayed signals, for which no values are available, zero values are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature Selection In the framework, several threshold-based feature selection methods are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' These methods analyze the input and target features based on a certain criterion, and then discard features with a low value of the criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A widely used criterion is the Pearson Correlation Coefficient, which is used to detect linear correlations between features [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The Pearson Correlation Coefficient calculates the correlation between two features for samples x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='.,N, y0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=',N with mean values ¯x and ¯y: rx,y = �N i=0(xi − ¯x)(yi − ¯y) ��N i=0(xi − ¯x)2 �N i=0(yi − ¯y)2 (5) While the Pearson correlation identifies linear correlations, non-linear dependencies are not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' To detect non-linear dependencies, criteria such as Maximum Information Coeffi- cient (MIC) [26], ennemi [27], dCor [28] or the Randomized Dependence Coefficient (RDC) [29] can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework provides classes for the criteria Pearson Corre- lation Coefficient, F-statistic based on the Pearson Correlation Coefficient, as well as thresholds based on the MIC, ennemi and RDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Transformation and Filtering Operations To transform features, the framework implements the Box- cox transformation as well as the square root and inverse transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In addition, the framework provides filtering operations, which were applied in timeseries prediction for instance in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Discrete-time based filters can be implemented in Python through the functions implemented in scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='signal library offers functions for calculating the coefficients for different types of digital filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A digital filter of order N can be defined through the transfer function H(z) in a direct form: H(z) = �N i=0 bizi �N i=0 aizi (6) The filter coefficients ai and bi define the behavior of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='signal library offers functions to compute the filter coefficients for filter types such as the Butterworth or Chebyshev filter [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='signal offers the compu- tation of analog and digital filter coefficients, the framework implementation focuses on digital filter implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework implements the Butterworth and Chebyshev fil- ter as scikit-learn TransformerMixin classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In addition, an envelope detection filter was implemented for demodulation of modulated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This filter was implemented using the pandas rolling average function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' For all filters, offset com- pensation before and after applying the filter operation and a mask for handling NaN values were implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The direct form filter classes of the framework offer a simple option for extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Different architectures can be implemented by re- defining the implemented method for coefficient calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This allows to create filters with different Finite Impulse Response (FIR) or Infinite Impulse Response (IIR) structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Composite Transformers In feature engineering, it is often the case that only a se- lected subset of features should be transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' To offer the possibility to transform only selected features, a composite transformer wrapper was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This wrapper offers to either automatically replace features through their transformed versions, or add transformed features separately to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Implementation The framework offers compatibility with the sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='Pipeline implementation, making it possible to use objects as part of a ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The parameters of each objects can be adapted through grid search, for instance using sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='model_selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='GridSearchCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In addition, every cre- ated object can be stored to and loaded from a Pickle file using the save_pkl or load_pkl method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' While the filtering, feature expansion and feature cre- ation methods support operations on a numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='ndarray or pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='Dataframe or pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='Series object, the feature creation methods require a pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='Dataframe or pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='Series object with a DateTimeIn- dex or TimedeltaIndex to create samples based on a certain date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' CASE STUDY The framework is demonstrated on a use case from prediction for energy systems modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' For this purpose, a mixed office- campus building is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A prediction model should be trained based on existing measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The data-driven model is created using a workflow based on the implemented methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Application In this case study, the energy demand of a mixed office-campus building should be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The data was provided from the research in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The energy demand of a building is subject to various factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Main factors that influence building energy demand are thermal characteristics and Heating, Ventilation, Air Conditioning and Cooling (HVAC) system behavior [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Additionally, building energy demand may be dependent on occupancy [3] or subject to seasonal trends [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Many of these factors show non-linear behavior, which makes it difficult to address them through a purely linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Therefore, feature engineering was used to model additional factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Data-driven Model For the selected application, a data-driven model of the build- ing energy demand should be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' To demonstrate the effect of feature engineering, two models were trained based on the existing measurement data: a basic regression model and a regression model with engineered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Measurement Data The energy demand was measured during a period from 05/2019 to 03/2020, with a sampling time of 1h [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The measurement data includes features based on weather data, such as temperature, as well as occupancy data, such as registrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The rest of the features are time-based, such as daytime or weekday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' TABLE I FEATURE SET FOR ENERGY CONSUMPTION PREDICTION Feature Name Unit Description temperature °C Outdoor Temperature daytime h Daytime weekday d Weekday from 0 to 6 holiday Public holiday daylight day or night registrations registrations for lectures Consumption kWh Energy Consumption Model Architecture For the energy demand, a linear regression model should be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The linear regression architecture was selected due to its simplicity and comprehensibility as a white- box ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Non-linear behavior of the underlying system should be incorporated through feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Feature Engineering To model the non-linear behavior of the energy demand, categorical features and cyclical features were used in combination with Butterworth Filtering, dynamic timeseries unrolling and feature selection through the Pearson Correlation Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' An overview of the implemented workflow is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Training Parameters For the model training, a train-test split of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='8 was selected together with a 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' For the model with engineered features, the parameters for the steps timeseries unrolling and feature selection were deter- mined through a grid search based on the metrics Coefficient of Determination (R2), mean squared error (MSE) and Mean Absolute Percentage Error (MAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Basic Feature Set Extended Feature Set Engineered Feature Set Feature Selection – Pearson Correlation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Implemented Workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Experimental Results The two models were trained on the measurement data and compared in terms of performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Additionally, anal- yses of the predicted values through timeseries analysis and prediction error plots were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Performance Metrics To evaluate the performance of the model, the metrics R2, Coefficient of Variation of the Root Mean Square Error (CV-RMSE) and MAPE were used [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Table II gives an overview of the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' TABLE II PERFORMANCE METRICS Model R2 CV-RMSE MAPE Basic Regression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='267 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='764 % Engineered Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='201 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content='493% From the performance metrics, an improvement in prediction accuracy for the linear regression model through the engi- neered features could be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Timeseries Analysis The improvement in prediction accuracy could also be observed from the timeseries analysis depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 2020-01-05 2020-01-06 2020-01-07 2020-01-08 2020-01-09 2020-01-10 2020-01-11 2020-01-12 2020-01-13 2020-01-14 2020-01-15 2020-01-16 2020-01-17 2020-01-18 2020-01-19 2020-01-20 2020-01-21 2020-01-22 2020-01-23 2020-01-24 2020-01-25 20 40 60 80 Time [Days] Consumption [kWh] Measurement value Basic Regression Engineered Features Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Timeseries Analysis for period of 25 days from test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The timeseries analysis showed that the cyclic behavior of the day-night changes in the energy demand could be more accurately replicated by the model with engineered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Additionally, the prediction using engineered features shows a higher accuracy in replicating low energy demand values than the basic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' This effect can be observed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 2020-01-15 2020-01-16 2020-01-17 2020-01-18 2020-01-19 2020-01-20 20 40 60 Time [Days] Consumption [kWh] Measurement value Basic Regression Engineered Features Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Timeseries Analysis for period of five days from test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' For both models, the residual error was analyzed through prediction error plots (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The prediction error plots show that the residual error is decreased for the model with engineered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In addition, the homogenity of the error distribution is improved through the applied feature engineer- ing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 20 30 40 50 60 70 80 20 30 40 50 60 70 80 True Value [kWh] Predicted Value [kWh] Basic Regression Optimal Prediction 20 25 30 35 40 45 50 55 20 25 30 35 40 45 50 55 True Value [kWh] Predicted Value [kWh] Engineered Features Optimal Prediction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Prediction Error Plots for Energy Consumption Since performance metrics, timeseries analysis and prediction error plots show an improvement in accuracy, the feature engi- neering steps are suggested to be beneficial for the prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' RELATED WORK In the creation of data-driven models in Python, many frame- works have been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' One of the most well-known Python ML frameworks is the scikit-learn framework, which provides methods such as data preprocessing, feature engineer- ing, clustering, and implementations of various ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The scikit-learn framework offers interfaces which can be used to implement additional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Due to the popularity of scikit-learn, various frameworks extending scikit-learn have been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' For instance, the imblearn framework [32] focuses on extending scikit-learn’s functionality to processing imbalanced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In addition, the imblearn framework offers different resampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The mlxtend framework [33] offers feature extraction methods such as PCA, or fea- ture selection methods such as sequential feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Additionally, different evaluation and utility functions are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In contrast, libraries such as statsmodels [34] provide their own interface for their regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The statsmodels framework provides models based on stochastic and statistical methods, such as the Weighted Least Squares (WLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' In the area of feature engineering, different Python packages have been created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The feature-engine [35] library contains a large collection of feature engineering methods, which are implemented based on scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The featuretools framework [36] allows the synthesis of features from relational databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' offers functionality for feature encoding, as well as different transformations or aggregate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Additionally, this framework offers transformations, feature encoding, ag- gregate functions, as well as coordinate transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' CONCLUSION This paper presents a Python framework for feature engineer- ing that provides different methods through a standardized interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework is based on the scikit-learn package and offers different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework offers classic feature engineering methods such feature expansion, as well as as feature creation, feature selection or transformation and filter operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' The framework is implemented as a Python package and can be included in different projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Through the specifically defined interfaces of the framework, additional methods can be added with low effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Finally, we demonstrate the framework on a case study of energy demand prediction, using a workflow created from a subset of the implemented methods for data-driven model creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' Future Work The current version of the framework gives many options for extensions.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} +page_content=' 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'} diff --git a/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf b/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4b8987c244d2571a2254117bd255015517bb7107 --- /dev/null +++ b/CdFJT4oBgHgl3EQfASzG/content/2301.11420v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0a95247a2a1200c7a860bb8ac0b0ef32461ad05e829c326fa4b1568651520484 +size 617189 diff --git a/CdFJT4oBgHgl3EQfASzG/vector_store/index.faiss b/CdFJT4oBgHgl3EQfASzG/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..26f1bab120e3ea014794b9332232e7438069f156 --- /dev/null +++ b/CdFJT4oBgHgl3EQfASzG/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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+played over T rounds, with a leader-follower hi- +erarchy described by a directed acyclic graph. +For each round, the graph structure dictates the +order of the players and how players observe +the actions of one another. By the end of each +round, all players receive a joint bandit-reward +based on their joint action that is used to update +the player strategies towards the goal of minimiz- +ing the joint pseudo-regret. We present a learn- +ing algorithm inspired by the single-player multi- +armed bandit problem and show that it achieves +sub-linear joint pseudo-regret in the number of +rounds for both adversarial and stochastic ban- +dit rewards. Furthermore, we quantify the cost +incurred due to the decentralized nature of our +problem compared to the centralized setting. +1. Introduction +Decentralized multi-agent online learning concerns agents +that, simultaneously, learn to behave over time in order +to achieve their goals. +Compared to the single-agent +setup, novel challenges are present as agents may not +share the same objectives, the environment becomes non- +stationary, and information asymmetry may exist between +agents (Yang & Wang, 2020). +Traditionally, the multi- +agent problem has been addressed by either relying on +a central controller to coordinate the agents’ actions or +to let the agents learn independently. +However, access +to a central controller may not be realistic and indepen- +dent learning suffers from convergence issues (Zhang et al., +2019). To circumvent these issues, a common approach +is to drop the central coordinator and allow informa- +tion exchange between agents (Zhang et al., 2018; 2019; +Cesa-Bianchi et al., 2021). +Decision-making that involves multiple agents is often +1AI Sweden, Gothenburg, Sweden. Correspondence to: Johan +¨Ostman . +modeled as a game and studied under the lens of game +theory to describe the learning outcomes.1 +Herein, we +consider games with a leader-follower structure in which +players act consecutively. For two players, such games +are known as Stackelberg games (Hicks, 1935). Stackel- +berg games have been used to model diverse learning situ- +ations such as airport security (Balcan et al., 2015), poach- +ing (Sessa et al., 2020), tax planning (Zheng et al., 2020), +and generative adversarial networks (Moghadam et al., +2021). +In a Stackelberg game, one is typically con- +cerned with finding the Stackelberg equilibrium, some- +times called Stackelberg-Nash equilibrium, in which the +leader uses a mixed strategy and the follower is best- +responding. A Stackelberg equilibrium may be obtained by +solving a bi-level optimization problem if the reward func- +tions are known (Sch¨afer et al., 2020; Aussel & Svensson, +2020) or, otherwise, it may be learnt via online learn- +ing techniques (Bai et al., 2021; Zhong et al., 2021), e.g., +no-regret algorithms (Shalev-Shwartz, 2012; Deng et al., +2019; Goktas et al., 2022). +No-regret algorithms have emerged from the single-player +multi-armed bandit problem as a means to alleviate +the exploitation-exploration trade-off (Bubeck & Slivkins, +2012). An algorithm is called no-regret if the difference be- +tween the cumulative rewards of the learnt strategy and the +single best action in hindsight is sublinear in the number +of rounds (Shalev-Shwartz, 2012). In the multi-armed ban- +dit problem, rewards may be adversarial (based on random- +ness and previous actions), oblivious adversarial (random), +or stochastic (independent and identically distributed) over +time (Auer et al., 2002). Different assumptions on the ban- +dit rewards yield different algorithms and regret bounds. +Indeed, algorithms tailored for one kind of rewards are +sub-optimal for others, e.g., the EXP3 algorithm due +to Auer et al. (2002) yields the optimal scaling for adversar- +ial rewards but not for stochastic rewards. For this reason, +best-of-two-worlds algorithms, able to optimally handle +both the stochastic and adversarial rewards, have recently +been pursued and resulted in algorithms with close to op- +timal performance in both settings (Auer & Chiang, 2016; +1The convention is to use agents in learning applications and +players in game theoretic applications, we shall use the game- +theoretic nomenclature in the remainder of the paper. + +Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds +Wei & Luo, 2018; Zimmert & Seldin, 2021). Extensions to +multiplayer multi-armed bandit problems have been pro- +posed in which players attempt to maximize the sum of +rewards by pulling an arm each, see, e.g., (Kalathil et al., +2014; Bubeck et al., 2021). +No-regret algorithms are a common element also when an- +alyzing multiplayer games. +For example, in continuous +two-player Stackelberg games, the leader strategy, based +on a no-regret algorithm, converges to the Stackelberg equi- +librium if the follower is best-responding (Goktas et al., +2022). In contrast, if also the follower adopts a no-regret al- +gorithm, the regret dynamics is not guaranteed to converge +to a Stackelberg equilibrium point (Goktas et al., 2022, +Ex. 3.2). +In (Deng et al., 2019), it was shown for two- +player Stackelberg games that a follower playing a, so- +called, mean-based no-regret algorithm, enables the leader +to achieve a reward strictly larger than the reward achieved +at the Stackelberg equilibrium. +This result does, how- +ever, not generalize to n-player games as demonstrated +by D’Andrea (2022). Apart from studying the Stackelberg +equilibrium, several papers have analyzed the regret. For +example, Sessa et al. (2020) presented upper-bounds on the +regret of a leader, employing a no-regret algorithm, playing +against an adversarial follower with an unknown response +function. Furthermore, Stackelberg games with states were +introduced by Lauffer et al. (2022) along with an algorithm +that was shown to achieve no-regret. +As the follower in a Stackelberg game observes the leader’s +action, there is information exchange. A generalization +to multiple players has been studied in a series of pa- +pers (Cesa-Bianchi et al., 2016; 2020; 2021). In this line +of work, players with a common action space form an ar- +bitrary graph and are randomly activated in each round. +Active players share information with their neighbors by +broadcasting their observed loss, previously received neigh- +bor losses, and their current strategy. The goal of the play- +ers is to minimize the network regret, defined with respect +to the cumulative losses observed by active players over +the rounds. The players, however, update their strategies +according to their individually observed loss. Although we +consider players connected on a graph, our work differs +significantly from (Cesa-Bianchi et al., 2016; 2020; 2021), +e.g., we allow only actions to be observed between players +and the players update their strategies based on a common +bandit reward rather than an individual reward. +Contributions: +We introduce the joint pseudo-regret, +defined with respect to the cumulative reward where +all the players observe the same bandit-reward in each +round. We provide an online learning-algorithmfor general +consecutive-play games that relies on no-regret algorithms +developed for the single-player multi-armed bandit prob- +lem. The main novelty of our contribution resides in the +joint analysis of players with coupled rewards where we +derive upper bounds on the joint pseudo-regret and prove +our algorithm to be no-regret in the adversarial setting. Fur- +thermore, we quantify the penalty incurred by our decen- +tralized setting in relation to the centralized setting. +2. Problem formulation +In this section, we formalize the consecutive structure of +the game and introduce the joint pseudo-regret that will +be used as a performance metric throughout. We consider +a decentralized setting where, in each round of the game, +players pick actions consecutively. The consecutive nature +of the game allows players to observe preceding players’ +actions and may be modeled by a DAG. For example, in +Fig. 1, a seven-player game is illustrated in which player 1 +initiates the game and her action is observed by players 2, 5, +and 6. The observations available to the remaining players +follow analogously. Note that for a two-player consecutive +game, the DAG models a Stackelberg game. +We let G = (V, E) denote a DAG where V denotes the ver- +tices and E denotes the edges. For our setting, V constitutes +the n different players and E = {(j, i) : j → i, j ∈ V, i ∈ +V} describes the observation structure where j → i indi- +cates that player i observes the action of player j. Accord- +ingly, a given player i ∈ V observes the actions of its direct +parents, i.e., players j ∈ Ei = {k : (k, i) ∈ E}. Further- +more, each player i ∈ V is associated with a discrete action +space Ai of size Ai. We denote by πi(t), the mixed strat- +egy of player i over the action space Ai in round t ∈ [T ] +such that πi(t) = a with probability pi,a for a ∈ Ai. In +the special case when pi,a = 1 for some a ∈ Ai, the strat- +egy is referred to as pure. Let AB denote the joint action +space of players in a set B given by the Cartesian product +AB = � +i∈B Ai. If a player i has no parents, i.e., Ei = ∅, +we use the convention |AEi| = 1. +We consider a collaborative setting with bandit rewards +given by a mapping rt : AV → [0, 1] in each round t ∈ [T ]. +The bandit rewards are assumed to be adversarial. Let C de- +note a set of cliques in the DAG (Koller & Friedman, 2009, +Def. 2.13) and let Nk ∈ C for k ∈ [|C|] denote the players +in the kth clique in C with joint action space ANk such that +Nk ∩ Nj = ∅ for j ̸= k. For a joint action a(t) ∈ AV, +we consider bandit rewards given by a linear combination +of the clique-rewards as +rt(a(t)) = +|C| +� +k=1 +βkrk +t (P k(a(t))), +(1) +where rk +t +: +ANk +→ +[0, 1], βk +≥ +0 is the weight +of the kth clique reward such that �|C| +k=1 βk += +1, +and P k(a(t)) denotes the joint action of the players in +Nk. As an example, Fig. 2 highlights the cliques C = + +Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds +1 +2 +3 +4 +5 +6 +7 +Figure 1. A game with seven players. +1 +2 +3 +4 +5 +6 +7 +Figure 2. Colored cliques comprising the bandit reward. +{{2, 3, 4}, {1, 5}, {6}, {7}} and we have, e.g., N1 += +{2, 3, 4}, and P 1(a(t)) = (a2(t), a3(t), a4(t)). Note that +each player influences only a single term in the reward (1). +In each round t ∈ [T ], the game proceeds as follows for +player i ∈ V: +1) the player is idle until the actions of all parents in Ei +have been observed, +2) the player picks an action ai(t) ∈ Ai according to its +strategy πi(t), +3) once all the n players in V have chosen an action, +the player observes the bandit reward rt(a(t)) and up- +dates its strategy. +The goal of the game is to find policies {πi(t)}n +i=1 that de- +pend on past actions and rewards in order to minimize the +joint pseudo-regret R(T ) which is defined similarly to the +pseudo regret (Shalev-Shwartz, 2012, Ch. 4.2) as +R(T ) = r(a⋆) − E +� T +� +t=1 +rt(a(t)) +� +, +(2) +where +r(a⋆) = max +a∈AV E +� T +� +t=1 +rt(a) +� +, +and the expectations are taken with respect to the rewards +and the player actions.2 Note that r(a⋆) corresponds to the +largest expected reward obtainable if all players use pure +strategies. Hence, the pseudo-regret in (2) quantifies the +difference between the expected reward accumulated by the +learnt strategies and the reward-maximizing pure strategies +in hindsight. +Our problem formulation pertains to a plethora of appli- +cations. +Examples include resource allocation in cog- +nitive radio networks where available frequencies are +obtained via channel sensing (Janatian et al., 2015) and +semi-autonomous vehicles with adaptive cruise control, +i.e., vehicles ahead are observed before an action is de- +cided (Marsden et al., 2001). +Also recently, the impor- +tance of coupled rewards and partner awareness through +implicit communications, e.g., by observation, has been +highlighted in human-robot and human-AI collaborative +settings (Bıyık et al., 2022). Furthermore, our formulation +is applicable in simple scenarios within to reinforcement +learning (Ibarz et al., 2021). +As will be shown in the next section, any no-regret al- +gorithm can be used as a building block for the games +considered herein to guarantee a sub-linear pseudo-regret +in the number of rounds T . +As our goal is to study +the joint pseudo-regret (2) for adversarial, we start from +a state-of-the-art algorithm for the adversarial multi- +armed bandit problem. In particular, we will utilize the +TSALLIS-INF algorithm that guarantees a pseudo-regret +with the optimal scaling in the adversarial single-player set- +ting (Zimmert & Seldin, 2021). +3. Analysis of the joint pseudo-regret +Our analysis of the joint pseudo-regret builds upon learning +algorithms for the single-player multi-armed bandit prob- +lem. First, let us build intuition on how to use a multi- +armed bandit algorithm in the DAG-based game described +in Section 2. Consider a 2-player Stackelberg game where +the players choose actions from A1 and A2, respectively, +and where player 2 observes the actions of player 1. For +simplicity, we let player 1 use a mixed strategy whereas +player 2 is limited to a pure strategy. Furthermore, con- +sider the rewards to be a priori known by the players and let +T = 1 for which the Stackelberg game may be viewed as a +bi-level optimization problem (Aussel & Svensson, 2020). +In this setting, the action of player 1 imposes a Nash game +on player 2 whom attempts to play optimally given the ob- +servation. Hence, player 2 has A1 pure strategies, one for +each of the A1 actions of player 1. +We may generalize this idea to the DAG-based multiplayer +game with unknown bandit-rewards and T ≥ 1 to achieve +2This is called pseudo-regret as r(a⋆) is obtained by a maxi- +mization outside of the expectation. + +Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds +no-regret. Indeed, a player i ∈ V may run |AEi| different +multi-armed bandit algorithms, one for each of the joint +actions of its parents. +Algorithm 1 illustrates this idea +in conjunction with the TSALLIS-INF update rule intro- +duced by Zimmert & Seldin (2021), which is given in Al- +gorithm 2 for completeness.3 In particular, for the 2-player +Stackelberg game, the leader runs a single multi-armed ban- +dit algorithm whereas the follower runs A1 learning algo- +rithms. For simplicity, Algorithm 1 assumes that player i +knows the size of the joint action space of its parents, i.e., +|AEi|. Dropping this assumption is straightforward: simply +keep track of the observed joint actions and initiate a new +multi-armed bandit learner upon a unique observation. +Algorithm 1 Learning algorithm of player i ∈ V +1: Input: for ease of notation, let the actions in AEi be +labeled as 1, 2, . . ., |AEi| +2: initialize cumulative loss Lk ← 0 ∈ RAi for k ∈ +[|AEi|] +3: initialize fixed-point xk ← 0 for k ∈ [|AEi|] +4: initialize counter nk ← 0 for k ∈ [|AEi|] +5: for t = 1, 2, . . ., T do +6: +observe the joint action j ∈ [|AEi|] of the preceding +players +7: +increase counter nj ← nj + 1 +8: +obtain +new +strategy +and +new +fixed-point +(πi(t), xj) ← TSALLIS-INF(nj, Lj, xj) +9: +play action ai(t) ∼ πi(t) +10: +observe the joint bandit-reward rt(a(t)) +11: +update the cumulative loss for all k ∈ [Ai] as +Lj,k ← Lj,k + 1{ai(t) = k}(1 − rt(a(t)))/pk +12: end for +Algorithm 2 Strategy update for player i ∈ V +1: Input: time step t, cumulative rewards L ∈ RAi ++ , pre- +vious fixed point x Output strategy πi(t), fixed point +x +2: set learning rate η ← 2 +� +1/t +3: repeat +4: +pj ← 4(η(Lj − x))−2 for all j ∈ [Ai] +5: +x ← x − +��Ai +j=1 pj − 1 +� +/ +� +η �Ai +j=1 p3/2 +j +� +6: until convergence +7: update strategy πi(t) ← (p1, . . . , pAi) +Next, we go on to analyze the joint pseudo-regret of Algo- +rithm 1. First, we present a result on the pseudo-regret for +the single-player multi-armed bandit problem that will be +used throughout. +3The original TSALLIS-INF Algorithm is given in terms of +losses. To use rewards, one may simply use the relationship l = +1 − r. +Theorem 3.1 (Pseudo-regret of TSALLIS-INF). Consider +a single-player multi-armed bandit problem with A1 arms, +played over T rounds. Let the player operate according to +Algorithm 1. Then, the pseudo-regret satisfies +R(T ) ≤ 4 +� +A1T + 1. +Proof. For a single player, E1 = ∅ and we have |AE1| = +1 by convention. Hence, our setting becomes equivalent +to that of Zimmert & Seldin (2021, Th 1) and the result +follows thereof. +Next, we consider a two-player Stackelberg game with +joint bandit-rewards defined over a two-player clique. We +have the following upper bound on the joint pseudo-regret. +Theorem 3.2 (Joint pseudo-regret over cliques of size 2). +Consider a 2-player Stackelberg game with bandit-rewards, +given by (1), defined over a single clique containing both +players. Furthermore, let each of the players follow Algo- +rithm 1. Then, the joint pseudo-regret satisfies +R(T ) ≤ 4 +� +A1A2T + 4 +� +A1T + A1 + 1. +Proof. Without loss of generality, let player 2 observe the +actions of player 1. Let a1(t) ∈ A1 and a2(t) ∈ A2 denote +the actions of player 1 and player 2, respectively, at time t ∈ +[T ] and let a⋆ +1 and a⋆ +2(a1) denote the reward-maximizing +pure strategies of the players in hindsight, i.e., +a⋆ +1 = arg max +a1∈A1 E +� T +� +t=1 +rt(a1, a⋆ +2(a1)) +� +, +(3) +a⋆ +2(a1) = arg max +a2∈A2 E +� T +� +t=1 +rt(a1, a2) +� +. +(4) +Note that the optimal joint decision in hindsight is given by +(a⋆ +1, a⋆ +2(a⋆ +1)). The joint pseudo-regret is given by +R(T ) = +T +� +t=1 +E [rt(a⋆ +1, a⋆ +2(a⋆ +1)) − rt(a⋆ +1, a2(t))] ++ E [rt(a⋆ +1, a2(t)) − rt(a1(t), a2(t))] +≤ +T +� +t=1 +max +at∈A1 E [rt(at, a⋆ +2(at)) − rt(at, a2(t))] ++ E +� T +� +t=1 +rt(a⋆ +1, a2(t)) − rt(a1(t), a2(t)) +� +. +(5) +Next, let +a+ +1 (t) = arg max +at∈A1 E [rt(at, a⋆ +2(at)) − rt(at, a2(t))] +and let Ta = {t : a+ +1 (t) = a}, for a ∈ A1, denote all the +rounds that player 1 chose action a and introduce Ta = |Ta|. + +Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds +Then, the first term in (5) is upper-bounded as +T +� +t=1 +max +at∈A1 E [rt(at, a⋆ +2(at)) − rt(at, a2(t))] += +� +a∈A1 +� +t∈Ta +E [rt(a, a⋆ +2(a)) − rt(a, a2(t))] +≤ +� +a∈A1 +4 +� +A2Ta + 1 +(6) +≤ +max +� +a Ta=T +� +a∈A1 +4 +� +A2Ta + 1 += 4 +� +A1A2T + A1 +(7) +where (6) follows from Theorem 3.1 and because player 2 +follows Algorithm 1. Note that the actions in Ta may not be +consecutive. However, as we consider adversarial rewards, +Theorem 3.1 is still applicable. +Next, we consider the second term in (5). Note that, accord- +ing to (4), a⋆ +1 is obtained from the optimal pure strategies +in hindsight of both the players. Let +a◦ +1 = arg max +a1∈A1 +T +� +t=1 +E [rt(a1, a2(t))] +and note that +E +� T +� +t=1 +rt(a⋆ +1, a2(t)) +� +≤ E +� T +� +t=1 +rt(a◦ +1, a2(t)) +� +. +By adding and subtracting rt(a◦ +1, a2(t)) to the second term +in (5), we get +E +� T +� +t=1 +rt(a⋆ +1, a2(t)) − rt(a1(t), a2(t)) +� +≤ E +� T +� +t=1 +rt(a◦ +1, a2(t)) − rt(a1(t), a2(t)) +� +≤ 4 +� +A1T + 1 +(8) +where the last equality follows from Theorem 3.1. The re- +sult follows from (7) and (8). +From Theorem 3.2, we note that the joint pseudo-regret +scales with the size of the joint action space as R(T ) = +O(√A1A2T). This is expected as a centralized version +of the cooperative Stackelberg game may be viewed as +a single-player multi-armed bandit problem with A1A2 +arms where, according to Theorem 3.1, the pseudo-regret +is upper-bounded by 4√A1A2T + 1. +Hence, from +Theorem 3.2, we observe a penalty of 4√A1T + A1 +due to the decentralized nature of our setup. +More- +over, in the single-player setting, Algorithm 2 was shown +in Zimmert & Seldin (2021) to achieve the same scaling as +the lower bound in Cesa-Bianchi & Lugosi (2006, Th. 6.1). +Hence, Algorithm 1 achieves the optimal scaling. Next, we +extend Theorem 3.2 to cliques of size larger than two. +Theorem 3.3 (Joint pseudo-regret over a clique of arbitrary +size). Consider a DAG-based game with bandit rewards +given by (1), defined over a single clique containing m +players. Let each of the players operate according to Al- +gorithm 1. Then, the joint pseudo-regret satisfies +R(T ) ≤ 4 +√ +T +m +� +i=1 +i� +k=1 +� +Ak + +m−1 +� +i=1 +i� +k=1 +Ak + 1. +Proof. Let Rub(T, m) denote an upper bound on the joint +pseudo-regret when the bandit-reward is defined over a +clique containing m players. From Theorem 3.1 and Theo- +rem 3.2, we have that +Rub(T, 1) = 4 +� +A1T + 1 +Rub(T, 2) = 4 +� +A1T + 4 +� +A1A2T + A1 + 1, +respectively. Therefore, we form an induction hypothesis +as +Rub(T, m) = 4 +√ +T +m +� +i=1 +i� +k=1 +� +Ak + +m−1 +� +i=1 +i� +k=1 +Ak + 1. (9) +Assume that (9) is true for a clique containing m − 1 +players and add an additional player, assigned player in- +dex 1, whose actions are observable to the original m − 1 +players. +The m players now form a clique C of size +m. +Let a(t) ∈ AC denote the joint action of all the +players in the clique at time t ∈ [T ] and let a−i(t) = +(a1(t), . . . , ai−1(t), ai+1(t), . . . , am(t)) ∈ AC\i denote +the joint action excluding the action of player i. Further- +more, let +a⋆ +1 = arg max +a1∈A1 E +� T +� +t=1 +rt(a1, a⋆ +−1(a1)) +� +a⋆ +−1(a1) = arg max +a∈AC\1 E +� T +� +t=1 +rt(a1, a) +� +denote the optimal actions in hindsight of player 1 and the +optimal joint action of the original m − 1 players given the +action of player 1, respectively. The optimal joint action in +hindsight is given as a⋆ = (a⋆ +1, a⋆ +−1(a⋆ +1)). Following the + +Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds +steps in the proof of Theorem 3.2 verbatim, we obtain +R(T ) = +T +� +t=1 +E [rt(a⋆) − rt(a⋆ +1, a−1(t))] ++ E [rt(a⋆ +1, a−1(t)) − rt(a(t))] +≤ +T +� +t=1 +max +a1 E +� +rt(a1, a⋆ +−1(a1)) − rt(a1, a−1(t)) +� ++ +T +� +t=1 +E [rt(a⋆ +1, a−1(t)) − rt(a1(t), a−1(t))] +≤ +� +a∈A1 +� +t∈Ta +E +� +rt(a, a⋆ +−1(a)) − rt(a, a−1(t)) +� ++ +T +� +t=1 +E [rt(a◦ +1, a−1(t)) − rt(a1(t), a−1(t))] +≤ +� +a∈A1 +Rub(Ta, m − 1) + 4 +� +A1T + 1 +≤ A1Rub(T/A1, m − 1) + 4 +� +A1T + 1 +(10) +where Ta, Ta, and a◦ +n are defined analogously as in the +proof of Theorem 3.2. By using the induction hypothe- +sis (9) in (10) and by accounting for the original m − 1 +players being indexed from 2 to m, we obtain +R(T ) ≤ A1 +� +4 +� +T/A1 +m +� +i=2 +i� +k=2 +� +Ak + +m−1 +� +i=2 +i� +k=2 +Ak + 1 +� ++ 4 +� +A1T + 1 += Rub(T, m) +which is what we wanted to show. +As in the two-player game, +the joint pseudo-regret +of +Algorithm 1 +achieves the +optimal scaling, +i.e., +R(T ) += +O( +√ +T �m +k=1 +√Ak), but exhibits a penalty +due to the decentralized setting which is equal to +4 +√ +T �m−1 +i=1 +�i +k=1 +√Ak + �m−2 +i=1 +�i +k=1 Ak. +Up until this point, we have considered the pseudo-regret +when the bandit-reward (1) is defined over a single clique. +The next theorem leverages the previous results to provide +an upper bound on the joint pseudo-regret when the bandit- +reward is defined over an arbitrary number of independent +cliques in the DAG. +Theorem 3.4 (Joint pseudo-regret in DAG-based games). +Consider a DAG-based game with bandit rewards given as +in (1) and let C contain a collection of independent cliques +associated with the DAG. Let each player operate accord- +ing to Algorithm 1. Then, the joint pseudo-regret satisfies +R(T ) = O +�� +T max +k∈[|C|] |ANk| +� +where ANk denotes the joint action-space of the players in +the kth clique Nk ∈ C. +Proof. Let Nk ∈ C denote the players belonging to the kth +clique in C with joint action space ANk. The structure of (1) +allows us to express the joint pseudo-regret as +R(T ) = E +� T +� +t=1 +rt(a⋆) − rt(a(t)) +� +≤ +|C| +� +k=1 +βkE +� T +� +t=1 +rk +t (a⋆ +k) − rk +t (P k(a(t))) +� +(11) +where +a⋆ = arg max +a∈AV E +� T +� +t=1 +rt(a) +� +, +a⋆ +k = arg max +a∈ANk +E +� T +� +t=1 +rk +t (a) +� +, +and the inequality follows since E +��T +t=1 rk +t (P k(a⋆)) +� +≤ +E +��T +t=1 rk +t (a⋆ +k) +� +. Now, for each clique Nk ∈ C, let the +player indices in Nk be ordered according to the order of +player observations within the clique. +As Theorem 3.3 +holds for any Nk ∈ C, we may, with a slight abuse of nota- +tion, bound the joint pseudo-regret of each clique as +R(T ) ≤ +|C| +� +k=1 +βkRub(T, Nk) ≤ max +k∈[|C|] βkRub (T, Nk) +where Rub(T, Nk) follows from Theorem 3.3 as +Rub(T, Nk) = 4 +√ +T +� +i∈Nk +� +j≤i,j∈Nk +� +Aj ++ +� +i∈N − +k +� +j≤i,j∈N − +k +Aj + 1 +where N − +k excludes the last element in Nk. The result fol- +lows as Rub(T, Nk) = O( +� +T |ANk|) where the βk has +been consumed in the prefactor. +4. Numerical results +The experimental setup in this section is inspired by the +socio-economic simulation in (Zheng et al., 2020).4 +We +consider a simple taxation game where one player acts as +a socio-economic planner and the remaining M players act +4The source code of our experiments is available on +https://anonymous.4open.science/r/bandit_optimization_dag-242C/. + +Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds +as workers that earn an income by performing actions, e.g., +constructing houses. The socio-economic planner divides +the possible incomes into N brackets where [βi−1, βi] de- +notes the ith bracket with β0 = 0 and βN = ∞. In each +round t ∈ [T ], the socio-economic planner picks an action +ap(t) = (ap,1(t), . . . , ap,N(t)) that determines the taxation +rate where ap,i(t) ∈ Ri denotes the the marginal taxation +rate in income bracket i and Ri is a finite set. We use the +discrete set Ap = �N +i=1 Ri of size Ap to denote the action +space of the planner. +In each round, the workers observe the taxation policy +ap(t) ∈ Ap and choose their actions consecutively, see +Fig. 3. Worker j ∈ [M] takes actions aj(t) ∈ Aj where +Aj is a finite set. A chosen action aj(t) ∈ Aj translates +into a tuple (xj(t), ˜lj(t)) consisting of a gross income and +a marginal default labor cost, respectively. Furthermore, +each worker has a skill level sj that serves as a divisor of +the default labor, resulting in an effective marginal labor +lj(t) = ˜lj(t)/sj. Hence, given a common action, high- +skilled workers exhibit less labor than low-skilled workers. +The gross income xj(t) of worker j in round t is taxed ac- +cording to ap(t) as +ξ(xj(t)) = +N +� +i=1 +ap,i(t)(βi − βi−1)1{xj(t) > βi} ++ (xj(t) − βi−1)1{xj(t) ∈ [βi−1, βi]} +where ap,i(t) is the taxation rate of the ith income bracket +and ξ(xj(t)) denotes the collected tax. Hence, worker j’s +cumulative net income zj(t) and cumulative labor ℓj(t) in +round t are given as +zj(t) = +t +� +u=1 +xj(u) − ξ(xj(u)), +ℓj(t) = +t +� +u=1 +lj(u). +In round t, the utility of worker j depends on the cumula- +tive net income and the cumulative labor as +rj +t (zj(t), ℓj(t)) = (zj(t))1−η − 1 +1 − η +− ℓj(t) +(12) +where η > 0 determines the non-linear impact of income. +An example of the utility function in (12) is shown in Fig. 4 +for η = 0.3, income xj(t) = 10, and a default marginal la- +bor ˜lj(t) = 1 at different skill levels. It can be seen that the +utility initially increases with income until a point at which +the cumulative labor outweighs the benefits of income and +the worker gets burnt out. +We consider bandit-rewards defined with respect to the +Socio-economic planner +1 +2 +3 +workers +Figure 3. Socio-economic setup with 4 players among which 3 are +designated workers. +100 +101 +102 +103 +104 +0 +200 +400 +600 +800 +1,000 +houses built +worker utility +s = 1 +s = 2 +s = 3 +Figure 4. Example of utility functions for different skill levels +when xj(t) = 10 and ˜ℓj(t) = 1. +worker utilities and the total collected tax as +rt(ap(t), a1(t), . . . , aM(t)) = +1 +(M + 1) + + +M +� +j=1 +wrj +t (zj(t), ℓj(t)) + wp +M +� +j=1 +ξ(xj(t)) + + +(13) +where the weights trade off worker utility for the col- +lected tax and satisfy Mw + wp += +M + 1. +The +individual rewards are all normalized to [0, 1], hence, +rt(ap(t), a1(t), . . . , aM(t)) ∈ [0, 1]. +For the numerical experiment, we consider N = 2 in- +come brackets where the boundaries of the income brackets +are {0, 14, ∞} and the socio-economic planner chooses a +marginal taxation rate from R = {0.1, 0.3, 0.5} in each +income bracket, hence, Ap = 9. We consider M = 3 work- +ers with the same action set A of size 3. Consequently, +the joint action space is of size 243. Furthermore, we let +the skill level of the workers coincide with the worker in- +dex, i.e., sj = j for j ∈ [M]. +Simply, workers able +to observe others have higher skill. The worker actions +translate to a gross marginal income and a marginal la- +bor as aj(t) → (xj(t), lj(t)) where xj(t) = 5aj(t) and + +Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds +lj(t) = aj(t)/sj for aj(t) ∈ {1, 2, 3}. Finally, we set +η = 0.3 and let w = 1/M and wp = M to model a sit- +uation where the collected tax is preferred over workers’ +individual utility. +The joint pseudo-regret of the socio-economic simulation is +illustrated in Fig. 5 and Fig. 6 (different scales) along with +the upper bound in Theorem 3.4. We collect 100 realiza- +tions of the experiment and, along with the pseudo-regret +R(T ), two standard deviations are also presented. It can +be seen that the players initially explore the action space +and are able to eventually converge on an optimal strategy +from a pseudo-regret perspective. The upper bound in the +figures is admittedly loose and does not exhibit the same +asymptotic decay as the simulation due to different con- +stants in the scaling law, see Fig. 6. However, it remains +valuable as it provides an asymptotic no-regret guarantee +for the learning algorithm. +100 +101 +102 +103 +104 +105 +106 +0.1 +0.2 +0.3 +0.4 +0.5 +T +Regret +Upper bound +R(T ) +Figure 5. Pseudo regret vs the upper bound in Theorem 3.4 (linear +scale). +5. Conclusion +We have studied multiplayer games with joint bandit- +rewards where players execute actions consecutively and +observe the actions of the preceding players. +We intro- +duced the notion of joint pseudo-regret and presented an +algorithm that is guaranteed to achieve no-regret for adver- +sarial bandit rewards. A bottleneck of many multi-agent +algorithms is that the complexity scales with the joint ac- +tion space (Jin et al., 2021) and our algorithm is no ex- +ception. An interesting venue of further study is to find +algorithms that have more benign scaling properties, see +e.g., (Jin et al., 2021; Daskalakis et al., 2021). +Further- +more, recent results on correlated multi-armed bandits have +demonstrated that multi-armed bandits with many arms +may become significantly more feasible if one is able to +101 +102 +103 +104 +105 +106 +10−3 +10−2 +10−1 +100 +101 +102 +T +Regret +Upper bound +R(T ) +Figure 6. Pseudo regret vs the upper bound in Theorem 3.4 (log- +scale). +exploit dependencies among arms (Gupta et al., 2021). It +would be interesting to explore how the scaling of our algo- +rithm is affected by modelling and exploiting dependencies +among players. +References +Auer, P. and Chiang, C.-K. 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Journal of +Machine Learning Research, 22(28):1–49, 2021. + diff --git a/DtFKT4oBgHgl3EQfZS4v/content/tmp_files/load_file.txt b/DtFKT4oBgHgl3EQfZS4v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77fef9514ab764ec93a2b3f38ca2d1e45d07427f --- /dev/null +++ b/DtFKT4oBgHgl3EQfZS4v/content/tmp_files/load_file.txt @@ -0,0 +1,600 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf,len=599 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content='11802v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content='LG] 27 Jan 2023 Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds Johan ¨Ostman 1 Ather Gattami 1 Daniel Gillblad 1 Abstract We consider a decentralized multiplayer game, played over T rounds, with a leader-follower hi- erarchy described by a directed acyclic graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' For each round, the graph structure dictates the order of the players and how players observe the actions of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' By the end of each round, all players receive a joint bandit-reward based on their joint action that is used to update the player strategies towards the goal of minimiz- ing the joint pseudo-regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' We present a learn- ing algorithm inspired by the single-player multi- armed bandit problem and show that it achieves sub-linear joint pseudo-regret in the number of rounds for both adversarial and stochastic ban- dit rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' Furthermore, we quantify the cost incurred due to the decentralized nature of our problem compared to the centralized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' Introduction Decentralized multi-agent online learning concerns agents that, simultaneously, learn to behave over time in order to achieve their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' Compared to the single-agent setup, novel challenges are present as agents may not share the same objectives, the environment becomes non- stationary, and information asymmetry may exist between agents (Yang & Wang, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' Traditionally, the multi- agent problem has been addressed by either relying on a central controller to coordinate the agents’ actions or to let the agents learn independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' However, access to a central controller may not be realistic and indepen- dent learning suffers from convergence issues (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' To circumvent these issues, a common approach is to drop the central coordinator and allow informa- tion exchange between agents (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' Cesa-Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' Decision-making that involves multiple agents is often 1AI Sweden, Gothenburg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'} +page_content=' Correspondence to: Johan ¨Ostman 1, this probability can be extended using the Markov property: +Pr(H | tin +i,n, ri,n = m) = Pr(I1, I2, ..., IJi,n,m | tin +i,n, ri,n = m) += Pr(I1 | tin +i,n, ri,n = m) · +Ji,n,m +� +j=2 +Pr(Ij | Ij−1, tin +i,n, ri,n = m) +(12) +In the following contents, we elaborate the derivation of Pr(I1 | tin +i,n, ri,n = m) and Pr(Ij | Ij−1, tin +i,n, ri,n = +m), respectively. +Note that Pr(I1 | tin +i,n, ri,n = m) is the probability of boarding Train I1 on the first segment of path +m. There are two different scenarios for this event to happen: 1) [No left behind] the passenger arrives at +the platform between the departure time of Trains I1 and I1 − 1 and boards Trains I1 without left behind. +2) [With left behind] the passenger arrives at the platform between the departures of Trains I1 − k and +I1 −k−1 and is able to board Train I1 after being left behind k times. Given the feasible itinerary set Ωi,n,m, +there is a maximum number of times the passenger is left behind to board Train I1. Denote the upper bound +of k as BI1 +i,n,m, where BI1 +i,n,m = arg maxk{k ∈ N | ∃ H′ ∈ Ωi,n,m s.t. I′ +1 = I1 − k, I′ +1 ∈ H′}, N is the set +of natural numbers (including zero). And Train I1 − BI1 +i,n,m represents the earliest train that passenger i can +board at the first segment. +Let tj +i,n,m be the walking time from the alighting platform of segment j − 1 to the boarding platform +of segment j in path m for passenger i, and t0 +i,n,m is the access walking time. Then, tin +i,n + t0 +i,n,m is the +passenger arrival time at the platform of his/her origin station. Hence, the probability of arriving at the +platform between the departure of Train I1 − k and I1 − k − 1 can be formulated as +Pr(Td(I1 − k − 1) ≤ tin +i,n + t0 +i,n,m ≤ Td(I1 − k) | tin +i,n, ri,n = m) = +� Td(I1−k)−tin +i,n +Td(I1−k−1)−tin +i,n +fAc +m (t)dt := ρI1,k +i,n,m +∀k = 0, 1, .., BI1 +i,n,m +(13) +where fAc +m (·) is the access walking time PDF for path m. Eq. 13 can be pre-calculated once fAc +m (·) is given +because it is a definite integral. +Let ηj,k +i,n,m be the probability of being left behind k times at the boarding station of the j-th segment +of path m for passenger i’s trip n. Let EIj,k +i,n,m be the event that “passenger i in the n-th trip arrives at the +boarding station of segment j of path m between the departure of Train Ij − k and Ij − k − 1 and is left +behind k times to board Train Ij”. We have: +Pr(EI1,k +i,n,m| tin +i,n, ri,n = m) = ρI1,k +i,n,m · η1,k +i,n,m +∀k = 0, 1, .., BI1 +i,n,m. +(14) +Then, Pr(I1 | tin +i,n, ri,n = m) can be rewritten as +Pr(I1 | tin +i,n, ri,n = m) = +BI1 +i,n,m +� +k=0 +Pr(EI1,k +i,n,m| tin +i,n, ri,n = m) = +BI1 +i,n,m +� +k=0 +ρI1,k +i,n,m · η1,k +i,n,m. +(15) +This finishes the derivation of Pr(I1 | tin +i,n, ri,n = m). η1,k +i,n,m can be estimated from AFC and AVL data +10 + +using a Gaussian Mixture model (Ma et al., 2019), which will be described in Appendix A. +Now, we derive Pr(Ij | Ij−1, tin +i,n, ri,n = m) in Eq. 12, the probability of boarding train Ij given that +the passenger has boarded train Ij−1 on the (j − 1)-th segment of path m. It is derived in a similar way as +Pr(I1 | tin +i,n, ri,n = m). Passenger i may arrive at the boarding station of segment j between the departure +times of Trains Ij − k and Ij − k − 1 and be left behind k times to board Train Ij (note that k = 0 means no +left behind). The probability of arriving at the platform between the departures of train Ij −k and Ij −k −1 +given he/she alights at Ta(Ij−1) is formulated as +Pr(Td(Ij − k − 1) ≤ Ta(Ij−1) + tj +i,n,m ≤ Td(Ij − k) | Ij−1, tin +i,n, ri,n = m) += +� Td(Ij−k)−Ta(Ij−1) +Td(Ij−k−1)−Ta(Ij−1) +fTr +m,j(t)dt = ˜ρIj,k +i,n,m +∀k = 0, 1, .., BIj +i,n,m. +(16) +where BIj +i,n,m is the maximum possible left behind times when boarding train Ij given the feasible itinerary +constraint, defined as arg maxk{k ∈ N | ∃H′ ∈ Ωi,n,m s.t. I′ +j = Ij − k, I′ +j ∈ H′}. tj +i,n,m is the transfer +walking time from the alighting of train Ij−1 to the next platform. fTr +m,j(·) is the PDF of tj +i,n,m. ˜ρIj,k +i,n,m is +defined for the simplicity of expression. Given the definition of EIj,k +i,n,m, we have +Pr(EIj,k +i,n,m| Ij−1, tin +i,n, ri,n = m) = ˜ρIj,k +i,n,m · ηj,k +i,n,m +∀k = 0, 1, .., BIj +i,n,m. +(17) +Then, Pr(Ij | Ij−1, tin +i,n, ri,n = m) can be rewritten as +Pr(Ij | Ij−1, tin +i,n, ri,n = m) = +B +Ij +i,n,m +� +k=0 +Pr(EIj,k +i,n,m| Ij−1, tin +i,n, ri,n = m) = +B +Ij +i,n,m +� +k=0 +˜ρIj,k +i,n,m · ηj,k +i,n,m. +(18) +With all parts of L(θ, β, σ) in Equation 8 derived, there are still two remaining challenges for the MLE +problem. First, the calculation of Pr(tout +i,n | tin +i,n, ri,n) requires the inputs of left behind probability ηj,k +i,n,m and +three PDF functions fAc +m (·), fEg +m (·), and fTr +m,j(·). The PDF functions can be obtained from field-experiments. +But obtaining the left behind probability is not trivial. In this study, we used the model proposed by Ma et al. +(2019) to estimate ηj,k +i,n,m from AFC and AVL data. Details can be found in Appendix A. The second challenge +is that, the calculation of Pr(vi) (Eq. 6) has an exponentially large number of summation over different paths, +and it requires the integral of a normally distributed random variable, which makes it numerically hard to +solve. In the following section, we derive a new conditional probability-based formulation to eliminate the +large number of summations, and use a numerical integration approach for the normal random variable, +which leads to a tractable likelihood function and enables an efficient model estimation. +2.3. Tractable log-likelihood function +To eliminate the exponentially large number of summation over different paths in Eq. 6, we observe that +given αk +i and gi, passenger i’s route choice for each trip becomes independent. Mathematically, +Pr(vi | αk +i , gi = Gk) = +Ni +� +n=1 +Pr(tout +i,n, tin +i,n | αk +i , gi = Gk) = +Ni +� +n=1 +� +mn∈Ri,n +Pr(tout +i,n | tin +i,n, ri,n = mn) · πk +i,n,mn[αk +i ] +(19) +11 + +Note that Eq.19 only has a total of Ni×|Ri,n| summation terms, while this number in Eq.6 is |Ri,n|Ni. Based +on Eq. 19, Pr(vi) can be obtained by integrating and summing over αk +i and gi, respectively. Since αk +i is a +normal random variable, an approximated numerical integration approach is used to get a tractable formula. +Note that there are a large class of quadrature rules for numerical integration (Davis and Rabinowitz, 2007). +In this paper, we use the simplest midpoint rule for the interpolation as this is not the focus of this study. +Let αU and αL be the upper and lower bounds for αk +i . We divide [αL, αU] into discrete intervals with equal +length ∆. Let S be the set of all middle points in each interval. Specifically, S = {αL + k·∆ +2 +| ∀k = +1, 3, 5, ..., |S|, |S| = 2(αU−αL) +∆ +− 1}. Hence, Pr(vi) can be rewritten as +Pr(vi) ≈ +K +� +k=1 +Pr(gi = Gk) · +� +αk +i ∈S +Pr(vi | αk +i , gi = Gk) · f(αk +i ) · ∆ +(20) +∆ is the parameter determining the trade-off between approximation accuracy and computational efficiency, +where a smaller ∆ indicates a more fine-grained integration, but higher computational cost. +Given the new formulation of Pr(vi), we can use L(θ, β, σ) = � +i∈P Pr(vi) to evaluate the likelihood +function with a tractable formulation. +2.4. Model estimation +The new log-likelihood function can be expressed as +LL(θ, β, σ) = +� +i∈P +log Pr(vi) = +� +i∈P +log +� +� +K +� +k=1 +� +αk +i ∈S +Pr(gi = Gk) · Pr(vi | αk +i , gi = Gk) · f(αk +i ) · ∆ +� +� (21) +As LL(θ, β, σ) is a combination of elementary functions, it is continuous and differentiable. Therefore, the +MLE can be solved with any first- or second-order optimization method. In this study, the BFGS algorithm +is used (Nocedal and Wright, 2006). BFGS is a quasi-Newton method. It uses only the first derivatives and +has demonstrated good performance for many optimization problems. However, as the function includes +the multiplication of several nonlinear terms, the convexity of this function is unknown. It is possible that +the LL is not concave and the BFGS algorithm may converge to different local minimums under different +initializations. Hence, we conduct a sensitivity analysis in Section 3.1 with respect to different initial values +and show that the model estimation results are stable. Besides, the numerical results show that LL is concave +within a reasonable range of path attribute values. +After obtaining the optimal parameters θ∗, β∗, and σ∗, we calculate the t-values of the estimated +parameters based on a numerically estimated Hessian matrix and the Cramer-Rao bound. Note that as LL +is second-order differentiable, the analytical Hessian matrix can also be derived. The numerical Hessian +matrix is used for simplification due to the complex function form. In this study, we adopt the formulation +with fourth-order approximation under uniform grid spacing to calculate the second derivative (Fornberg, +1988). The exact formulas are attached in Appendix B (other approximation formulas can also be used). +With the second derivative formulas, we can calculate the numerical Hessian matrix of LL(θ, β, σ) at point +(θ∗, β∗, σ∗). Denote the Hessian matrix as ˆH. Note that, from the second-order optimality conditions, ˆH is +negative semi-definite, which is the algebraic equivalent of the local concavity of the log-likelihood function +(Bierlaire, 2020). +12 + +Let Θ = (θ, β, σ) be a vector of all parameters. Using the Cramer-Rao bound (Cramér, 2016; Rao, +1992), the variance of an estimated parameter ˆΘk is +Var[ˆΘk] = − ˆH−1 +k,k +(22) +where ˆH−1 is the inverse of the Hessian matrix and ˆH−1 +k,k is its k-th diagonal element. Then, the corresponding +t-value is calculated as: +t-value[ˆΘk] = +ˆΘk +� +Var[ˆΘk] +(23) +3. Case study +3.1. Model validation and sensitivity test +It is difficult to collect passengers’ actual path choices in reality. To validate the proposed approach, we +use synthetic data generated by simulating passengers’ route choices, train operations, and their interactions +(Mo et al., 2022; Zhu et al., 2021; Mo et al., 2020a). +Figure 3 shows the configuration of the synthetic urban rail network, where there are 7 stations (A∼G) +and 3 lines (red, green, and blue). The number on each link represents the in-vehicle travel time. This +network is extracted from the MTR metro system in Hong Kong. It is also representative of typical metro +network structures in terms of lines and transfers. The platform of station C of the red line in the up direction +is assumed to be crowded with extensive left behind. All the other platforms are assumed to have no left +behind. +Figure 3: Synthetic urban rail network +To generate the synthetic data, we assume that passengers’ path choice behavior is based on four path +attributes: 1) in-vehicle time (i.e., the train run time of a path), 2) out-of-vehicle time (i.e., the sum of access, +egress, transfer, and waiting time without left behind), 3) the number of transfers (i.e., the number of times +transferring on the path), and 4) denied waiting time (i.e., the waiting time due to left behind at the crowded +13 + +Gplatforms). +We also assume passenger’s latent groups can be characterized by two sociodemographic +variables x(1) and x(2), where x(1) is drawn from U[−4, 4] and x(2) is drawn from U[−2, 2]. Suppose there +are two latent groups for the synthetic passengers: “time-sensitive” (TS) and “comfort-aware” (CA). The TS +passengers, when making path choices, tend to minimize their total travel time, meaning that the impact of +in-vehicle time, out-of-vehicle time, and denied waiting time are similar to passenger’s path choice utility. +CA passengers prefer paths with less walking or waiting time though the in-vehicle time could be longer. +That is, the out-of-vehicle time and denied waiting time have a higher impact on these passengers’ path +choice utilities than the in-vehicle time. +Table 3 shows the parameters for the latent class path choice model for generating the synthetic data. +These parameters are chosen based on the survey results in Jin et al. (2017). We set the in-vehicle time +and path size factor parameters to be the same for TS and CA passengers according to the survey modeling +results. In addition, we set the choice parameters to be 0.7× (1.5×) of the parameters from the survey results +for the TS (CA) group. The parameters for the latent class model are set as 1.5 and 0.6 for x(1) and x(2), +respectively. +Table 3: Path choice parameters for synthetic data generation +Variables +TS +CA +Jin et al. (2017) +Path choice parameters +In-veh time +-0.2676 +-0.2676 +-0.2676 +Out-of-veh time +-0.2980 +-0.6386 +-0.4257 +Num of transfer +-1.3068 +-3.1737 +-1.8669 +Denied waiting time +-0.3222 +-0.7825 +-0.4603 +log PSm +0.5815 +0.5815 +0.5815 +σk +1 +1 +N/A +Latent class parameters +x(1) +1.5 +x(2) +0.6 +Table 4 summarizes the parameters of the network, train operations, and passengers. The synthetic data +is generated for 9 OD pairs (origin stations A, B, D, and destination stations E, F, G) by simulating the tap-out +time given tap-in time. All the OD pairs have 2 paths. For example, the possible paths for OD pair (A, +E) are A-B-E and A-B-C-E. Without loss of generality, we assume that there are 2,700 passengers, each of +whom performs 3 trips (i.e., Ni = 3). Each trip is associated with a randomly selected OD pair. Algorithm +1 describes the detailed synthetic data generation procedure. +14 + +Table 4: System settings for synthetic data generation +Entity +Settings +Network +Walk distance 30-50 meters (access, egress, transfer) +Train operations +Headway 2+δH minutes. +δH is drawn uniformly from [−10, 10] seconds +In-vehicle time+δV (see Figure 3) +δV is drawn uniformly from [−20, 20] seconds +Passengers +Walk speed distribution follows a lognormal distribution +with mean of 1.2m/s and standard deviation of 0.5m/s. +Left behind probabilities at station C, red line, up +direction are 20% no left behind, 50% left behind once +and 30% left behind twice +Algorithm 1 Synthetic data generation +Input: Path choice parameters; Passenger set P. +Output: Synthetic AFC data (tap-in/tap-out stations/times) +1: Initialize the number of sample instances N. +2: for i ∈ P do +3: +Sample x(1) +i +∼ U[−4, 4], x(2) +i +∼ U[−2, 2]. +4: +Sample group gi ∼ Pr(gi; θ). Denote the group as Gk. +5: +Sample αk +i ∼ N(0, θk). +6: +for n = 1 to Ni do +7: +Sample tin +i,n ∼ U[7:00AM, 10:00AM]. +8: +Randomly sample an OD pair for this trip. +9: +Calculate the path choice probability πk +i,n,m[αk +i ]. +10: +Sample a path m ∈ Ri,n based on πk +i,n,m[αk +i ]. +11: +Sample the actual travel information (i.e., train run time, headway, access, egress, transfer, and denied +waiting times) for this trip based on path m. Obtain tout +i,n. +12: +Save tin +i,n and tout +i,n and the OD as a trip record. +13: Combine all trip records as the synthetic AFC data. +In total, 8,100 trips from the 2,700 passengers were generated. The synthetic AFC data is then used for +model estimation. As we have the “true” value of choice parameters (Table 3), we can validate the model’s +performance. The MLE is solved using the BFGS algorithm (Fletcher, 2013) in the Python Scipy package. +αL = −3, αU = 3, and ∆ = 1 are used for numerical integration. +Table 5 shows the estimation results of the path choice parameters. The percentage values in the brackets +quantify the relative errors compared to the “true” parameters. Note that the actual walking speed distribution +and left behind probabilities are used in the model estimation. And the sensitivity analysis on these inputs +is shown in Section 3.1. For comparison purposes, we also estimate a baseline model without latent classes. +The latent class model can estimate the actual parameters with a mean percentage error of around 10%. It +outperforms the baseline model in estimation accuracy. The out-of-vehicle time parameter for the TS group +has the maximum error (-33.2%), which may be due to the fact that the out-of-vehicle time is highly correlated +with the number of transfers, making the numerical estimation harder. Note that as the absolute values for +these parameters are relatively small, the absolute errors of the estimated parameters are acceptable. +15 + +In terms of the goodness-of-fit, the initial log-likelihood (denoted as LL0) for the null model (with all +parameters zero) is −52, 219.17, the final latent-class model log-likelihood (denoted as LL∗) is −51, 526.13, +and the final baseline model log-likelihood (denoted as LLB) is −51, 571.67. We conduct the log-likelihood +ratio test (Wilks, 1938) and obtain the statistic χ2 = −2(LLB − LL∗) = 91.08, which suggests a p-value +of 0 given 5 degrees-of-restrictions (i.e., number of parameters of latent class model minus that of baseline +model). +This indicates that the latent-class model specification is significantly better than the baseline +model. We also calculate the log-likelihood with “true” parameters (referred to as LLTrue-para) and the +value is −51, 535.16. It is smaller than the LL∗, which means that the estimated parameters have a better +goodness-of-fit than the “true” parameters. This suggests that the estimation errors may mostly come from +random errors in the data generation process, instead of the model estimation. +All parameters have absolute t-values greater than 1.96, showing significant impacts of these param- +eters on passengers’ path choices. This is reasonable because the synthetic data are generated with those +parameters. We also observe that the in-vehicle time shows the highest significance compared to other cost +parameters, which is consistent with survey results (Jin et al., 2017). +Table 5: Estimation results for the synthetic data +Variables +Latent class +Baseline +Estimation (Error) +t-value +Estimation (Error) +t-value +Choice model +In-veh time +-0.2599 (-2.9%) +-10.62 +-0.2254 (-15.8%) +-10.03 +TS: Out-of-veh time +-0.1991 (-33.2%) +-2.88 +-0.3010 (1.0%) +-6.88 +TS: Num of transfer +-1.3327 (+1.9%) +-4.91 +-1.8777 (+43.7%) +-16.77 +TS: Denied waiting time +-0.3789 (+17.6%) +-6.18 +-0.4901 (+52.1%) +-18.80 +CA: Out-of-veh time +-0.5419 (-15.14%) +-3.82 +-0.3010 (-52.9%) +-6.88 +CA: Num of transfer +-2.8071 (-11.5%) +-4.98 +-1.8777 (-40.8%) +-16.77 +CA: Denied waiting time +-0.7298 (-6.7%) +-6.35 +-0.4901 (-37.4%) +-18.80 +log PSm +0.6758 (+16.2%) +12.72 +0.6054 (+4.1%) +12.31 +σk +0.9191 (-8.1%) +11.05 +0.9122 (-8.8%) +14.40 +Latent group model +x(1) +0.6213 (+3.6%) +4.99 +N/A +N/A +x(2) +1.8637 (+24.3%) +7.01 +N/A +N/A +Number of passengers: 2,700. Number of observations: 8,100 +LL0: -52,219.17; LL∗: -51,526.13; LLB: -51,571.67; LLTrue-para: -51,535.16 +χ2 = −2(LLB − LL∗) = 91.08, likelihood ratio test p-value: 0 +To further validate the model performance, sensitivity analysis was conducted to explore the impacts +of parameter initialization on the model’s performance. Moreover, we also evaluate whether the inaccurate +estimation of walking speed distribution and left behind distribution would affect the model’s performance. +Figure 4 shows the sensitivity analysis on the initialization of the parameters. A total of 20 experiments +are conducted. In each experiment, the initial values of all parameters are drawn uniformly from U[−5, 5]. +We observe that the final estimated parameters all converged to the same values regardless of initialized +parameter values, showing the estimation robustness against the parameter initialization. +16 + +Figure 4: Estimated parameters with different initializations +Figure 5 illustrates the LL value as a function of variable values. The log-likelihood function is concave +around the optimal values1, which further indicates that the estimation results are robust. +(a) LL vs. in-veh time and out-of-veh time (TS) +(b) LL vs. x(1) and x(2) +Figure 5: Log-likelihood function surface +Figure 6 shows the model estimation results with respect to different inputs of walking speeds. We +evaluates the model’s robustness with respect to errors in walking speed distribution because the estimation +passenger’s walking speed may not be accurate in the real world. Let µWS and σWS be the actual walking speed +mean and standard deviation when generating the synthetic data (i.e., µWS = 1.2m/s and σWS = 0.5m/s). +When estimating the model, we set the speed distribution parameters as (Γ1 · µWS, Γ2 · σWS), where +1Due to space limitations, we only show the function curves with respect to four variables +17 + +2 + parameters +1 +Y +Y +Y +丫 +Y +Y +Y ++(1) ++(2) +In-veh time +logPSm +0 +estimated +人人 +丫 +人人 +人人 +gk +<人 +《人 +人 +Out-of-veh time (TS) +. +. +. +Num of transfer (TS) +.1 +Denied waiting time (TS) +of +Out-of-veh time (CA) +Values +Num of transfer (CA) +Denied waiting time (CA) +2 + 1７ 18 19 20 +X +Random initialization ID-51600 +-51700 +-51800 +-51900 +-52000 +-52100 +-1.0 +-0.8 +0.6 +0.0 +0.2 +-0.4 +-0.4 +0.2 +0.6 +0.8 +0.0 +-1.0 +In-veh time (TS)51535.0 +-51537.5 +51540.0 +-51542.5 +-51547.5 +-51550.0 +51552.5 +0.0 +0.2 +51555.0 +0.4 +2.0 ++0.6 +1.8 +1.6 +0.8 +1.4 +1.0 +1.2 +1.0 ++(2)Γ1, Γ2 ∈ {0.8, 1, 1.2}, which represents different perturbations in the speed parameter inputs (Γ1 = Γ2 = 1 +means no errors). Figure 6 shows that the variability of the walking speed distribution does not show much +impact on the model performance. +Figure 6: Sensitivity analysis on walking speed inputs +Figure 7 shows the estimation results with respect to different input left behind probabilities at Station C, +red line, up direction, which indicates the model’s performance if there are estimation errors in left behind +probabilities. Similarly, left behind probabilities are chosen for sensitivity analysis because they are values +estimated from data and may suffer from errors. Three scenarios are compared: actual crowding (20% no +left behind, 50% left behind once and 30% left behind twice, which means no errors), less crowding (80% +no left behind, 20% left behind once and 0% left behind twice), and more crowding (10% no left behind, +20% left behind once and 70% left behind twice). It can be seen that the parameters of in-vehicle time, x(1), +and x(2) are not sensitive to the left behind inputs. But other parameters (such as the number of transfers and +out-of vehicle time) are highly affected. The reason may be that errors in left behind estimation can affect +the model’s evaluation of other factors’ impacts on travel time. That is, an additional 10-minute trip time +can be caused by more transfers, or high out-of vehicle time, or left behind. If left behind is not estimated +accurately, the impact of other factors on total travel time (and passenger choices) may be biased. Hence, +the results highlight the importance of incorporating crowding and capacity constraints in the estimation of +path choices. +18 + +2 +i = 1, 「2 = 1 +「1 = 0.8, 「2 = 1 +Values of estimated parameters +1 = 1.2, 「2 = 1 +「1 = 1, 「2 = 0.8 +「1 = 1,「2 = 1.2 +0 +-3 +In-veh time +K +Out-of-veh time (TS) +Num of transfer (TS) +Denied waiting time (TS) +Out-of-veh time (CA) +Num of transfer (CA) +Denied waiting time (CA) +9 +X +ParametersFigure 7: Sensitivity analysis on left behind inputs +3.2. MTR empirical case study +The proposed method is also applied using actual AFC and AVL data from the Hong Kong MTR network. +Figure 8 shows the MTR network and select OD pair areas (origins in the black dashed box and destinations +in the red). These OD pairs are selected because 1) there are multiple paths between each OD pair, which +supports the application of the path choice modeling, and 2) these stations have high enough OD passenger +flows to allow the estimation of the left behind probability distribution (Ma et al., 2019). +We randomly select 3,425 passengers with trips between these OD pairs. We consider trips with departure +times in the evening peak (5:30 PM - 7:30 PM). Finally, a total of 6,425 trips were collected from the AFC +data in July 2018 (i.e., on average each passenger had 1.88 trips). The walking time is assumed to follow +the log-normal distribution with mean and variance calibrated by MTR employees. αL = −9, αU = 9, and +∆ = 1 are used for numerical integration. +19 + +2 +Actual crowding +Less crowding +Values of estimated parameters +More crowding +0 +-1 +-2 +-3 +In-veh time +logPSm +(TS) +Num of transfer (TS) +Denied waiting time (TS) +Out-of-veh time (CA) +Num of transfer (CA) +Denied waiting time (CA) +X +X +Out-of-veh time ( +ParametersFigure 8: Hong Kong MTR network +Table 6: Descriptive statistics of the MTR data +Variables +Mean +Std. Dev. +Max +Min +Individual characteristics +Avg # travel days per week +4.31 +2.06 +7.00 +0.12 +Std. of 1st trip dept. time (hr) +0.68 +0.51 +3.54 +0.01 +Min # stations with 70% trips +3.22 +0.15 +15 +1 +If student (Yes = 1) +0.10 +0.30 +1 +0 +Path attributes +In-veh time (min) +34.0 +15.5 +85.8 +3.50 +Out-of-veh time (min) +9.20 +1.11 +24.6 +0.50 +Denied waiting time (min) +1.03 +2.48 +16.3 +0.0 +Number of passengers: 3,425. Number of trips: 6,425 +We assume passenger’s latent groups can be characterized by the following attributes, readily extracted +from AFC data: 1) travel frequency, defined as the average number of days with travel per week, 2) schedule +flexibility, measured by the standard deviation of the first trip’s departure time on weekdays, 3) spatial +concentration of trips, defined as the minimum number of stations that covers 70% of trips in a month, 4) +whether the cardholder is a student or not (dummy variable). All these attributes are calculated based on +the AFC data in July 2018. The descriptive statistics of the data are shown in Table 6. Two latent groups +are considered for the experiment. The reason for considering two groups instead of more is that two latent +groups are more interpretable in terms of estimation results. The path attributes are the same as the synthetic +data experiment except for the “number of transfers”. The “number of transfers” is dropped from the model +due to its high correlation with the “out-of-vehicle time”. Similar to the synthetic data experiment, we make +the parameters of in-veh-time the same for the two groups so that we can compare the scales of out-of-veh +time and the number of transfers. +20 + +O +Origin stations +Destination stationsTable 7: Estimation results for the real-world data +Variables +Group 1 (CA) +Group 2 (TS) +Estimation +t-value +Estimation +t-value +Choice model +In-veh time +-0.2785 +-18.13 +Same as Group 1 +Out-of-veh time +-1.1320 +-2.09 +-0.7457 +-9.04 +Denied waiting time +-3.0450 +-1.95 +-0.4517 +-9.90 +log PSm +1.2611 +4.38 +Same as Group 1 +σk +2.7294 +11.99 +Same as Group 1 +Latent group model (Group 2 is set as the base group) +ASC1 +-0.9644 +-6.83 +0 (fixed) +Avg # travel days per week +-0.0987 +-3.91 +0 (fixed) +Std. of 1st trip dept. time +0.8667 +4.19 +0 (fixed) +Min # stations with 70% trips +0.1865 +0.40 +0 (fixed) +If student (Yes = 1) +1.0192 +2.14 +0 (fixed) +1: ASC: alternative specific constant +Number of passengers: 3,425. Number of observations: 6,425 +LL0: -32,157.84; Final LL∗: -30,867.48 +χ2 = −2(LL0 − LL∗) = 2, 580.72, likelihood ratio test p-value: 0 +The estimation results are shown in Table 7. The two latent groups are referred to as Group 1 and 2, +respectively. Group 2 is set as the base alternative in the group-assignment multinomial logit model (Eq. +1). Results show that the signs and scales of all parameters are reasonable. All time-related parameters +are significantly negative. The value of the in-vehicle time parameter is -0.2785, which is similar to the +survey results (-0.2676) (Jin et al., 2017). For both Groups 1 and 2, the absolute values of the parameters for +out-of-vehicle time and denied waiting time are both greater than that of the in-vehicle travel time, reflecting +that passengers were more sensitive to walking and waiting times compared to the time seated in vehicles. +And these results are also consistent with the survey. +Comparing the results of Group 1 and 2, we observe that the out-of-vehicle and denied waiting times +show a larger impact on the path choice utility for passengers in group 1, which implies that Group 1 is +possibly comfort-aware (CA) and Group 2 is time-sensitive (TS). The parameters determining the latent +groups indicate that CA passengers have less travel frequency (i.e., the effect of avg. # travel days per week +is negative) and more schedule flexibility (i.e., the effect for the std. of the 1st trip departure time is positive), +and students are more likely to belong to this group. These suggest that CA passengers are most likely to +be irregular users and they may mainly use the MTR system for non-work activities (such as entertainment). +Hence, they care more about the trip comfort and prefer paths with less walking and waiting times. In +contrast, TS passengers have higher travel frequency and less schedule flexibility and they may use the metro +system mostly for regular commuting trips. Besides, they care more about saving the total travel time to +arrive at their destinations on time. The spatial concentration variable (i.e., min # stations with 70% trips), +though having a positive impact on being in the CA group, is not statistically significant (t-value 0.4). σk +is significant for both groups, which means that the panel effects are diverse across populations (i.e., some +have more stable travel patterns but are not). +21 + +4. Conclusion +Understanding passenger path choices are important for operations management in urban rail systems, +especially those with crowded conditions. This paper presents a probabilistic approach for the path choice +model estimation with train capacity constraints using AFC (tap-in and tap-out) and AVL data. The choice +heterogeneity and longitudinal behavioral correlations are captured by a latent class model with panel effects. +Passenger’s movement is formulated using a passenger to train assignment model with explicit modeling of +the processes of access/egress, left behind (crowding), and transfer. A tractable likelihood function is derived +to facilitate the model estimation. The t-value of estimated parameters is calculated based on the numerically +estimated Hessian matrix and Cramer–Rao bound. The method is data-driven, flexible to accommodate +different choice models, and easy to solve using non-linear optimizers. +The model performance is validated using synthetic data to estimate the individual choice parameters. +The sensitivity analysis affirms its robustness to parameter initialization and small errors in inputs (walking +speed and left behind distributions). It also highlights that neglecting capacity constraints (left behind) can +lead to biased estimation of path choice parameters under crowding conditions. The model is also applied +using real-world data from the MTR system in Hong Kong. It reveals two different groups of passengers: +time-sensitive (TS) and comfort-aware (CA). TS passengers are generally regular commuters with high travel +frequency and small schedule flexibility. They are more likely to choose paths with less trip travel times. CA +passengers care more about travel comfort and prefer choosing paths with less walking and waiting times. +Example policy implications can be derived from the case study. As these two groups of passengers +value in-vehicle and out-of-vehicle times differently, transit agencies can use this insight to design customized +route recommendation systems with better passenger acceptance. Moreover, route recommendations can +help to relieve congestion by recommending TS and CA passengers to use different routes during peak +hours. Interesting future research directions include: 1) Explore the evolution of choice preferences and +learning behavior over time under network interventions (such as network extension and demand management +policies). 2) Develop a downstream model to utilize the latent passenger group information for better route +recommendations or fare policies. +5. Authors’ contribution +Baichuan Mo: Conceptualization, Methodology, Software, Formal analysis, Data Curation, Writing - +Original Draft, Writing - Review & Editing, Visualization. Zhenliang Ma: Conceptualization, Methodology, +Supervision, Formal analysis, Data Curation, Writing - Original Draft, Writing - Review & Editing. Haris +N. Koutsopoulos: Conceptualization, Supervision, Formal analysis. Jinhua Zhao: Conceptualization, +Supervision, Project administration, Funding acquisition. +6. Acknowledgement +The authors would like to thank Chicago Transit Authority (CTA) for their support and data availability +for this research. +22 + +Appendices +Appendix A. Left behind probability calibration +The left-behind probability can be estimated from a Gaussian mixture model proposed by Ma et al. +(2019). The main idea is that passengers being left behind different times would have different journey +times. Let tJn +i be the random variable indicating the journey time of passenger i (i.e., tap-out time minus +the tap-in time). Figure A.9 shows an example of journey time distribution between a specific OD pair. We +observe there are three clusters, indicating passengers being left behind 0, 1, and 2 times at the origin station. +Hence, we can model the journey time distribution as a Gaussian mixture model: +Pr(tJn +i ; µ, σ, w) = +C +� +c=0 +wc · Φ(µc, σc) +(A.1) +where wc is the (unknown) fraction of passengers in cluster c, wc > 0 and �C +c=0 wc = 1. C is the total +number of clusters (i.e., the maximum number of left behind times at the origin station). Φ(µc, σc) is the +PDF for the Gaussian distribution N(µc, σc). +Figure A.9: Example of journey time distribution +The Gaussian mixture model can be estimated by solving the following problem: +max +µ,σ,w +� +i∈P′ +log(Pr(tJn +i ; µ, σ, w)) +(A.2a) +s.t. +Auxiliary Constraints +(A.2b) +� +c∈C +wc = 1 +(A.2c) +wc ≥ 0 +∀c = 0, 1, ..., C +(A.2d) +where P′ is the passenger set used for the left behind probability estimation. The auxiliary constraints are +used for model stability. These constraints contain prior human knowledge on the journey time distribution, +such as the difference between µc and µc+1 should be close to a headway, the mean journey time without being +left behind (i.e., µ0) should be close to the sum of access, egress, and in-vehicle times. More information on +the model can be found in Ma et al. (2019). +23 + +on +02The model is station and time-specific, which enables the calibration of left behind probabilities for each +station at different time intervals in the system (by adjusting P′). The estimated wc (i.e., the fraction of +passengers in cluster c) is the probability of being left behind c times at the corresponding station and time +period. +Appendix B. 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Transportation Research Part C: Emerging Technologies 123, 102896. +26 + diff --git a/H9E2T4oBgHgl3EQfUAdV/content/tmp_files/load_file.txt b/H9E2T4oBgHgl3EQfUAdV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f45f37c2d6bab234a4702740efd09c54751685c --- /dev/null +++ b/H9E2T4oBgHgl3EQfUAdV/content/tmp_files/load_file.txt @@ -0,0 +1,1153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf,len=1152 +page_content='Passenger Path Choice Estimation Using Smart Card Data: A Latent Class Approach with Panel Effects Across Days Baichuan Moa, ZhenLiang Mab,∗, Haris N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Koutsopoulosc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Jinhua Zhaod aDepartment of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' MA 02139 bDepartment of Civil and Architectural Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' KTH Royal Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Stockholm 10044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Sweden cDepartment of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' MA 02115 dDepartment of Urban Studies and Planning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' MA 20139 Abstract Understanding passengers’ path choice behavior in urban rail systems is a prerequisite for effective operations and planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The area witnesses active developments in two broad but separate fields, including behaviour modeling using ‘small’ survey data in transport and mobility pattern using ‘big’ data in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This paper attempts bridging the gap by proposing a probabilistic approach to infer passengers’ path choice behavior in urban rail systems using a large-scale smart card data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The model uses latent classes and panel effects to capture passengers’ implicit behavior heterogeneity and longitudinal correlations, key research gaps in big data driven behavior studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We formulate the probability of each individual’s arrival time at a destination based on their path choice behavior, and estimate corresponding path choice model parameters as a maximum likelihood estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The original likelihood function is intractable due to the exponential computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We derive a tractable likelihood function and propose a numerical integral approach to efficiently estimate the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Also, we propose a method to calculate the t-statistic of the estimated choice parameters based on the numerically estimated Hessian matrix and Cramer–Rao bound (the lower bound on the coefficient variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Case studies using synthetic data validate the model performance and its robustness against parameter initialization and input errors, and highlight the importance of incorporating crowding impact in path choice estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Applications using actual data from the Mass Transit Railway, Hong Kong reveal two latent groups of passengers: time-sensitive (TS) and comfort-aware (CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' TS passengers are those who are more likely to choose paths with short travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Most of them are regular commuters with high travel frequency and less schedule flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' CA passengers care more about the travel comfort experience and choose paths with less walking and waiting times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The proposed approach is data-driven and general to accommodate other discrete choice structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It provides the same outputs as traditional choice modeling and facilities a deep understanding of passengers choice behaviors in both a cost-effective and timely way, based on which more informed planning and management strategies could be designed, evaluated, and monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Keywords: Path choices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Urban railway systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Smart card data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Latent passenger groups, Panel effects 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Introduction Increases in ridership are outpacing capacity in many large urban rail transit systems, including Hong Kong’s Mass Transit Railway (MTR), the London Underground, and the New York subway system (Zhu ∗Corresponding author Preprint submitted to Elsevier January 11, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='03808v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='AP] 10 Jan 2023 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2017b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Crowding at stations and on trains is a concern due to its impact on safety, service quality, and operating efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Various studies have measured passengers’ willingness to pay for less crowded conditions (Li and Hensher, 2011) and suggested incorporating the crowding disutility in investment appraisals (Haywood and Koning, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Given the interest in dealing with crowding-related problems, understanding passengers’ route choice behavior under crowding situations is important for both operations management and planning practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, estimating path choice fractions or individual choice behavior is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' As passenger’s path choices in an urban rail system are not directly observed, most of the previous studies rely on revealed and stated preference survey data (Raveau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2011, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Surveys are a powerful tool to facilitate behavior analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, they are constrained by high costs, reporting accuracy, and survey coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Automated Fare Collection (AFC) and Automatic Vehicle Location (AVL) data provide opportunities for analysis in areas such as travel behavior, demand modeling, transit operations planning, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (Pelletier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Bagchi and White, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Koutsopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In addition to aggregate trends of when and where passengers travel, AFC data provides detailed information on the travel patterns of individuals and/or specific groups (Goulet-Langlois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Table 1 summarizes the existing route choice studies using AFC or/and AVL data in metro systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Several studies have used AFC data to estimate passengers’ path choice probabilities (Sun and Xu, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Sun and Schonfeld, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' They provide useful insights on the aggregate choice behavior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', path fractions) under existing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For example, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2021) develops a data-driven approach for the inference of passenger itineraries in urban heavy rail systems, where the path fractions can be estimated using AFC and AVL data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, these results cannot be used for operations planning applications without modeling the individual path choice behavior, such as timetable design, network expansion, operating strategies and policy interventions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This is because the new timetable or network expansion may change the service attributes, causing changes in an individual’s choice behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Inference of path fractions cannot capture the impact of these changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This study focuses on the estimation of path choice models as a function of attributes of alternative paths using AFC and AVL data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Relevant to this context, Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2015) developed an integrated Bayesian approach to infer network attributes and passenger route choice behavior using AFC data from the Singapore Mass Rapid Transit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2018) developed a data fusion model to estimate individual path choices by combining revealed preference (RP) survey data and AFC data and modeled the risk attitudes of passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, both studies imposed a strong assumption on link travel times (independent normal distribution) ignoring the fact that under congested conditions passengers may experience left behind at major stations due to capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' During peak periods, a (usually) shorter travel time route may have passengers who are left behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' But the models above cannot distinguish whether the longer travel time is due to choosing a longer route or being left behind multiple times on a shorter route (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2017a, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' To incorporate the left behind phenomenon, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2017b) proposed a passenger-to-train assignment model (PTAM) by decomposing the journey time into access, waiting, in-vehicle, egress walking times, and considering the dynamics of being left behind at origin stations explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The model was applied to estimate the left behind at key stations for non-transfer trips with capacity constraints and validated using both synthetic and actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Hörcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2017) extended the PTAM to the case with transfers and presented a discrete choice model (DCM) to estimate the user cost of crowding in urban rail systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, the model identified the “actual” path used by passengers based on predetermined probability thresholds, which 2 Table 1: Summary of literature on urban rail path choice inference and modeling using AFC data Reference Data Behavior Characteristics Method AFC AVL Aggregate Individual Crowding Heterogeneity Panel effect Optimization Probabilistic Sun and Xu (2012) ✓ ✓ ✓ ✓ ✓ Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2016) ✓ ✓ ✓ ✓ ✓ Sun and Schonfeld (2016) ✓ ✓ ✓ ✓ ✓ Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2015) ✓ ✓ ✓ ✓ ✓ Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2021) ✓ ✓ ✓ ✓ ✓ Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2022) ✓ ✓ ✓ ✓ ✓ Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2015) ✓ ✓ ✓ Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2018) ✓ ✓ ✓ Hörcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2017) ✓ ✓ ✓ ✓ ✓ This study ✓ ✓ ✓ ✓ ✓ ✓ ✓ may eventually impact the estimation quality of the choice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' These data-driven behavioral studies provide a good attempt to bridge the gap between ’small’ (survey) and ’big’ (AFC) data studies in different domains but answering the same question regarding passengers’ path choices under crowding (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, existing AFC/AVL data driven path choice estimation studies are basically designed towards calibrating parameters of standard discrete choice models in DCM studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' They lack of systematic consideration of common and unique characteristics of the behavior choice problem itself and model them correspondingly in the context of big data, comparing with the survey data based DCM studies, including: Choice Heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It is commonly known that passengers may have different perceptions of service performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', travel time, waiting time) and show different choice strategies for travels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Ignoring the population heterogeneity may lead to estimation bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The choice heterogeneity is usually captured by latent class or mixture models in the DCM literature (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, to the best of the authors’ knowledge, none of the previous AFC data-based studies have considered passenger heterogeneity in modeling individual path choice behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Panel Effect (choice correlations across time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' AFC data records passengers travels across days, and thus one unique challenge is about which days data should be used to estimate passengers’ routine behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Considering individual travels on multiple days is important for robust choice behavior estimation, as passengers may occasionally deviate from their habitual behavior on some days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, this brings challenges to model the temporal correlations of individuals’ route choices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', panel effect) across times and days, which is not considered or even discussed in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Choice Model Coefficient Significance (t-statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In DCM studies, the significance levels of model coefficients are important for deriving behavioral and policy insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, no AFC/AVL driven study is found on reporting the t-statistics of calibrated model coefficients, thus limiting its capability in facilitating comprehensive behavioral interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' To fill the research gaps, this paper develops a latent class approach with panel effects to estimate individual path choice behavior from AFC (tap-in and tap-out) and AVL data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The proposed framework explicitly captures capacity constraints (crowding), choice heterogeneity, and panel effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We formulate the probability of each individual’s arrival time at its destination station based on their path choice behavior, and estimate the corresponding path choice parameters as a maximum likelihood estimation (MLE) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 3 The original likelihood function is intractable due to the exponentially large number of summations and the integration over a normally distributed variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We derive a new conditional probability-based formulation to eliminate a large number of summations and use a numerical integration approach for the normal random variable, which leads to a tractable likelihood function and enables an efficient model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Given the difficulty in deriving the analytical Hessian matrix, the t-values of estimated parameters are calculated based on the numerically estimated Hessian matrix and the Cramer–Rao bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Case studies using synthetic data validate the model performance and highlight the importance of incorporating crowding impact in path choice estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Applications using actual data from the Mass Transit Railway (MTR), Hong Kong reveal two latent groups of passengers in the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The main contributions of this study are as follows: Introducing and model the passenger path choice problem using smart card data in closed public transport systems considering system crowding, choice heterogeneity and panel effects across times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Formulating a MLE based latent-class path choice estimation problem with panel effects and deriving a tractable likelihood function for efficient model coefficients estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Proposing a numerical method to calculate the t-statistic of estimated choice coefficients based on numerically estimated Hessian matrix and Cramer–Rao bound (lower bound of coefficient variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Validating the model performance using both synthetic and real-world data in Hong Kong, and identi- fying latent groups of passengers with heterogeneous preference over travel times and comfortableness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The rest of the paper is organized as follows: Section 2 formulates the route choice problem and develops the MLE estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Section 3 validates the proposed approach using synthetic and real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The final section summarizes the main findings and discusses future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Problem description Consider a closed AFC system where both tap-in and tap-out records of passengers over time are available, and train arrivals and departures at stations are available from the AVL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Define a passenger i with a series of observed AFC records vi = {(oi,1, di,1, tin i,1, tout i,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', (oi,Ni, di,Ni, tin i,Ni, tout i,Ni)}, where oi,n, di,n, tin i,n, tout i,n represent the passenger’s origin, destination, tap-in time, and tap-out time of the n-th trip, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The set of all passengers is defined as P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', i ∈ P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' To capture passengers’ choice heterogeneity, we assume that there are K latent groups in the population and passengers in the same group share the same choice preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let gi be a random variable indicating the group that passenger i belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The probability that passenger i belongs to a latent group Gk is formulated as a multinomial logit model: Pr(gi = Gk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' θ) = exp(θk · xi) �K k′=1 exp(θk′ · xi) (1) where xi is the vector of the characteristics of passenger i, including variables (extracted from smart card data) such as travel frequency, card type, travel regularity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' G = {Gk | k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', K} is a set of latent groups to be estimated (K need to be pre-specified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' θ = (θk)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=',K is the parameter vector to be estimated, associated individual’s characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 4 According to the random utility maximization (RUM) assumption (Ben-Akiva and Lerman, 2018), the utility of passenger i choosing path m at the n-th trip, given that passenger i is in group Gk, can be formulated as: U k i,n,m = βk · zn,m + αk i + εk i,n,m (2) whereβk arethe unknown parameters to beestimated, associatedwithpathattributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' zn,m := [yn,m, log PSm] and yn,m is the vector of path attributes, including variables such as in-vehicle time, out-of-vehicle time, left behind waiting time, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' PSm is the “path size” factor, which is used to capture the correlation in error terms caused by path overlapping (Hoogendoorn-Lanser and Bovy, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The formulation with the path size factor is known as the “path-size logit model” (Prato, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' PSm is defined as PSm = 1 Lk � a∈Am la � m′∈Ri,n δa,m′ ∀ m ∈ Ri,n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Ni, i ∈ P (3) where Am is the set of all links of path m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' δa,m′ = 1 if link a is in path m′, otherwise δa,m′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Lk is the length of path m and la is the length of link a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Ri,n is the set of all available paths for passenger i’s n-th trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' To capture the individual’s behavior correlation over time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', panel effect), the utility function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2) also includes an individual specific unobserved factor αk i (a random variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The panel effect is assumed to be persistent over time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', no subscript n) (Ben-Akiva and Lerman, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' αk i is assumed to be independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=') for all passengers in group k and follows a normal distribution N(0, (σk)2) (the zero-mean is due to the fact that the mean value can be estimated as a part of the alternative specific constant), where σk is the standard deviation to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Given αk i , the unobserved error term εk i,n,m is assumed to be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Gumbel distributed across all i, n, and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let πk i,n,m[αk i ] be the probability of passenger i choosing path m at the n-th trip given that passenger i is in group Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' According to RUM theory, πk i,n,m[αk i ] = exp(βk · zn,m + αk i ) � m′∈Ri,n exp(βk · zn,m′ + αk i ) (4) Since there are a total of Ni trip records for passenger i, we can formulate the series choice probability as (Arellano and Honoré, 2001): Pr(ri,1 = m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', ri,Ni = mNi) = K � k=1 Pr(gi = Gk) · � αk i � Ni � n=1 πk i,n,mn[αk i ] � f(αk i ) dαk i ∀m1 ∈ Ri,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', mNi ∈ Ri,Ni (5) where ri,n is a random variable indicating the path used by passenger i in the n-th trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' And f(αk i ) is the probability density function of αk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The goal of this study is to develop an approach to simultaneously estimate β = (βk)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=',K, θ, σ = (σk)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=',K, which specify passenger’s path choice behavior, choice heterogeneity, and panel effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We formulate an MLE problem to estimate these parameters in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The structure of the methodology is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The notation used across this paper is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 5 Figure 1: Methodology framework Table 2: Notation summary Notation Description Model Parameters vi A series of AFC data records for passenger i (oi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' tin i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' tout i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n) Passenger’s origin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' destination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' tap-in time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' and tap-out time of the n-th trip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' respec- tively P The set of all passengers xi Vector of the characteristics of passenger i Ni Total number of trips for passenger i Gk The k-th latent group G The set of all latent groups K The number of latent groups zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m Vector of path attributes for path m and trip n αk i A random variable to capture panel effect for individual i in latent group k U k i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m The utility of passenger i choosing path m at the n-th trip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' given that passenger i is in group Gk PSm Path size factor for path m Am The set of all links of path m Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n The set of all available paths for passenger i’s n-th trip τ The time duration that each time index represents Ru,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='v The set of feasible paths for OD pair (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' v) πk i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m[αk i ] The probability of passenger i choosing path m at the n-th trip given that passenger i is in group Gk ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n A random variable indicating the path used by passenger i in the n-th trip LL(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' σ) Log-likelihood function of all observations f(αk i ) The probability density function of αk i Λj i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m the set of all trains associated with the j-th segment of path m for passenger i’s trip n Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m Number of path segments for path m of passenger i’s trip n Ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content="m The set of possible itineraries for path m in the n-th trip of passenger i 6 Raw Data Hyper-parameters Passenger's latent class Model Path choice estimation Log-likelihood (LL) of observed AFC records parameters and t-values Latent-class Path choice modeTd(·) A function which returns the train’s departure time at the boarding (resp." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' alighting) station of the corresponding segment Ta(·) A function which returns the train’s arrival time at the boarding (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' alighting) station of the corresponding segment fEg m (·) Egress walking time probability density function (PDF) for path m fAc m (·) Access walking time PDF for path m BI1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m Maximum number of times that passenger i is left behind to board Train I1 in trip n for path m ηj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='k i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m The probability of being left behind k times at the boarding station of the j-th segment of path m for passenger i’s trip n Iu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='r The set of legs for path r of OD pair (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' v) EIj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='k i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m The event that “passenger i in the n-th trip arrives at the boarding station of segment j of path m between the departure of Train Ij − k and Ij − k − 1 and is left behind k times to board Train Ij” tj i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='m The transfer walking time from the alighting of train Ij−1 to the next platform for passenger i’s n-th trip using path m ˆH−1 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='k k-th diagonal element the inverse Hessian matrix for the log-likelihood function Parameters to estimate βk Parameters associated with path attributes in latent group Gk σk The standard deviation of αk i θk Parameter vector associated individual’s characteristics in latent group Gk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Model formulation Given the set of passenger i’s AFC data records vi, the probability of observing vi can be expressed as Pr(vi) = � ri,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=',ri,Ni Pr(vi | ri,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', ri,Ni)Pr(ri,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', ri,Ni) = � ri,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=',ri,Ni �� Ni � n=1 Pr(oi,n, di,n, tin i,n, tout i,n | ri,n) � × Pr(ri,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', ri,Ni) � (6) where the second equality follows from the Bayesian theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' As the origin and destination are known given path m, Pr(oi,n, di,n, tin i,n, tout i,n | ri,n = m) is equivalent to the probability that passenger i enters the origin at time tin i,n and exits the destination at time tout i,n (denoted as Pr(tin i,n, tout i,n | ri,n = m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Notice that Pr(tin i,n, tout i,n | ri,n = m) = Pr(tout i,n | tin i,n, ri,n = m) · Pr(tin i,n | ri,n) ∝ Pr(tout i,n | tin i,n, ri,n = m) (7) The “proportional to” is due to the fact that we do not model the tap-in time choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Therefore, the likelihood function becomes L(θ, β, σ) = � i∈P Pr(vi) = � i∈P � � � ri,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=',ri,Ni Ni � n=1 � Pr(tout i,n | tin i,n, ri,n) � Pr(ri,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', ri,Ni) � � (8) 7 The only unknown part in the likelihood function (Equation 8) is Pr(tout i,n | tin i,n, ri,n = m), which is the probability that passenger i taps out at his/her destination at time tout i,n given that he/she uses path m ∈ Ri,n and taps in at time tin i,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It can be derived by integrating over different itinerary scenarios, where each scenario is associated with a specific walking, boarding, and left behind possibility (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' To illustrate the derivation of Pr(tout i,n | tin i,n, ri,n = m), we consider an example journey involving one transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 2 shows, in a time-space diagram, all possible movements of a passenger tapping in at the origin station on line 1 and tapping out at the destination station on line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The movement along a specific line is referred to as a “path segment”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' A path segment is characterized by the line, the boarding station, and the transfer/alighting station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Each path segment is associated with a set of trains with run IDs indicating the dispatching time sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For example, the first path segment in Figure 2 has Trains 1, 2, 3, and 4 numbered in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let the set of all trains associated with the j-th segment of path m for passenger i’s trip n be Λj i,n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For example, for the first path segment in Figure 1, we have Λj i,n,m = {Line 1 Train 1, Line 1 Train 2, Line 1 Train 3, Line 1 Train 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' With slight abuse of notation, for a Train I ∈ Λj i,n,m, Train I + k represents the train in the same line with ID+k (k ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For example, if Train I is Line 1 Train 1, then Train I + 1 is Line 1 Train 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' After the passenger taps in, he/she walks directly to the platform at the origin station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The walking time from the entry gate to the origin station platform is referred to as the “access walking time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Depending on the available capacity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', potentially left behind), this passenger may board Trains 2 or 3 on Line 1 for the first path segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Note that Train 1 is not feasible because the passenger arrives on the platform after the departure of Train 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' After alighting at the transfer station, the passenger walks to the boarding platform for the next path segment on Line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The walking time from the alighting platform to the next boarding platform is referred to as the “transfer time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Similarly, depending on the available capacity, the passenger may board Trains 2 or 3 on Line 2 (Train 4 is not feasible because the passenger cannot exit at his/her current tap-out time if boarding Train 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' After alighting at the platform of the destination station, the passenger walks directly to the exit gate and taps out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The walking time from the alighting platform to the exit gate is referred to as the “egress walking time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Generally, let us consider passenger i ∈ P who uses path m ∈ Ri,n in his/her n-th trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let Ji,n,m be the number of path segments for path m of this trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' An itinerary H = {I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', IJi,n,m} is defined by “a sequence of train IDs” (each train ID is associated with a path segment) representing a possible movement of the passenger in the system, where Ij ∈ Λj i,n,m indicates Train Ij for the j-th path segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For example, in Figure 2, a feasible itinerary is H = {Line 1 Train 2, Line 2 Train 3}, which indicates the itinerary that the passenger first boards Train 2 on Line 1 and then boards Train 3 on Line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It is worth noting that for a specific passenger i, there are a limited number of feasible itineraries given his/her tap-in and tap-out time and the feasibility of transfer times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For example, in Figure 2, any itineraries with trains in Line 1 departing before Line 1 Train 2 are not feasible because passengers cannot board those trains given their tap-in times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let Ωi,n,m be the set of possible itineraries for path m in the n-th trip of passenger i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We have Ωi,n,m = {{I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', IJi,n,m}, ∀ Ij ∈ Λj i,n,m, | Td(I1) ≥ tin i,n, Ta(IJi,n,m) ≤ tout i,n, Td(Ij) ≥ Ta(Ij+1), ∀ j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Ji,n,m} (9) where Td(·) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Ta(·)) is a function which returns the train’s departure (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' arrival) time at the boarding (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' alighting) station of the corresponding segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This information is available from the AVL data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 8 9 means that a feasible itinerary needs to satisfy that 1) Train I1 departs after tin i,n so that the passenger is able to board it (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Td(I1) ≥ tin i,n, assuming the minimum access walking time is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2) The last train (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Train IJi,n,m) arrives earlier than tout i,n so that the passenger is able to tap-out at tout i,n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Ta(IJi,n,m) ≤ tout i,n, assuming the minimum egress walking time is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 3) Train Ij+1 departs later than the arrival of the train Ij so that the passenger can successfully transfer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Td(Ij) ≥ Ta(Ij+1), assuming the minimum transfer time is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 2: Time-space diagram for a journey involving one transfer (adapted from Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2017b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The red lines indicate feasible itineraries Given the feasible itinerary set Ωi,n,m, Pr(tout i,n | tin i,n, ri,n = m) can be rewritten as: Pr(tout i,n | tin i,n, ri,n = m) = � H∈Ωi,n,m Pr(tout i,n | H, tin i,n, ri,n = m) · Pr(H | tin i,n, ri,n = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (10) We first consider the derivation of Pr(tout i,n | H, tin i,n, ri,n = m), the probability of tap out at time tout i,n given itinerary H, path m and tap-in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Since the itinerary includes the information of the last train’s arrival time Ta(IJi,n,m), this probability is equivalent to the probability that the egress walking time is equal to tout i,n − Ta(IJi,n,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let the egress walking time probability density function (PDF) for path m be fEg m (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Then, Pr(tout i,n | H, tin i,n, ri,n = m) can be expressed as Pr(tout i,n | H, tin i,n, ri,n = m) = fEg m (tout i,n − Ta(IJi,n,m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (11) Note that Equation 11 uses the density to represent the probability, which does not affect the parameter estimations in the MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Now let us consider the derivation of Pr(H | tin i,n, ri,n = m), the probability of choosing itinerary H given path m and the tap-in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Since the boarded train on segment j only depends on the boarded train 9 on segment j − 1, but not j − k for all k > 1, this probability can be extended using the Markov property: Pr(H | tin i,n, ri,n = m) = Pr(I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', IJi,n,m | tin i,n, ri,n = m) = Pr(I1 | tin i,n, ri,n = m) · Ji,n,m � j=2 Pr(Ij | Ij−1, tin i,n, ri,n = m) (12) In the following contents, we elaborate the derivation of Pr(I1 | tin i,n, ri,n = m) and Pr(Ij | Ij−1, tin i,n, ri,n = m), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Note that Pr(I1 | tin i,n, ri,n = m) is the probability of boarding Train I1 on the first segment of path m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' There are two different scenarios for this event to happen: 1) [No left behind] the passenger arrives at the platform between the departure time of Trains I1 and I1 − 1 and boards Trains I1 without left behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2) [With left behind] the passenger arrives at the platform between the departures of Trains I1 − k and I1 −k−1 and is able to board Train I1 after being left behind k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Given the feasible itinerary set Ωi,n,m, there is a maximum number of times the passenger is left behind to board Train I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Denote the upper bound of k as BI1 i,n,m, where BI1 i,n,m = arg maxk{k ∈ N | ∃ H′ ∈ Ωi,n,m s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' I′ 1 = I1 − k, I′ 1 ∈ H′}, N is the set of natural numbers (including zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' And Train I1 − BI1 i,n,m represents the earliest train that passenger i can board at the first segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let tj i,n,m be the walking time from the alighting platform of segment j − 1 to the boarding platform of segment j in path m for passenger i, and t0 i,n,m is the access walking time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Then, tin i,n + t0 i,n,m is the passenger arrival time at the platform of his/her origin station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Hence, the probability of arriving at the platform between the departure of Train I1 − k and I1 − k − 1 can be formulated as Pr(Td(I1 − k − 1) ≤ tin i,n + t0 i,n,m ≤ Td(I1 − k) | tin i,n, ri,n = m) = � Td(I1−k)−tin i,n Td(I1−k−1)−tin i,n fAc m (t)dt := ρI1,k i,n,m ∀k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='., BI1 i,n,m (13) where fAc m (·) is the access walking time PDF for path m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 13 can be pre-calculated once fAc m (·) is given because it is a definite integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let ηj,k i,n,m be the probability of being left behind k times at the boarding station of the j-th segment of path m for passenger i’s trip n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let EIj,k i,n,m be the event that “passenger i in the n-th trip arrives at the boarding station of segment j of path m between the departure of Train Ij − k and Ij − k − 1 and is left behind k times to board Train Ij”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We have: Pr(EI1,k i,n,m| tin i,n, ri,n = m) = ρI1,k i,n,m · η1,k i,n,m ∀k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='., BI1 i,n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (14) Then, Pr(I1 | tin i,n, ri,n = m) can be rewritten as Pr(I1 | tin i,n, ri,n = m) = BI1 i,n,m � k=0 Pr(EI1,k i,n,m| tin i,n, ri,n = m) = BI1 i,n,m � k=0 ρI1,k i,n,m · η1,k i,n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (15) This finishes the derivation of Pr(I1 | tin i,n, ri,n = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' η1,k i,n,m can be estimated from AFC and AVL data 10 using a Gaussian Mixture model (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2019), which will be described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Now, we derive Pr(Ij | Ij−1, tin i,n, ri,n = m) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 12, the probability of boarding train Ij given that the passenger has boarded train Ij−1 on the (j − 1)-th segment of path m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It is derived in a similar way as Pr(I1 | tin i,n, ri,n = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Passenger i may arrive at the boarding station of segment j between the departure times of Trains Ij − k and Ij − k − 1 and be left behind k times to board Train Ij (note that k = 0 means no left behind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The probability of arriving at the platform between the departures of train Ij −k and Ij −k −1 given he/she alights at Ta(Ij−1) is formulated as Pr(Td(Ij − k − 1) ≤ Ta(Ij−1) + tj i,n,m ≤ Td(Ij − k) | Ij−1, tin i,n, ri,n = m) = � Td(Ij−k)−Ta(Ij−1) Td(Ij−k−1)−Ta(Ij−1) fTr m,j(t)dt = ˜ρIj,k i,n,m ∀k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='., BIj i,n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (16) where BIj i,n,m is the maximum possible left behind times when boarding train Ij given the feasible itinerary constraint, defined as arg maxk{k ∈ N | ∃H′ ∈ Ωi,n,m s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' I′ j = Ij − k, I′ j ∈ H′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' tj i,n,m is the transfer walking time from the alighting of train Ij−1 to the next platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' fTr m,j(·) is the PDF of tj i,n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' ˜ρIj,k i,n,m is defined for the simplicity of expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Given the definition of EIj,k i,n,m, we have Pr(EIj,k i,n,m| Ij−1, tin i,n, ri,n = m) = ˜ρIj,k i,n,m · ηj,k i,n,m ∀k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='., BIj i,n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (17) Then, Pr(Ij | Ij−1, tin i,n, ri,n = m) can be rewritten as Pr(Ij | Ij−1, tin i,n, ri,n = m) = B Ij i,n,m � k=0 Pr(EIj,k i,n,m| Ij−1, tin i,n, ri,n = m) = B Ij i,n,m � k=0 ˜ρIj,k i,n,m · ηj,k i,n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (18) With all parts of L(θ, β, σ) in Equation 8 derived, there are still two remaining challenges for the MLE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' First, the calculation of Pr(tout i,n | tin i,n, ri,n) requires the inputs of left behind probability ηj,k i,n,m and three PDF functions fAc m (·), fEg m (·), and fTr m,j(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The PDF functions can be obtained from field-experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' But obtaining the left behind probability is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In this study, we used the model proposed by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2019) to estimate ηj,k i,n,m from AFC and AVL data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Details can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The second challenge is that, the calculation of Pr(vi) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 6) has an exponentially large number of summation over different paths, and it requires the integral of a normally distributed random variable, which makes it numerically hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In the following section, we derive a new conditional probability-based formulation to eliminate the large number of summations, and use a numerical integration approach for the normal random variable, which leads to a tractable likelihood function and enables an efficient model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Tractable log-likelihood function To eliminate the exponentially large number of summation over different paths in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 6, we observe that given αk i and gi, passenger i’s route choice for each trip becomes independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Mathematically, Pr(vi | αk i , gi = Gk) = Ni � n=1 Pr(tout i,n, tin i,n | αk i , gi = Gk) = Ni � n=1 � mn∈Ri,n Pr(tout i,n | tin i,n, ri,n = mn) · πk i,n,mn[αk i ] (19) 11 Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='19 only has a total of Ni×|Ri,n| summation terms, while this number in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 is |Ri,n|Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 19, Pr(vi) can be obtained by integrating and summing over αk i and gi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Since αk i is a normal random variable, an approximated numerical integration approach is used to get a tractable formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Note that there are a large class of quadrature rules for numerical integration (Davis and Rabinowitz, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In this paper, we use the simplest midpoint rule for the interpolation as this is not the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let αU and αL be the upper and lower bounds for αk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We divide [αL, αU] into discrete intervals with equal length ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let S be the set of all middle points in each interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Specifically, S = {αL + k·∆ 2 | ∀k = 1, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', |S|, |S| = 2(αU−αL) ∆ − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Hence, Pr(vi) can be rewritten as Pr(vi) ≈ K � k=1 Pr(gi = Gk) · � αk i ∈S Pr(vi | αk i , gi = Gk) · f(αk i ) · ∆ (20) ∆ is the parameter determining the trade-off between approximation accuracy and computational efficiency, where a smaller ∆ indicates a more fine-grained integration, but higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Given the new formulation of Pr(vi), we can use L(θ, β, σ) = � i∈P Pr(vi) to evaluate the likelihood function with a tractable formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Model estimation The new log-likelihood function can be expressed as LL(θ, β, σ) = � i∈P log Pr(vi) = � i∈P log � � K � k=1 � αk i ∈S Pr(gi = Gk) · Pr(vi | αk i , gi = Gk) · f(αk i ) · ∆ � � (21) As LL(θ, β, σ) is a combination of elementary functions, it is continuous and differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Therefore, the MLE can be solved with any first- or second-order optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In this study, the BFGS algorithm is used (Nocedal and Wright, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' BFGS is a quasi-Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It uses only the first derivatives and has demonstrated good performance for many optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' However, as the function includes the multiplication of several nonlinear terms, the convexity of this function is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It is possible that the LL is not concave and the BFGS algorithm may converge to different local minimums under different initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Hence, we conduct a sensitivity analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1 with respect to different initial values and show that the model estimation results are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Besides, the numerical results show that LL is concave within a reasonable range of path attribute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' After obtaining the optimal parameters θ∗, β∗, and σ∗, we calculate the t-values of the estimated parameters based on a numerically estimated Hessian matrix and the Cramer-Rao bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Note that as LL is second-order differentiable, the analytical Hessian matrix can also be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The numerical Hessian matrix is used for simplification due to the complex function form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In this study, we adopt the formulation with fourth-order approximation under uniform grid spacing to calculate the second derivative (Fornberg, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The exact formulas are attached in Appendix B (other approximation formulas can also be used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' With the second derivative formulas, we can calculate the numerical Hessian matrix of LL(θ, β, σ) at point (θ∗, β∗, σ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Denote the Hessian matrix as ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Note that, from the second-order optimality conditions, ˆH is negative semi-definite, which is the algebraic equivalent of the local concavity of the log-likelihood function (Bierlaire, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 12 Let Θ = (θ, β, σ) be a vector of all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Using the Cramer-Rao bound (Cramér, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Rao, 1992), the variance of an estimated parameter ˆΘk is Var[ˆΘk] = − ˆH−1 k,k (22) where ˆH−1 is the inverse of the Hessian matrix and ˆH−1 k,k is its k-th diagonal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Then, the corresponding t-value is calculated as: t-value[ˆΘk] = ˆΘk � Var[ˆΘk] (23) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Case study 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Model validation and sensitivity test It is difficult to collect passengers’ actual path choices in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' To validate the proposed approach, we use synthetic data generated by simulating passengers’ route choices, train operations, and their interactions (Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 3 shows the configuration of the synthetic urban rail network, where there are 7 stations (A∼G) and 3 lines (red, green, and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The number on each link represents the in-vehicle travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This network is extracted from the MTR metro system in Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It is also representative of typical metro network structures in terms of lines and transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The platform of station C of the red line in the up direction is assumed to be crowded with extensive left behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' All the other platforms are assumed to have no left behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 3: Synthetic urban rail network To generate the synthetic data, we assume that passengers’ path choice behavior is based on four path attributes: 1) in-vehicle time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the train run time of a path), 2) out-of-vehicle time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the sum of access, egress, transfer, and waiting time without left behind), 3) the number of transfers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the number of times transferring on the path), and 4) denied waiting time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the waiting time due to left behind at the crowded 13 Gplatforms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We also assume passenger’s latent groups can be characterized by two sociodemographic variables x(1) and x(2), where x(1) is drawn from U[−4, 4] and x(2) is drawn from U[−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Suppose there are two latent groups for the synthetic passengers: “time-sensitive” (TS) and “comfort-aware” (CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The TS passengers, when making path choices, tend to minimize their total travel time, meaning that the impact of in-vehicle time, out-of-vehicle time, and denied waiting time are similar to passenger’s path choice utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' CA passengers prefer paths with less walking or waiting time though the in-vehicle time could be longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' That is, the out-of-vehicle time and denied waiting time have a higher impact on these passengers’ path choice utilities than the in-vehicle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Table 3 shows the parameters for the latent class path choice model for generating the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' These parameters are chosen based on the survey results in Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We set the in-vehicle time and path size factor parameters to be the same for TS and CA passengers according to the survey modeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In addition, we set the choice parameters to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='7× (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5×) of the parameters from the survey results for the TS (CA) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The parameters for the latent class model are set as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 for x(1) and x(2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Table 3: Path choice parameters for synthetic data generation Variables TS CA Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2017) Path choice parameters In-veh time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2676 Out-of-veh time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4257 Num of transfer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3068 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1737 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8669 Denied waiting time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='7825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4603 log PSm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5815 σk 1 1 N/A Latent class parameters x(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5 x(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 Table 4 summarizes the parameters of the network, train operations, and passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The synthetic data is generated for 9 OD pairs (origin stations A, B, D, and destination stations E, F, G) by simulating the tap-out time given tap-in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' All the OD pairs have 2 paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For example, the possible paths for OD pair (A, E) are A-B-E and A-B-C-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Without loss of generality, we assume that there are 2,700 passengers, each of whom performs 3 trips (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Ni = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Each trip is associated with a randomly selected OD pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Algorithm 1 describes the detailed synthetic data generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 14 Table 4: System settings for synthetic data generation Entity Settings Network Walk distance 30-50 meters (access, egress, transfer) Train operations Headway 2+δH minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' δH is drawn uniformly from [−10, 10] seconds In-vehicle time+δV (see Figure 3) δV is drawn uniformly from [−20, 20] seconds Passengers Walk speed distribution follows a lognormal distribution with mean of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2m/s and standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Left behind probabilities at station C, red line, up direction are 20% no left behind, 50% left behind once and 30% left behind twice Algorithm 1 Synthetic data generation Input: Path choice parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Passenger set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Output: Synthetic AFC data (tap-in/tap-out stations/times) 1: Initialize the number of sample instances N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2: for i ∈ P do 3: Sample x(1) i ∼ U[−4, 4], x(2) i ∼ U[−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 4: Sample group gi ∼ Pr(gi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Denote the group as Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 5: Sample αk i ∼ N(0, θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 6: for n = 1 to Ni do 7: Sample tin i,n ∼ U[7:00AM, 10:00AM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 8: Randomly sample an OD pair for this trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 9: Calculate the path choice probability πk i,n,m[αk i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 10: Sample a path m ∈ Ri,n based on πk i,n,m[αk i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 11: Sample the actual travel information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', train run time, headway, access, egress, transfer, and denied waiting times) for this trip based on path m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Obtain tout i,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 12: Save tin i,n and tout i,n and the OD as a trip record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 13: Combine all trip records as the synthetic AFC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In total, 8,100 trips from the 2,700 passengers were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The synthetic AFC data is then used for model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' As we have the “true” value of choice parameters (Table 3), we can validate the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The MLE is solved using the BFGS algorithm (Fletcher, 2013) in the Python Scipy package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' αL = −3, αU = 3, and ∆ = 1 are used for numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Table 5 shows the estimation results of the path choice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The percentage values in the brackets quantify the relative errors compared to the “true” parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Note that the actual walking speed distribution and left behind probabilities are used in the model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' And the sensitivity analysis on these inputs is shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For comparison purposes, we also estimate a baseline model without latent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The latent class model can estimate the actual parameters with a mean percentage error of around 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It outperforms the baseline model in estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The out-of-vehicle time parameter for the TS group has the maximum error (-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2%), which may be due to the fact that the out-of-vehicle time is highly correlated with the number of transfers, making the numerical estimation harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Note that as the absolute values for these parameters are relatively small, the absolute errors of the estimated parameters are acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 15 In terms of the goodness-of-fit, the initial log-likelihood (denoted as LL0) for the null model (with all parameters zero) is −52, 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='17, the final latent-class model log-likelihood (denoted as LL∗) is −51, 526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='13, and the final baseline model log-likelihood (denoted as LLB) is −51, 571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We conduct the log-likelihood ratio test (Wilks, 1938) and obtain the statistic χ2 = −2(LLB − LL∗) = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='08, which suggests a p-value of 0 given 5 degrees-of-restrictions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', number of parameters of latent class model minus that of baseline model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This indicates that the latent-class model specification is significantly better than the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We also calculate the log-likelihood with “true” parameters (referred to as LLTrue-para) and the value is −51, 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It is smaller than the LL∗, which means that the estimated parameters have a better goodness-of-fit than the “true” parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This suggests that the estimation errors may mostly come from random errors in the data generation process, instead of the model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' All parameters have absolute t-values greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='96, showing significant impacts of these param- eters on passengers’ path choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This is reasonable because the synthetic data are generated with those parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We also observe that the in-vehicle time shows the highest significance compared to other cost parameters, which is consistent with survey results (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Table 5: Estimation results for the synthetic data Variables Latent class Baseline Estimation (Error) t-value Estimation (Error) t-value Choice model In-veh time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2599 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2254 (-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='03 TS: Out-of-veh time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1991 (-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3010 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0%) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='88 TS: Num of transfer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3327 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8777 (+43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='7%) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='77 TS: Denied waiting time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3789 (+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6%) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4901 (+52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1%) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='80 CA: Out-of-veh time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5419 (-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='14%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3010 (-52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9%) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='88 CA: Num of transfer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8071 (-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8777 (-40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8%) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='77 CA: Denied waiting time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='7298 (-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='7%) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4901 (-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4%) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='80 log PSm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6758 (+16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2%) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6054 (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1%) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='31 σk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9191 (-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1%) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9122 (-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8%) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='40 Latent group model x(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6213 (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='99 N/A N/A x(2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8637 (+24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='01 N/A N/A Number of passengers: 2,700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Number of observations: 8,100 LL0: -52,219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' LL∗: -51,526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' LLB: -51,571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='67;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' LLTrue-para: -51,535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='16 χ2 = −2(LLB − LL∗) = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='08, likelihood ratio test p-value: 0 To further validate the model performance, sensitivity analysis was conducted to explore the impacts of parameter initialization on the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Moreover, we also evaluate whether the inaccurate estimation of walking speed distribution and left behind distribution would affect the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 4 shows the sensitivity analysis on the initialization of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' A total of 20 experiments are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In each experiment, the initial values of all parameters are drawn uniformly from U[−5, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We observe that the final estimated parameters all converged to the same values regardless of initialized parameter values, showing the estimation robustness against the parameter initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 16 Figure 4: Estimated parameters with different initializations Figure 5 illustrates the LL value as a function of variable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The log-likelihood function is concave around the optimal values1, which further indicates that the estimation results are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (a) LL vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' in-veh time and out-of-veh time (TS) (b) LL vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' x(1) and x(2) Figure 5: Log-likelihood function surface Figure 6 shows the model estimation results with respect to different inputs of walking speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We evaluates the model’s robustness with respect to errors in walking speed distribution because the estimation passenger’s walking speed may not be accurate in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let µWS and σWS be the actual walking speed mean and standard deviation when generating the synthetic data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', µWS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2m/s and σWS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' When estimating the model, we set the speed distribution parameters as (Γ1 · µWS, Γ2 · σWS), where 1Due to space limitations, we only show the function curves with respect to four variables 17 2 parameters 1 Y Y Y 丫 Y Y Y +(1) +(2) In-veh time logPSm 0 estimated 人人 丫 人人 人人 gk <人 《人 人 Out-of-veh time (TS) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Num of transfer (TS) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1 Denied waiting time (TS) of Out-of-veh time (CA) Values Num of transfer (CA) Denied waiting time (CA) 2 1７ 18 19 20 X Random initialization ID-51600 51700 51800 51900 52000 52100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 In-veh time (TS)51535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 51537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5 51540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 51542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5 51547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5 51550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 51552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2 51555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 +(2)Γ1, Γ2 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2}, which represents different perturbations in the speed parameter inputs (Γ1 = Γ2 = 1 means no errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 6 shows that the variability of the walking speed distribution does not show much impact on the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 6: Sensitivity analysis on walking speed inputs Figure 7 shows the estimation results with respect to different input left behind probabilities at Station C, red line, up direction, which indicates the model’s performance if there are estimation errors in left behind probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Similarly, left behind probabilities are chosen for sensitivity analysis because they are values estimated from data and may suffer from errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Three scenarios are compared: actual crowding (20% no left behind, 50% left behind once and 30% left behind twice, which means no errors), less crowding (80% no left behind, 20% left behind once and 0% left behind twice), and more crowding (10% no left behind, 20% left behind once and 70% left behind twice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It can be seen that the parameters of in-vehicle time, x(1), and x(2) are not sensitive to the left behind inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' But other parameters (such as the number of transfers and out-of vehicle time) are highly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The reason may be that errors in left behind estimation can affect the model’s evaluation of other factors’ impacts on travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' That is, an additional 10-minute trip time can be caused by more transfers, or high out-of vehicle time, or left behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' If left behind is not estimated accurately, the impact of other factors on total travel time (and passenger choices) may be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Hence, the results highlight the importance of incorporating crowding and capacity constraints in the estimation of path choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 18 2 i = 1, 「2 = 1 「1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8, 「2 = 1 Values of estimated parameters 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2, 「2 = 1 「1 = 1, 「2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8 「1 = 1,「2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2 0 3 In-veh time K Out-of-veh time (TS) Num of transfer (TS) Denied waiting time (TS) Out-of-veh time (CA) Num of transfer (CA) Denied waiting time (CA) 9 X ParametersFigure 7: Sensitivity analysis on left behind inputs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' MTR empirical case study The proposed method is also applied using actual AFC and AVL data from the Hong Kong MTR network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure 8 shows the MTR network and select OD pair areas (origins in the black dashed box and destinations in the red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' These OD pairs are selected because 1) there are multiple paths between each OD pair, which supports the application of the path choice modeling, and 2) these stations have high enough OD passenger flows to allow the estimation of the left behind probability distribution (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We randomly select 3,425 passengers with trips between these OD pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We consider trips with departure times in the evening peak (5:30 PM - 7:30 PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Finally, a total of 6,425 trips were collected from the AFC data in July 2018 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', on average each passenger had 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='88 trips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The walking time is assumed to follow the log-normal distribution with mean and variance calibrated by MTR employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' αL = −9, αU = 9, and ∆ = 1 are used for numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 19 2 Actual crowding Less crowding Values of estimated parameters More crowding 0 1 2 3 In-veh time logPSm (TS) Num of transfer (TS) Denied waiting time (TS) Out-of-veh time (CA) Num of transfer (CA) Denied waiting time (CA) X X Out-of-veh time ( ParametersFigure 8: Hong Kong MTR network Table 6: Descriptive statistics of the MTR data Variables Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Max Min Individual characteristics Avg # travel days per week 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='12 Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' of 1st trip dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' time (hr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='01 Min # stations with 70% trips 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='15 15 1 If student (Yes = 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='30 1 0 Path attributes In-veh time (min) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='50 Out-of-veh time (min) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='11 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='50 Denied waiting time (min) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='48 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0 Number of passengers: 3,425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Number of trips: 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='425 We assume passenger’s latent groups can be characterized by the following attributes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' readily extracted from AFC data: 1) travel frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' defined as the average number of days with travel per week,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2) schedule flexibility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' measured by the standard deviation of the first trip’s departure time on weekdays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 3) spatial concentration of trips,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' defined as the minimum number of stations that covers 70% of trips in a month,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 4) whether the cardholder is a student or not (dummy variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' All these attributes are calculated based on the AFC data in July 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The descriptive statistics of the data are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Two latent groups are considered for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The reason for considering two groups instead of more is that two latent groups are more interpretable in terms of estimation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The path attributes are the same as the synthetic data experiment except for the “number of transfers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The “number of transfers” is dropped from the model due to its high correlation with the “out-of-vehicle time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Similar to the synthetic data experiment, we make the parameters of in-veh-time the same for the two groups so that we can compare the scales of out-of-veh time and the number of transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 20 O Origin stations Destination stationsTable 7: Estimation results for the real-world data Variables Group 1 (CA) Group 2 (TS) Estimation t-value Estimation t-value Choice model In-veh time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2785 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='13 Same as Group 1 Out-of-veh time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1320 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='7457 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='04 Denied waiting time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0450 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4517 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='90 log PSm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2611 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='38 Same as Group 1 σk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='7294 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='99 Same as Group 1 Latent group model (Group 2 is set as the base group) ASC1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9644 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='83 0 (fixed) Avg # travel days per week 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0987 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='91 0 (fixed) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' of 1st trip dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='8667 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='19 0 (fixed) Min # stations with 70% trips 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='40 0 (fixed) If student (Yes = 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='0192 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='14 0 (fixed) 1: ASC: alternative specific constant Number of passengers: 3,425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Number of observations: 6,425 LL0: -32,157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='84;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Final LL∗: -30,867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='48 χ2 = −2(LL0 − LL∗) = 2, 580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='72, likelihood ratio test p-value: 0 The estimation results are shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The two latent groups are referred to as Group 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Group 2 is set as the base alternative in the group-assignment multinomial logit model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Results show that the signs and scales of all parameters are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' All time-related parameters are significantly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The value of the in-vehicle time parameter is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2785, which is similar to the survey results (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2676) (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' For both Groups 1 and 2, the absolute values of the parameters for out-of-vehicle time and denied waiting time are both greater than that of the in-vehicle travel time, reflecting that passengers were more sensitive to walking and waiting times compared to the time seated in vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' And these results are also consistent with the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Comparing the results of Group 1 and 2, we observe that the out-of-vehicle and denied waiting times show a larger impact on the path choice utility for passengers in group 1, which implies that Group 1 is possibly comfort-aware (CA) and Group 2 is time-sensitive (TS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The parameters determining the latent groups indicate that CA passengers have less travel frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the effect of avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' # travel days per week is negative) and more schedule flexibility (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the effect for the std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' of the 1st trip departure time is positive), and students are more likely to belong to this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' These suggest that CA passengers are most likely to be irregular users and they may mainly use the MTR system for non-work activities (such as entertainment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Hence, they care more about the trip comfort and prefer paths with less walking and waiting times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' In contrast, TS passengers have higher travel frequency and less schedule flexibility and they may use the metro system mostly for regular commuting trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Besides, they care more about saving the total travel time to arrive at their destinations on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The spatial concentration variable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', min # stations with 70% trips), though having a positive impact on being in the CA group, is not statistically significant (t-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' σk is significant for both groups, which means that the panel effects are diverse across populations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', some have more stable travel patterns but are not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Conclusion Understanding passenger path choices are important for operations management in urban rail systems, especially those with crowded conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' This paper presents a probabilistic approach for the path choice model estimation with train capacity constraints using AFC (tap-in and tap-out) and AVL data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The choice heterogeneity and longitudinal behavioral correlations are captured by a latent class model with panel effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Passenger’s movement is formulated using a passenger to train assignment model with explicit modeling of the processes of access/egress, left behind (crowding), and transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' A tractable likelihood function is derived to facilitate the model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The t-value of estimated parameters is calculated based on the numerically estimated Hessian matrix and Cramer–Rao bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The method is data-driven, flexible to accommodate different choice models, and easy to solve using non-linear optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The model performance is validated using synthetic data to estimate the individual choice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The sensitivity analysis affirms its robustness to parameter initialization and small errors in inputs (walking speed and left behind distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It also highlights that neglecting capacity constraints (left behind) can lead to biased estimation of path choice parameters under crowding conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The model is also applied using real-world data from the MTR system in Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' It reveals two different groups of passengers: time-sensitive (TS) and comfort-aware (CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' TS passengers are generally regular commuters with high travel frequency and small schedule flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' They are more likely to choose paths with less trip travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' CA passengers care more about travel comfort and prefer choosing paths with less walking and waiting times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Example policy implications can be derived from the case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' As these two groups of passengers value in-vehicle and out-of-vehicle times differently, transit agencies can use this insight to design customized route recommendation systems with better passenger acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Moreover, route recommendations can help to relieve congestion by recommending TS and CA passengers to use different routes during peak hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Interesting future research directions include: 1) Explore the evolution of choice preferences and learning behavior over time under network interventions (such as network extension and demand management policies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 2) Develop a downstream model to utilize the latent passenger group information for better route recommendations or fare policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Authors’ contribution Baichuan Mo: Conceptualization, Methodology, Software, Formal analysis, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Zhenliang Ma: Conceptualization, Methodology, Supervision, Formal analysis, Data Curation, Writing - Original Draft, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Haris N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Koutsopoulos: Conceptualization, Supervision, Formal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Jinhua Zhao: Conceptualization, Supervision, Project administration, Funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Acknowledgement The authors would like to thank Chicago Transit Authority (CTA) for their support and data availability for this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 22 Appendices Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Left behind probability calibration The left-behind probability can be estimated from a Gaussian mixture model proposed by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The main idea is that passengers being left behind different times would have different journey times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Let tJn i be the random variable indicating the journey time of passenger i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', tap-out time minus the tap-in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9 shows an example of journey time distribution between a specific OD pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' We observe there are three clusters, indicating passengers being left behind 0, 1, and 2 times at the origin station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Hence, we can model the journey time distribution as a Gaussian mixture model: Pr(tJn i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' µ, σ, w) = C � c=0 wc · Φ(µc, σc) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1) where wc is the (unknown) fraction of passengers in cluster c, wc > 0 and �C c=0 wc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' C is the total number of clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the maximum number of left behind times at the origin station).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Φ(µc, σc) is the PDF for the Gaussian distribution N(µc, σc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='9: Example of journey time distribution The Gaussian mixture model can be estimated by solving the following problem: max µ,σ,w � i∈P′ log(Pr(tJn i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' µ, σ, w)) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Auxiliary Constraints (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2b) � c∈C wc = 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2c) wc ≥ 0 ∀c = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', C (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2d) where P′ is the passenger set used for the left behind probability estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The auxiliary constraints are used for model stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' These constraints contain prior human knowledge on the journey time distribution, such as the difference between µc and µc+1 should be close to a headway, the mean journey time without being left behind (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', µ0) should be close to the sum of access, egress, and in-vehicle times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' More information on the model can be found in Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 23 on 02The model is station and time-specific, which enables the calibration of left behind probabilities for each station at different time intervals in the system (by adjusting P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The estimated wc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', the fraction of passengers in cluster c) is the probability of being left behind c times at the corresponding station and time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Numerical calculation of Hessian matrix According to Fornberg (1988), given a general function f(x, y), the second derivative with a fourth-order accuracy can be calculated as ∂2f(x, y) ∂x2 |x0,y0 = 1 h2x �−1 12 f(x−2, y0) + 4 3f(x−1, y0) +−5 2 f(x0, y0) + 4 3f(x+1, y0) + −1 12 f(x+2, y0) � + O(h4 x) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='1) and ∂2f(x, y) ∂x∂y |x0,y0 = 1 hxhy �−1 48 f(x−2, y−2) + 1 3f(x−1, y−1) + −1 3 f(x−1, y+1) + −1 3 f(x+1, y−1) + 1 3f(x+1, y+1) + 1 48f(x+2, y−2) + 1 48f(x−2, y+2) + −1 48 f(x+2, y+2) � + O(h2 xh2 y) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content='2) where hx and hy are small perturbations for x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' xk (yk) represents x0 + khx (y0 + khy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' The derivation is based on Taylor’s series expansion with uniform grid spacing (Fornberg, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' References Arellano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', Honoré, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=', 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Panel data models: some recent developments, in: Handbook of economet- rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} 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model for congested urban rail networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies 123, 102896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E2T4oBgHgl3EQfUAdV/content/2301.03808v1.pdf'} diff --git a/HtAzT4oBgHgl3EQfxv41/vector_store/index.faiss b/HtAzT4oBgHgl3EQfxv41/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3837a14594d2f264bce95a0b9773dcc489ee7511 --- /dev/null +++ b/HtAzT4oBgHgl3EQfxv41/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:60648c8e257bb2a7b99dae374fa482ad940be71fd32762bf41d4a86cb182f2da +size 852013 diff --git a/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf 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irreducibility in +terms of the exactness of functorial correspondence between a category of data structures and elementary +computations and a corresponding category of (1-dimensional) cobordisms. We proceed to demonstrate +that, by equipping both categories with a symmetric monoidal structure and considering the case of +higher-dimensional cobordism categories, we obtain a natural extension of this formalism that serves +also to encompass non-deterministic or “multiway” computations, in which one quantifies not only the +irreducibility in the behavior of a single (deterministic) computation path, but in the branching and +merging behavior of an entire “multiway system” of such paths too. +We finally outline how, in the +most general case, the resulting symmetric monoidal functor may be considered to be adjoint to the +functor characterizing the Atiyah-Segal axiomatization of a functorial quantum field theory. Thus, we +conclude by arguing that the irreducibility of (multi)computations may be thought of as being dual to +the locality of time evolution in functorial approaches to quantum mechanics and quantum field theory. +In the process, we propose an extension of the methods of standard (monoidal) category theory, in +which morphisms are effectively equipped with intrinsic computational complexity data, together with +an algebra for how those complexities compose (both in sequence and in parallel, subject to the monoidal +structure). Some possible extensions of this formalism (for instance to encompass notions of causality, +space complexity, irreversibility, complexity of index manipulation in tensor networks, etc.), as well as +potential implications for physics (for instance by providing an ability to distinguish formally between +certain computational and multicomputational definitions of entropy) are briefly discussed. +∗gorardj@cardiff.ac.uk +†jg865@cantab.ac.uk +1 +arXiv:2301.04690v1 [cs.CC] 13 Oct 2022 + +1 +Introduction +Computational irreducibility, as first proposed by Stephen Wolfram in the 1980s[1], explored empirically +within A New Kind of Science[2] (NKS) and subsequently refined and formalized by the author in later +work[3], refers to the general phenomenon in which the outcome of any sufficiently sophisticated computa- +tional process cannot be predicted (or “shortcut”) using any less computational effort than the system itself +requires for its own explicit evolution. In this way, computational irreducibility may be considered to be an +extension of the standard recursion-theoretic concepts of universality and undecidability[4][5][6] (indeed, it +is straightforward to prove by diagonalization that any Turing-complete computation must be irreducible, +and undecidability may be thought of as corresponding to a limiting case of irreducibility in which certain +computations become infinite in length). Within the NKS paradigm (in which all natural processes are con- +ceived of as corresponding to computations), traditional methods of theoretical science thus correspond to +those cases in which computations are reducible, and can therefore be preempted by means of sufficiently so- +phisticated tools of scientific and mathematical analysis. The abstract phenomenon of complexity in natural +systems may, in turn, be thought of as corresponding to those cases in which computations are fundamentally +irreducible, and therefore in which the only reasonable course of action available to the theoretical scientist +is to simulate the behavior of the system explicitly[7][8][9]. +The present article seeks to recast the author’s previous formal definition of computational irreducibility +within the broader conceptual framework of functoriality[10]. In Section 2, by considering a category whose +objects are data structures and whose morphisms are elementary computations (in the first instance, the +article employs Turing machines as its chosen formal model of computation, whose corresponding category +has as its objects configurations of the Turing machine’s tape/head and head position, and as its morphisms +applications of the Turing machine’s partial transition function), we are able to reformulate the definition +of reducibility/irreducibility in terms of the compositional properties of a certain map from this category to +a category whose objects are time steps/step numbers and whose morphisms are discrete intervals between +these step numbers. This has the effect of equipping the morphisms of our original category of Turing machine +states and computations with intrinsic computational complexity data, along with a natural algebra (encoded +by the action of the map on the composition operation) describing how the corresponding time complexities +compose. Computational irreducibility then corresponds to the case where these complexities compose purely +additively, and therefore in which this map is a pure functor; this allows us to say, in a rather precise sense, +that irreducibility is functoriality. Computational reducibility is then a measure of the extent to which this +map distorts pure additive composition of time complexities, i.e. it is a measure of deviation away from +2 + +pure functoriality. The codomain of this map (i.e. the category of step numbers and discrete intervals) +may be interpreted as being a certain discretization of a (1-dimensional) category of cobordisms/“gluings of +boundaries” between (0-dimensional) manifolds[11][12]. +In Section 3, we proceed to demonstrate that, when our category of data structures and elementary com- +putations is also equipped with a symmetric tensor product structure (thereby promoting it to a symmetric +monoidal category[13]), which allows us to describe a multiway system of many different branching and +merging (singleway) computation paths, with the branchial graphs of that multiway system describing the +tensor product structure of the corresponding monoidal category, then there exists a very natural extension +of the previous irreducibility formalism for ordinary categories and singleway computations to encompass +the multicomputational case too. Just as (singleway) computational reducibility may be formulated as a +measure of distortion of the additivity of time complexity under ordinary (sequential) composition of com- +putations, multicomputational reducibility may therefore be formulated as a measure of distortion of the +additivity of time complexity under parallel (tensor product) composition of computations, and hence, a +multicomputationally irreducible computation is one under which the map from the (monoidal) category of +data structures and elementary computations to the (monoidal) category consisting of parallel compositions +of step numbers and parallel compositions of discrete intervals (effectively describing the coordinatization of +branchial graphs), is a pure symmetric monoidal functor. This idea formally encodes and concretizes the +intuition that a singleway computation is irreducible if one must explicitly trace every intermediate step +in the composition in order to determine the final result, whereas a multiway computation is irreducible +if one must explicitly trace every singleway computation path in order to determine the system’s overall +branchial structure. The codomain of this map between symmetric monoidal categories is now interpretable +as a certain discretization of a higher-dimensional category of cobordisms between higher-dimensional man- +ifolds (equipped with a Riemannian structure). We discuss how ordinary computational irreducibility is a +byproduct of the complexity of the state evolution function used in the specification of a multiway system, +whilst multicomputational irreducibility is a byproduct of the complexity of its state equivalence function, +and emphasize that the two concepts are therefore essentially orthogonal (e.g. one can easily build mul- +ticomputationally irreducible systems out of tensor products of many computationally reducible singleway +paths). To illustrate this fact, we also analyze the case of multiway hypergraph rewriting systems, in which +the state evolution function has comparable computational complexity to that of Turing machine evolution, +but the state equivalence function (based on hypergraph isomorphism) is now far more non-trivial. +In Section 4, we illustrate how, in a certain general sense, the functor from data structures and compu- +3 + +tations to step numbers and intervals that characterizes an irreducible computation may be considered to +be the left adjoint of the functor encoding the passage from moments of time and intervals to vector spaces +and linear isomorphisms in the context of categorical quantum mechanics[14]. Thus, the same functoriality +that characterizes the irreducibility of a computation may be thought of as being formally dual/adjoint to +the functoriality that characterizes the locality of time evolution in non-relativistic quantum mechanics (in +which any global unitary evolution over an interval may be decomposed into several local unitary evolutions +over subintervals). In a fairly natural extension of this idea, we show how the symmetric monoidal func- +tor encoding an irreducible multicomputation can be viewed as the left adjoint of the symmetric monoidal +functor from manifolds and cobordisms to vector spaces and linear isomorphisms that defines a functorial +quantum field theory, with the functoriality of multicomputational irreducibility now being dual/adjoint to +the Atiyah-Segal sewing laws[15][16][17], in which the path integral over any global interval may be decom- +posed into local path integrals over subintervals. Hence, we argue for the existence of a general adjunction +relationship between computational complexity theory and categorical quantum mechanics, and between +multicomputational complexity theory and functorial quantum field theory. We discuss how some of the +other algebraic structure inherent to categorical quantum mechanics and functorial quantum field theory +models (in particular, the involutive dagger and compact structures of the corresponding categories) might +therefore have potential complexity-theoretic interpretations (for instance, in terms of the irreducibility of +reversing computations, or of swapping arguments and values in a composition of multi-argument, multi- +valued functions as described by a tensor network or monoidal string diagram, etc.), although a complete +analysis of these interpretations lies beyond the scope of the present work. +Finally, in Section 5, we outline some of the broader implications of the formalism developed herein, +including the generalization of techniques of ordinary (monoidal) category theory to accommodate the case +where morphisms designate explicit computations with associated computational complexity classes, and thus +in which composition operations must respect the underlying algebra for how the complexities of those various +computations interact. We also discuss the possibility of using these new algebraic techniques to conduct +a more systematic investigation of computational complexity classes and (multi)computational complexity +classes that are in some way “wilder” or less structured (in the sense that their algebra of composition is +less well-behaved) than those studied within the setting of traditional computational complexity theory; we, +moreover, indicate the various ways in which multicomputational complexity classes differ from traditional +non-deterministic complexity classes (namely by considering the inherent computational complexity of the +branching and merging operations of the multiway system, rather than solely considering the result yielded +4 + +non-deterministically by a single multiway branch). We summarize some of the potential implications for +physics, including a plausible disambiguation of two different computational definitions of entropy - one +in which microstates are treated as being possible instantaneous states of a system (and thus based on +computational irreducibility), and one in which microstates are treated as being possible paths of evolution +history for the system (and thus based on multicomputational irreducibility) - that are often otherwise +conflated. We propose some possible future directions of investigation, including the extension of the relevant +maps/functors to also encompass preservation of dagger structure and/or compact structure, the extension +of the overall formalism to encompass the additional information encoded within the structure of causal +relations via certain higher categories/higher functors and the extension to multi-argument computations as +encoded through glocal multiway systems and their description as tensor networks/string diagrams. +Note that the majority (though by no means all) of the categories considered within this article are +small (and all of the ones which are not are sufficiently widely studied that their behavior is known to be at +least somewhat well-behaved). For this reason, we shall henceforth neglect all considerations of set-theoretic +issues, and shall use terms such as “object set” and “hom set” irrespective of whether the structures involved +are sets or proper classes. Moreover, note that the code necessary to reproduce the results, proofs and figures +from this article is open source and freely exposed through the Wolfram Function Repository, for instance +via MultiwayTuringMachine, TuringMachineGlocalMultiwaySystem and MultiwaySystem for system evo- +lution, AbstractCategory, AbstractFunctor and AbstractStrictMonoidalCategory for representation of the +underlying category-theoretic structures (and for automating the process of theorem-proving over them), +etc. +2 +(Singleway) Irreducibility as Functoriality +We begin by adopting the formal definition of computational irreducibility proposed by the author in [3]: +Definition 1 If f : N → N is a function on natural numbers and T is a Turing machine that computes the +value of f (i) for some fixed input i in n steps, then T’s computation is reducible if and only if there exists +a Turing machine T ∗ that computes f (i) in m steps, where m < n[5][6]. +The “degree” of reducibility of the computation may thus be quantified in terms of the discrepancy, i.e. the +value of n − m (the converse value, i.e. m − n for the case of irreducible computations in which m ≥ n, is +known as the slowdown of T ∗’s simulation of T). Note that we are here and henceforth assuming that all +Turing machines are 1-tape Turing machines,[7] which we can do without loss of generality since any k-tape +5 + +Turing machine M operating in time f (n) may be simulated by a 1-tape Turing machine M ′ operating +in time O +� +[f (n)]2� +, i.e. for any input x, M ′ (x) = M (x)[18][19]. We have also (implicitly) adopted the +Hopcroft-Ullman formalization[6] of a 1-tape Turing machine T as a 7-tuple T = ⟨Q, Γ, b, Σ, δ, q0, F⟩, with +finite alphabet set Γ ̸= ∅, blank symbol b ∈ Γ, input symbols Σ ⊆ Γ \ {b}, finite state set Q ̸= ∅, initial state +q0 ∈ Q, accepting states F ⊆ Q and (partial) transition function: +δ : (Q \ F) × Γ ↛ Q × Γ × {L, F} . +(1) +We are now in a position to be able to construct a category[10] representing a particular formal (abstract) +model of computation, whose objects are data structures and whose morphisms are elementary/primitive +computations. For the specific case considered here of the category generated by the Turing machine T, +which we shall denote T , we choose as its object set ob (T ) the set Γℵ0 × Q × N of possible ordered triples +consisting of tape state, head state and head position (assuming an infinite-length tape), and we wish for +its morphism set hom (T ) to consist of all possible valid transitions of the Turing machine T. In order to +construct this morphism set, we therefore begin by constructing a quiver (i.e. a directed multigraph) whose +arrows/edges correspond to applications of the (partial) transition function δ; for our present purposes, it +suffices to think of the set of all possible such arrows as being N × +� +Γℵ0 × Q × N +�2, i.e. each arrow is treated +as an ordered triple (f, X, Y ), denoted f : X → Y , consisting of a name f ∈ N, and a pair of elements +X, Y ∈ ob (T ) (X being the arrow’s source/domain and Y being its target/codomain). This quiver freely +generates a category if we populate the morphism set hom (T ) initially with the set of arrows/transitions, +and proceed to introduce a binary composition operator ◦ such that: +∀ (f : X → Y ) , (g : Y → Z) ∈ hom (T ) , +(g ◦ f : X → Z) ∈ hom (T ) , +(2) +i.e. any pair of morphisms with matching codomain and domain can be composed by means of the operator +◦. +If X, Y and Z represent possible Turing machine states (including tape state, head state and head +position), and f and g represent possible Turing machine transitions, this procedure of generating a category +from the underlying quiver can be illustrated diagrammatically as follows: +Y +Y +X +Z +X +Z. +g +g +f +f +g◦f +(3) +The resulting algebraic structure is indeed a category, since the operator ◦ inherits associativity: +6 + +∀ (f : X → Y ) , (g : Y → Z) , (h : Z → W) ∈ hom (T ) , +((h ◦ g) ◦ f : X → W) = (h ◦ (g ◦ f) : X → W) , +(4) +from the fact that (partial) function composition is associative, and the identity axiom, namely: +∀X ∈ ob (T ) , +∃ (idX : X → X) ∈ hom (T ) , +(5) +such that: +∀ (f : X → Y ) ∈ hom (T ) , +(f ◦ idX : X → Y ) = (f : X → Y ) , +(6) +and: +∀ (g : Y → X) ∈ hom (T ) , +(idX ◦ g : Y → X) = (g : Y → X) , +(7) +holds by virtue of the fact that the (partial) transition function may be augmented by a neutral (“no shift”) +operation id as follows: +δ : (Q \ F) × Γ ↛ Q × Γ × {L, R, id} . +(8) +Note that, since the Turing machines considered here are all deterministic/classical, meaning that the (par- +tial) transition function is always single-valued, it follows that the resulting quiver must take the form of +either a path graph, a cycle graph or a union of path and cycle graphs. An explicit example of this construc- +tion, for the 2-state, 2-color Turing machine shown in Figure 1 (known as rule number 2506 in the canonical +Turing machine enumeration) is shown in Figure 2. +Figure 1: A graphical representation of the 2-state, 2-color Turing machine rule number 2506, with the black +icon representing the location and state of the head. +The process of obtaining the full category T representing the Turing machine T may consequently be +7 + +Figure 2: On the left, a graph-theoretic representation of the evolution of the 2-state, 2-color Turing machine +rule 2506, starting from the tape state {0, 1, 0, 0}, for 4 evolution steps; each arrow/edge of the quiver +represents a single application of the Turing machine’s transition function. On the right, a graph-theoretic +representation of the category that is freely generated from this quiver. +thought of graph-theoretically as taking a reflexive transitive closure of the underlying transition quiver. +However, this operation of taking transitive closures appears, at least conceptually, to be very much against +the spirit of computational irreducibility - it seems intuitively to imply that whenever there exists a com- +putation from X to Y , and another computation from Y to Z, then one can necessarily always “jump +ahead” to get directly from X to Z with the same amount of computational effort. Ideally, we would like to +consider some form of “decorated” category in which morphisms are tagged with some additional metadata +corresponding to the computational complexity of the underlying computation that the morphism signifies; +we could then proceed to define an algebra to describe formally how these complexities behave under the +action of the composition operator ◦. Under such an algebra, irreducible computations would correspond +to those computations whose complexities behave purely additively under composition, i.e. if the computa- +tion underlying morphism f : X → Y requires at least n steps to execute, and the computation underlying +morphism g : Y → Z requires at least m steps to execute, then the composite computation g ◦ f : X → Z is +irreducible (under the definition given above) if and only if it cannot be executed in fewer than n + m steps. +Conversely, if the complexities of the computations behave strictly subadditively under composition, then +8 + +the composite computation is reducible (with the “degree” of subadditivity corresponding to the “degree” +of reducibility). This intuition is illustrated in Figure 3 for the case of the 2-state, 2-color Turing machine +rule considered above, with each edge/morphism tagged with the number of transitions necessary to per- +form the corresponding computation; the composition operation here is purely additive, indicating that all +computations shown are irreducible. +Figure 3: A graph-theoretic representation of the category that is freely generated from the quiver repre- +senting the evolution of the 2-state, 2-color Turing machine rule 2506, with edges/morphisms tagged with +additional metadata corresponding to the number of “steps” (i.e. transitions) required to perform the req- +uisite computation. +In order to formalize this intuition, let us proceed on the assumption that all computations are fully irre- +ducible. We can now consider a function Z′ mapping from our category of data structures and computations +(for the case of Turing machines, this is the category T defined above) to a category B whose objects are +natural numbers (i.e. ob (B) = N) corresponding to step numbers/moments of time, and whose morphisms +are discrete intervals between these step numbers/moments of time, i.e: +hom (B) = {[n, m] ∩ N |n, m ∈ N and n ≤ m} . +(9) +If we now equip B with a composition operation given by the union of discrete (contiguous) intervals ∪, then it +9 + +trivially forms a category (with [n, n] ∩ N = {n} being the identity morphism for any n ∈ N), and, moreover, +since all computations are irreducible (and therefore computational complexities are purely additive under +composition) by hypothesis, the function Z′ : T → B trivially forms a functor[20]. More precisely, Z′ is a +map between categories T and B, i.e: +∀X ∈ ob (T ) , +∃Z′ (X) ∈ ob (B) , +(10) +and: +∀ (f : X → Y ) ∈ hom (T ) , +∃ (Z′ (f) : Z′ (X) → Z′ (Y )) ∈ hom (B) , +with (Z′ (f) : Z′ (X) → Z′ (Y )) = [Z′ (X) , Z′ (Y )] ∩ N, +(11) +such that the structure of composition is preserved: +∀ (f : X → Y ) , (g : Y → Z) ∈ hom (T ) , +(Z′ (g ◦ f) : Z′ (X) → Z′ (Z)) = (Z′ (g) ∪ Z′ (f) : Z′ (X) → Z′ (Z)) = [Z′ (X) , Z′ (Z)] ∩ N, +(12) +and all identity morphisms are preserved: +∀X ∈ ob (T ) , +(Z′ (idX) : Z′ (X) → Z′ (X)) = +� +idZ′(X) : Z′ (X) → Z′ (X) +� += [Z′ (X) , Z′ (X)] ∩ N = {Z′ (X)} . +(13) +If X, Y and Z represent Turing machine states reached at step numbers t1, t2 and t3 respectively, and +f : X → Y and g : Y → Z represent the transitions between states X and Y and states Y and Z respectively, +then this functoriality between categories T and B can be illustrated diagrammatically as follows: +10 + +Y +t2 +X +Z +t1 +t3. +g +[t2,t3]∩N +f +g◦f +[t1,t2]∩N +([t2,t3]∪[t1,t2])∩N +=[t1,t3]∩N +Z′ +(14) +In this way, the cardinality of any morphism in B (e.g. |[t1, t2] ∩ N|) represents, up to an additive constant, the +time complexity of the corresponding computation in T (e.g. f : X → Y ). Since this functorial relationship +holds only in the case where all computations in T are irreducible, we are consequently able to conclude +that, in some precise sense, computational irreducibility is functoriality, and, moreover, that computational +reducibility corresponds precisely to a deformation of the map Z′ away from being a pure functor. This +construction is demonstrated in Figure 4 for the 2-state, 2-color Turing machine described previously, with +each vertex/object tagged with its step number and each edge/morphism tagged with a list of step numbers +traversed throughout the course of its corresponding computation. +Figure 4: A graph-theoretic representation of the category that is freely generated from the quiver repre- +senting the evolution of the 2-state, 2-color Turing machine rule 2506, with vertices/objects tagged with +additional metadata corresponding to the step number on which they occur (shown in blue) and with +edges/morphisms tagged with additional metadata corresponding to all intermediate step numbers traversed +as part of the requisite computation (shown in black). +Note that the axioms obeyed by this “algebra of complexity” are formally almost identical to those +11 + +1, 2,3, 4 +3 +[2. 3, 4] +3,4,5 +[1, 2, 3, 4, 5] +[3. 4, 5]of a metric space: the complexity of applying the identity transition id mapping from a state to itself is +always 0 (or possibly 1, depending upon precisely how the intervals are defined), the complexity of applying +any transition between two distinct states is always strictly positive, and the complexities always obey the +triangle inequality (i.e. non-strict subadditivity) under composition; the only metric space axiom for which +there exists no immediate analog is the symmetry axiom, since transitions need not necessarily be reversible, +and the complexity of reversing a transition (when such a reversal exists) need not necessarily be equal to the +complexity of performing the transition. Note, moreover, that, if we modify the definition of the category B +such that its objects are not merely natural numbers but real ones (i.e. ob (B) = R), and its morphisms are +not merely discrete intervals but continuous ones, i.e: +hom (B) = {[x, y] |x, y ∈ R and x ≤ y } , +(15) +with the composition operation still being the standard union of contiguous intervals ∪, then nothing in the +above argument is substantively modified: the category T is still in functorial correspondence with the new +version of category B if and only if it was in functorial correspondence with the old category B. The only +material difference that this modification makes is that it guarantees that the functor/map Z′ (functor in +the irreducible case, map in the reducible one) cannot be surjective on objects, but there was no requirement +for it to be so in the first place. However, this slightly generalized definition of the category B does carry the +definite advantage of making the underlying topological intuition of this construction manifest; the objects +of R (real numbers, representing moments of time) can be thought of as being 0-dimensional manifolds, and +its morphisms can be thought of as being 1-dimensional cobordisms between those manifolds. Thus, B is +really a cobordism category[21]. +Here and henceforth, in relation to category B and its descendants, we employ the formal definition of a +cobordism category as an ordered triple ⟨B, ∂, i⟩, where B has all finite coproducts and is equipped with an +initial object ∅, ∂ : B → B is an additive endofunctor that preserves all coproducts and i : ∂ ⇒ IdB (with IdB +denoting the identity functor on B that sends every object/morphism to itself) is a natural transformation +satisfying: +∀M ∈ ob (B) , +∂ (∂ (M)) = ∅. +(16) +In the above, the coproduct of a pair of objects M1, M2 ∈ ob (B) is taken to be an object M1 ⊕ M2 ∈ ob (B) +equipped with a pair of canonical injection morphisms: +12 + +(i1 : M1 → M1 ⊕ M2) , (i2 : M2 → M1 ⊕ M2) ∈ hom (B) , +(17) +that are universal, in the sense that, for any object M ∗ ∈ ob (B) equipped with morphisms: +(i∗ +1 : M1 → M ∗) , (i∗ +2 : M2 → M ∗) ∈ hom (B) , +(18) +there exists a unique morphism (u : M1 ⊕ M2 → M ∗) ∈ hom (B) such that: +(i∗ +1 : M1 → M ∗) = (u ◦ i1 : M1 → M ∗) , +and +(i∗ +2 : M2 → M ∗) = (u ◦ i2 : M2 → M ∗) . +(19) +This condition may be restated concisely by means of the following commutative diagram: +∀M ∗ +M1 +M1 ⊕ M2 +M2. +∀i∗ +1 +i1 +∃!u +∀i∗ +2 +i2 +(20) +This definition can be extended in the obvious way to any finite collection of objects Mj ∈ ob (B) to yield a +finite coproduct � +j +Mj ∈ ob (B) equipped with a finite collection of (universal) injection morphisms: +� +ij : Mj → +� +k +Mk +� +∈ hom (B) . +(21) +An initial object ∅ ∈ ob (B) is a distinguished object such that, for any object M ∈ ob (B), there exists a +unique morphism (u : ∅ → M) ∈ hom (B), i.e: +∅ +∀M. +∃!u +(22) +An endofunctor is any functor from a category to itself, and an additive functor is any functor that preserves +finite coproducts (or, in certain contexts, finite biproducts, though this is not the case considered here); in +other words, zero objects are preserved (up to isomorphism): +∂ (∅) ∼= ∅ ∈ ob (B) , +(23) +and, for any pair of objects M1, M2 ∈ ob (B), there exists an isomorphism: +13 + +∂ (M1 ⊕ M2) ∼= ∂ (M1) ⊕ ∂ (M2) , +(24) +that preserves the canonical injection morphisms of the coproduct construction: +M1 ⊕ M2 +∂ (M1 ⊕ M2) ∼= ∂ (M1) ⊕ ∂ (M2) +M1 +M2 +∂ (M1) +∂ (M2) . +i1 +i2 +∂(i1) +∂(i2) +∂ +(25) +An isomorphism (indicated by ∼=) here refers to any morphism (f : X → Y ) ∈ hom (B) for which there exists +a corresponding morphism +� +f −1 : Y → X +� +∈ hom (B) that acts as both a left and right inverse of f, i.e: +� +f −1 ◦ f : X → X +� += (idX : X → X) , +and +� +f ◦ f −1 : Y → Y +� += (idY : Y → Y ) . +(26) +Finally, a natural transformation η : F ⇒ G between functors F : C → D and G : C → D is characterized by +a family of component morphisms: +∀X ∈ ob (C) , +∃ (ηX : F (X) → G (X)) ∈ hom (D) , +(27) +such that one has: +∀ (f : X → Y ) ∈ hom (C) , +(ηY ◦ F (f) : F (X) → G (Y )) = (G (f) ◦ ηX : F (X) → G (Y )) , +(28) +or, represented in the form of a commutative diagram: +F (X) +F (Y ) +G (X) +G (Y ) . +F (f) +ηX +ηY +G(f) +(29) +The intuition lying behind this formalization of cobordism categories can be articulated as follows. Every +object M ∈ ob (B) is interpreted as a (generalized) manifold; every morphism (f : M1 → M2) ∈ hom (B) is +interpreted as a (generalized) cobordism between those manifolds, i.e. a “gluing” together of those manifolds +along their boundaries. +The initial object ∅ plays the role of the empty set/empty manifold, and the +14 + +additive endofunctor ∂ represents the boundary relation: ∂ (M) (for some M ∈ ob (B)) corresponds to the +(generalized) boundary of manifold M, with the condition that ∂ (∂ (M)) = ∅ thus designating that the +boundary of a boundary is always empty. Coproducts ⊕ in the cobordism category B play the role of the +direct sum/disjoint union of manifolds. A pair of objects/manifolds M1, M2 ∈ ob (B) may be said to be +cobordant (in the sense of their disjoint union forming the boundary of a manifold in one dimension higher), +written M1 ∼ M2, if and only if: +∃V1, V2 ∈ ob (B) , +such that +M1 ⊕ ∂ (V1) ∼= M2 ⊕ ∂ (V2) , +(30) +where ∼= is an isomorphism in B; the cobordism relation ∼ is hence an equivalence relation in which all +isomorphic objects/manifolds are cobordant: +∀M1, M2 ∈ ob (B) , +M1 ∼= M2 =⇒ M1 ∼ M2, +(31) +and where: +∀M ∈ ob (B) , +∂ (M) ∼ ∅. +(32) +This connection to cobordism categories may appear to be a rather trivial technical point, but it will +proceed to play a crucial role in the forthcoming discussion on formal correspondences with categorical +quantum mechanics and functorial quantum field theory[14], and the relationship between computational +irreducibility and the locality of time evolution. +3 +Multicomputational Irreducibility as Monoidal Functoriality +We now proceed to consider the case of non-deterministic (multiway) computations, in which the (partial) +transition function acting on data structures/computational states is not necessarily single-valued. Such +computations may be parameterized by means of a multiway system[22], or, more precisely, a multiway +evolution graph[23][24], namely a directed acyclic graph whose vertices represent computational states and +whose directed edges represent single-step transitions between those states. +An explicit example of the +multiway system construction, for a pair of 2-state, 2-color Turing machine rules shown in Figure 5 (rule +numbers 2506 and 3506 in the canonical Turing machine enumeration) is shown in Figure 6. By considering +the resulting multiway evolution graph as a quiver, we are able to construct the category that this quiver +15 + +freely generates via the same procedure outlined in the previous section, as shown in Figure 7. +Figure 5: A graphical representation of two 2-state, 2-color Turing machine rules (rule numbers 2506 and +3506), with the black icons representing the locations and states of the heads. +Figure 6: A multiway evolution graph corresponding to the non-deterministic evolution of a 2-state, 2-color +Turing machine constructed from the (parallel) composition of the Turing machine transition functions for +rules 2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 4 steps; each edge represents a single +application of one of the two Turing machine transition functions. +However, we are now able to construct a deterministic/singleway computation from this non-deterministic/multiway +one by exploiting the formalism of branchial graphs, through a procedure that is more-or-less directly anal- +ogous to the Rabin-Scott powerset/subset construction[25] for converting non-deterministic finite automata +into deterministic ones. More concretely, we “foliate” the multiway evolution graph into an ordered sequence +of non-intersecting “branchlike hypersurfaces” Σt that cover the entire multiway evolution graph, with the +ordering relation defined by a universal time function: +16 + +Figure 7: A graph-theoretic representation of the category that is freely generated by the multiway evolution +graph corresponding to the non-deterministic evolution of a 2-state, 2-color Turing machine constructed from +the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506 (considered as +a quiver), starting from the single tape state {0, 1, 0, 0}, for 3 steps. +t : V → Z, +such that +∆t ̸= 0 everywhere, +(33) +where V is the vertex set of the multiway evolution graph (i.e. the set of reachable Turing machine states), +and where each “branchlike hypersurface” is now a level set of this function, satisfying: +∀t1, t2 ∈ Z, +Σt1 = {p ∈ V : t (p) = t1} , +and +Σ1 ∩ Σ2 = ∅ ⇐⇒ t1 ̸= t2. +(34) +Branchial graphs constitute a discrete/combinatorial representation of these abstract branchlike hypersur- +faces, in which the common ancestry distance between state vertices is represented for any given value of +the universal time function, i.e. vertices X and Y in the branchial graph for a given time step are connected +by an undirected edge in the branchial graph if and only if they share a common ancestor state Z in the +multiway evolution graph. The default choice of foliation for the multiway evolution graph corresponding +to the evolution of the non-deterministic 2-state, 2-color Turing machine discussed above is shown in Figure +8, with the associated sequence of branchial graphs shown in Figure 9. We can therefore think of every +multiway system as being equipped with a certain tensor product structure, in which certain pairs of states +occur in parallel (as defined by the simultaneity surfaces of the universal time function, i.e. the branchlike +hypersurfaces), and are therefore considered to be “tensored” together, such that the entire multiway system +can be decomposed into a tensor product of single branches (i.e. of individual deterministic/singleway com- +17 + +putations). The branchial graphs thus provide a combinatorial description of the tensor product structure of +the multiway computation at each time step, and the overall non-deterministic/multiway computation can +be recast as a deterministic/singleway computation over these tensor products/branchial graphs. +Figure 8: The default “foliation” of the multiway evolution graph for the non-deterministic evolution of the +2-state, 2-color Turing machine constructed from parallel composition of the transition functions for rules +2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 4 steps. +Figure 9: The corresponding sequence of “branchlike hypersurfaces” associated to the default “foliation” +of the multiway evolution graph for the non-deterministic evolution of the 2-state, 2-color Turing machine +constructed from parallel composition of the transition functions for rules 2506 and 3506. +We can make this intuition mathematically rigorous by noting that our category T of Turing machine +states and transitions is now (in the non-deterministic/multiway case) a monoidal category, i.e. a category +equipped with a tensor product structure, as proved for the case of generic multiway systems based on +symbolic rewriting in [26] and [27]; concretely, this entails that T is really an ordered triple ⟨T, ⊗, I⟩ consisting +of an underlying category, a tensor product operation ⊗ and a distinguished unit object I ∈ ob (T ). The +tensor product operation is encoded as a bifunctor of the form: +⊗ : T × T → T , +(35) +18 + +in other words a functor whose domain is the product category T × T , whose object set is given by ordered +pairs of objects in T : +ob (T × T ) = {(X, Y ) |X, Y ∈ ob (T )} , +(36) +whose morphism set is given by ordered pairs of morphisms in T : +hom (T × T ) = {((f, g) : (X1, X2) → (Y1, Y2)) |(f : X1 → Y1) , (g : X2 → Y2) ∈ hom (T )} , +(37) +and in which composition and identity are defined component-wise, i.e: +∀ ((f1, g1) : (X1, X2) → (Y1, Y2)) , ((f2, g2) : (Y1, Y2) → (Z1, Z2)) ∈ hom (T × T ) , +((f2, g2) ◦ (f1, g1) : (X1, X2) → (Z1, Z2)) = ((f2 ◦ f1, g2 ◦ g1) : (X1, X2) → (Z1, Z2)) , +(38) +and: +∀ (X, Y ) ∈ ob (T × T ) , +� +id(X,Y ) : (X, Y ) → (X, Y ) +� += ((idX, idY ) : (X, Y ) → (X, Y )) , +(39) +respectively. +In order to encode the fact that the tensor product operation should be (weakly) associa- +tive, there should exist a natural isomorphism (i.e. +a natural transformation whose components are all +isomorphisms) α, which we call the associator[28][29], of the general form: +α : (−) ⊗ ((−) ⊗ (−)) ∼= ((−) ⊗ (−)) ⊗ (−) , +(40) +with components: +∀X, Y, Z ∈ ob (T ) , +αX,Y,Z : X ⊗ (Y ⊗ Z) ∼= (X ⊗ Y ) ⊗ Z, +(41) +such that the following diagram (the associator coherence) commutes for all X, Y, Z, W ∈ ob (T ): +19 + +X ⊗ (Y ⊗ (Z ⊗ W)) +(X ⊗ Y ) ⊗ (Z ⊗ W) +((X ⊗ Y ) ⊗ Z) ⊗ W +X ⊗ ((Y ⊗ Z) ⊗ W) +(X ⊗ (Y ⊗ Z)) ⊗ W, +αX,Y,Z⊗W +idX⊗αY,Z,W +αX⊗Y,Z,W +αX,Y ⊗Z,W +αX,Y,Z⊗idZ +(42) +i.e: +∀X, Y, Z, W ∈ ob (T ) , +αX,Y,Z ⊗ idZ ◦ (αX,Y ⊗Z,W ◦ idX ⊗ αY,Z,W ) = αX⊗Y,Z,W ◦ αX,Y,Z⊗W . +(43) +Moreover, in order to encode the fact that the tensor product is (weakly) unital, i.e. that the distinguished +unit object I acts as both a left and right identity for ⊗, there should exist a further pair of natural +isomorphisms λ and ρ, which we call the left and right unitor isomorphisms respectively, of the general form: +λ : I ⊗ (−) ∼= (−) , +and +ρ : (−) ⊗ I ∼= (−) , +(44) +with components: +∀X ∈ ob (T ) , +λX : I ⊗ X ∼= X, +and +ρX : X ⊗ I ∼= X, +(45) +such that the following diagram (the unitor coherence) commutes for all X, Y ∈ ob (T ): +X ⊗ (I ⊗ Y ) +(X ⊗ I) ⊗ Y +X ⊗ Y +, +αX,I,Y +idX⊗λY +ρX⊗idY +(46) +i.e: +∀X, Y ∈ ob (T ) , +ρX ⊗ idY ◦ αX,I,Y = idX ⊗ λY . +(47) +Our monoidal category ⟨T , ⊗, I⟩ inherits the associativity and unitality of its tensor product structure +⊗ from the associativity and unitality of the disjoint union operation ⊔ (with the halt state HALT of the +Turing machine playing the role of the unit object I, since, by definition, parallel composition with the +halt state does not substantively modify the structure of any multiway computation because the halt state +20 + +does not evolve). Furthermore, since the disjoint union operation is also commutative, it follows that our +monoidal category is, in fact, symmetric[30][31], in the sense that it is also equipped with an additional +natural isomorphism σ, called the symmetry or braiding isomorphism, of the general form: +σ : (−) ⊗ (−) ∼= (−) ⊗ (−) , +(48) +with components: +∀X, Y ∈ ob (T ) , +σX,Y : X ⊗ Y ∼= Y ⊗ X, +(49) +such that the symmetry/braiding isomorphism is compatible with the associator, meaning that the following +diagram (essentially an additional associator coherence condition) commutes for all X, Y, Z ∈ ob (T ): +(X ⊗ Y ) ⊗ Z +(Y ⊗ X) ⊗ Z +X ⊗ (Y ⊗ Z) +Y ⊗ (X ⊗ Z) +(Y ⊗ Z) ⊗ X +Y ⊗ (Z ⊗ X) , +σX,Y ⊗idZ +α−1 +X,Y,Z +α−1 +Y,X,Z +σX,Y ⊗Z +idY ⊗σX,Z +α−1 +Y,Z,X +(50) +i.e: +∀X, Y, Z ∈ ob (T ) , +idY ⊗ σX,Z ◦ +� +α−1 +Y,X,Z ◦ σX,Y ⊗ idZ +� += α−1 +Y,Z,X ◦ +� +σX,Y ⊗Z ◦ α−1 +X,Y,Z +� +. +(51) +Additionally, the symmetry/braiding isomorphism should be compatible with the left and right unitor iso- +morphisms, meaning that the following diagram (an additional unitor coherence condition) commutes for all +X ∈ ob (T ): +X ⊗ I +I ⊗ X +X +, +σX,I +ρX +λX +(52) +21 + +i.e: +∀X ∈ ob (T ) , +λX ◦ σX,I = ρX. +(53) +Finally, the symmetry/braiding isomorphism should be involutive/self-inverse, meaning that the following +diagram commutes for all X, Y ∈ ob (T ): +Y ⊗ X +X ⊗ Y +X ⊗ Y, +σY,X +idX⊗Y +σX,Y +(54) +i.e: +∀X, Y ∈ ob (T ) , +σY,X ◦ σX,Y = idX⊗Y . +(55) +Note that, if we relax the last condition (namely that the natural isomorphism σ should be involutive/self- +inverse), then we obtain not a symmetric monoidal category but the weaker notion of a braided monoidal +category[32][33], in which the action of σ on an n-fold tensor product factors not through the symmetric +group but through the braid group. For the case of braided monoidal categories, we must impose one further +associator coherence condition, wherein the following diagram commutes for all X, Y, Z ∈ ob (T ): +X ⊗ (Y ⊗ Z) +X ⊗ (Z ⊗ Y ) +(X ⊗ Y ) ⊗ Z +(X ⊗ Z) ⊗ Y +Z ⊗ (X ⊗ Y ) +(Z ⊗ X) ⊗ Y, +idX⊗σY,Z +αX,Y,Z +αX,Z,Y +σX⊗Y,Z +σX,Z⊗idY +αZ,X,Y +(56) +i.e: +∀X, Y, Z ∈ ob (T ) , +σX,Z ⊗ idY ◦ (αX,Z,Y ◦ idX ⊗ σY,Z) = αZ,X,Y ◦ (σX⊗Y,Z ◦ αX,Y,Z) ; +(57) +however, this coherence condition is unnecessary for the case of symmetric monoidal categories, since it can +be derived from a combination of the first associator coherence law and the involutive/self-inverse property +22 + +of the symmetry/braiding isomorphism. +As before, we can now consider equipping the morphisms of the category T with additional semantic +metadata specifying the computational complexities of the corresponding computations that the morphisms +signify symbolically. However, since we are now able to compose computations not only in sequence (using +the standard composition operator ◦) but also in parallel (using the tensor product operation ⊗), the algebra +describing how these complexities compose is now somewhat more complicated. As previously described, +(singleway) computational irreducibility is characterized by additivity of sequential composition ◦, i.e. if +morphism f : X → Y requires at least n computational steps to enact and morphism g : Y → Z requires at +least m computational steps to enact, then their sequential composition g ◦ f : X → Z requires at least n + m +computational steps to enact. On the other hand, due to the presence of the tensor product structure, we can +now characterize a form of multicomputational irreducibility in terms of additivity of parallel composition +⊗, i.e. if morphism f : X1 → Y1 requires at least n computational steps to enact and morphism g : X2 → Y2 +requires at least m computational steps to enact, then their parallel composition f ⊗ g : X1 ⊗ X2 → Y1 ⊗ Y2 +(acting on the branchial graph consisting initially of states X1 and X2) is multicomputationally irreducible +if and only if it cannot be enacted in fewer than n + m computational steps. Likewise, if the complexi- +ties of computations behave subadditively under parallel composition/tensor product, then the composite +(multiway) computation is multicomputationally reducible (with the degree of subadditivity quantifying the +degree of multicomputational reducibility). The intuition behind this characterization is that, much as “or- +dinary” computational irreducibility is intended to describe the case in which the outcome of a deterministic +computation cannot be preempted without explicitly tracing out all intermediate steps, multicomputational +irreducibility is intended to describe the case in which the outcome of a non-deterministic/multiway compu- +tation cannot be preempted without explicitly tracing out all parallel deterministic/singleway computations +of which it is the tensor product. This intuition is illustrated in Figure 10 for the case of the non-deterministic +2-state, 2-color Turing machine discussed above, with each edge/morphism tagged with the number of tran- +sitions necessary to perform the corresponding computation; neither the sequential composition operation ◦ +nor the parallel tensor product operation ⊗ is purely additive, indicating that the system is, at least in part, +multicomputationally reducible. +We can consequently extend our previous formalization of computational irreducibility in terms of the +functoriality of the map Z′ : T → B to deal with the new case of multicomputational irreducibility too. If +we suppose, in the first instance, that all singleway computation paths through the multiway system for the +non-deterministic Turing machine T are computationally irreducible, then Z′ is a functor (as proved above). +23 + +Figure 10: A graph-theoretic representation of the category that is freely generated by the multiway evolution +graph corresponding to the non-deterministic evolution of a 2-state, 2-color Turing machine constructed from +the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506 (considered as +a quiver), with edges/morphisms tagged with additional metadata corresponding to the number of “steps” +(i.e. transitions) required to perform the requisite computation. +Furthermore, the (discrete) cobordism category B of step numbers/moments of time and (discrete) intervals +between them can trivially be equipped with a commutative tensor product structure (namely the disjoint +union of sets and intervals ⊔) with a unit object given by the empty set ∅; recall that, for more abstract and +general cobordism categories, the coproduct ⊕ plays the role of the disjoint union ⊔ (i.e. the direct sum of +manifolds), with the initial object ∅ playing the role of the empty set/empty manifold. The objects in the +symmetric monoidal category ⟨B, ⊕, ∅⟩ therefore represent not merely the time steps associated to individual +computational states (as in the singleway case considered within the previous section), but parallel compo- +sitions of time steps associated to multiple states on the same “branchlike hypersurface”; we can, as such, +think of the category B as providing a “coordinatization” of the time-ordered sequence of branchial graphs +computed by the multiway system. Thus, both ⟨T , ⊗, I = HALT⟩, and ⟨B, ⊕, ∅⟩ are symmetric monoidal +categories, and therefore, subject to the additional hypothesis that all parallel compositions of singleway +computations are multicomputationally irreducible (and therefore all computational complexities behave +purely additively under tensor products), the map Z′ : T → B forms a symmetric monoidal functor[34]. +More precisely, Z′ is a monoidal functor in the sense that it is a functor between monoidal categories: +Z′ : ⟨T , ⊗, I = HALT⟩ → ⟨B, ⊕, ∅⟩ , +(58) +that preserves the tensor product structure, meaning concretely that Z′ is equipped with a morphism +24 + +20(ε : ∅ → Z′ (I)) ∈ hom (B), along with a natural transformation µ between functors from T × T to B, with +components: +∀X, Y ∈ ob (T ) , +µX,Y : Z′ (X) ⊕ Z′ (Y ) → Z′ (X ⊗ Y ) . +(59) +Together, ε and µ are known as the coherence maps of the monoidal functor Z′, and satisfy coherence +conditions with the associators αT , αB, left unitor isomorphisms λT , λB and right unitor isomorphisms ρT , ρB. +The associator coherence condition is represented by the assertion that the following diagram commutes for +all X, Y, Z ∈ ob (T ): +(Z′ (X) ⊕ Z′ (Y )) ⊕ Z′ (Z) +Z′ (X) ⊕ (Z′ (Y ) ⊕ Z′ (Z)) +Z′ (X ⊗ Y ) ⊕ Z′ (Z) +Z′ (X) ⊕ Z′ (Y ⊗ Z) +Z′ ((X ⊗ Y ) ⊗ Z) +Z′ (X ⊗ (Y ⊗ Z)) , +αB +Z′(X),Z′(Y ),Z′(Z) +µX,Y ⊕idB +Z′(Z) +idB +Z′(Z) +µX⊗Y,Z +µX,Y ⊗Z +Z′(αT +X,Y,Z) +(60) +i.e. (recalling that the composition operation in the cobordism category B is given by the ordinary union of +contiguous intervals/cobordisms ∪): +∀X, Y, Z ∈ ob (T ) , +µX,Y ⊗Z ∪ +� +idZ′(Z) ∪ αB +Z′(X),Z′(Y ),Z′(Z) +� += Z′ � +αT +X,Y,Z +� +∪ +� +µX⊗Y,Z ∪ µX,Y ⊕ idB +Z′(Z) +� +, +(61) +while the left and right unitor coherence conditions are represented by the assertion that the following +diagrams commute for all X ∈ ob (T ): +25 + +Z′ (X) ⊕ ∅ +Z′ (X) ⊕ Z′ (I) +Z′ (X) +Z′ (X ⊗ I) , +idB +Z′(X)⊕ε +ρB +Z′(X) +µX,I +Z′(ρT +X) +(62) +i.e: +∀X ∈ ob (T ) , +Z′ � +ρT +X +� +∪ +� +µX,I ∪ idB +Z′(X) ⊕ ε +� += ρB +Z′(X), +(63) +and: +∅ ⊕ Z′ (X) +Z′ (I) ⊕ Z′ (X) +Z′ (X) +Z′ (I ⊗ X) , +ϵ⊕idB +Z′(X) +λB +Z′(X) +µI,X +Z′(λT +X) +(64) +i.e: +∀X ∈ ob (T ) , +Z′ � +λT +X +� +∪ +� +µI,X ∪ ϵ ⊕ idB +Z′(X) +� += λB +Z′(X), +(65) +respectively. Note that the definition presented here is for a lax monoidal functor; if the coherence maps ε +and µX,Y were, additionally, either isomorphisms or identities for all X, Y ∈ ob (T ), then one would obtain +a strong or a strict monoidal functor, respectively, instead. The monoidal functor Z′ : ⟨T , ⊗, I⟩ → ⟨B, ⊕, ∅⟩ +is also symmetric, in the sense that the coherence maps ε and µ also satisfy a further coherence condition +with the braiding/symmetry isomorphisms σT , σB, represented by the assertion that the following diagram +commutes for all X, Y ∈ ob (T ): +Z′ (X) ⊕ Z′ (Y ) +Z′ (X) ⊕ Z′ (Y ) +Z′ (X ⊗ Y ) +Z′ (Y ⊗ X) , +σB +Z′(X),Z′(Y ) +µX,Y +µY,X +Z′(σT +X,Y ) +(66) +i.e: +26 + +∀X, Y ∈ ob (T ) , +µY,X ∪ σB +Z′(X),Z′(Y ) = Z′ � +σT +X,Y +� +∪ µX,Y . +(67) +If ⟨T , ⊗, I⟩ and ⟨B, ⊕, ∅⟩ were merely braided monoidal rather than symmetric monoidal categories, then a +monoidal functor Z′ satisfying the above coherence condition would instead be a braided monoidal functor. +Thus, if X1, Y1, X2 and Y2 represent Turing machine states reached at step numbers t1, t2, s1 and s2 +respectively, and f : X1 → Y1 and g : X2 → Y2 represent the transitions between states X1 and Y1 and states +X2 and Y2 respectively, then the (symmetric) monoidal functor Z′: +X1 +X2 +t1 +s1 +Y1 +Y2 +t2 +s2, +f +g +[t1,t2]∩N +[s1,s2]∩N +Z′ +(68) +preserves the (symmetric) tensor product structure in the sense depicted by the following diagram: +X1 ⊗ X2 +t1 ⊕ s1 +Y1 ⊗ Y2 +t2 ⊕ s2. +f⊗g +([t1,t2]∩N)⊕([s1,s2]∩N) +Z′ +(69) +Hence, by equipping B with a tensor product structure, the resulting symmetric monoidal category ⟨B, ⊕, ∅⟩ +is actually a higher-dimensional cobordism category, in which the objects are potentially higher-dimensional +manifolds (consisting of direct sums/disjoint unions of several 0-dimensional manifolds) and the morphisms +are potentially higher-dimensional cobordisms (consisting of direct sums/disjoint unions of several 1-dimensional +cobordisms), since there now exist two distinct directions in which cobordisms can be “glued” (either se- +quentially, via the standard union of contiguous intervals ∪, or in parallel, via the tensor product/direct +sum of intervals ⊔). For any given morphism in B, the cardinalities of the various individual (1-dimensional) +intervals of which it is composed represent, up to an additive constant, the time complexities of the cor- +responding deterministic/singleway computations in T (as in the singleway case analyzed in the previous +section), whereas the number of distinct (1-dimensional) intervals appearing in the tensor product/direct sum +within that morphism represents, up to an additive constant, the time complexity of the resulting composite +non-deterministic/multiway computation in T . As a consequence, we can conclude that, just as (singleway) +computational irreducibility is reflected in the functoriality of Z′, multicomputational irreducibility is re- +flected in the symmetric monoidal functoriality of Z′, and just as computational reducibility corresponds +27 + +precisely to a deformation of Z′ away from a pure functor, multicomputational reducibility corresponds +to a deformation of Z′ from being symmetric monoidal. Another, perhaps cleaner, way to articulate this +would be to say that computational reducibility corresponds to how much the map Z′ distorts the sequential +composition (◦) of computations in T , while multicomputational reducibility corresponds to how much the +map Z′ distorts the parallel composition (⊗) of computations in T . This construction is demonstrated in +Figure 11 for the case of the non-deterministic 2-state, 2-color Turing machine discussed previously, with +each vertex/object tagged with its step number and each edge/morphism tagged with a list of step numbers +traversed throughout the course of its corresponding computation. Note that the same algebraic axioms that +describe how the time complexities attached to morphisms compose (namely identity, strict positivity and +subadditivity/triangle inequality) in the sequential case under ◦ also hold in the parallel case under ⊗. +Figure 11: A graph-theoretic representation of the category that is freely generated by the multiway evolution +graph corresponding to the non-deterministic evolution of a 2-state, 2-color Turing machine constructed from +the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506 (considered as +a quiver), with vertices/objects tagged with additional metadata corresponding to the step number on which +they occur (shown in blue) and with edges/morphisms tagged with additional metadata corresponding to +all intermediate step numbers traversed as part of the requisite computation. +From this rather abstract description, it should be apparent that computational irreducibility and mul- +ticomputational irreducibility are essentially orthogonal concepts: the map Z′ can distort sequential com- +positions to an arbitrary extent whilst keeping the tensor product structure entirely intact, or vice versa +(or anything in between). This is a fairly intuitive consequence of the fact that computational irreducibility +is a byproduct of the state evolution function of a given multiway system (i.e. the function that speci- +fies which states are obtainable in a single step from which other states, and which therefore provides the +rules for constructing morphisms in T ), while multicomputational irreducibility is a byproduct of the state +equivalence function of a given multiway system (i.e. the function that specifies which pairs of states are +28 + +[1. 2] +1, 2,3, 4 +[1,2,3] +{1, 2, 3, 4 +[1, 2. 3] +51, 2,3, 4) +[1, 2, 3, 4] +31.,2,3. 4) +2,3 +[2, 3] +[2. 3. 4] +[2, 3,4 +2,3. 4 +[3.4]to be considered equivalent, and which therefore provides the rules for merging/equating objects in T ), +and the evolution and equivalence functions have entirely independent definitions: one can define a state +evolution function with an arbitrarily high computational complexity whilst keeping the state equivalence +function essentially trivial, or vice versa (or, once again, any reasonable intermediate). For instance, in the +case of non-deterministic/multiway Turing machines considered thus far, the state equivalence function is +elementary (two Turing machine states are considered equivalent if their tape states, head states and head +positions are both identical, which is trivial to determine algorithmically), with all of the computational +complexity originating from the state evolution function. On the other hand, in the case of (hyper)graph +rewriting, as considered in the context of the Wolfram model[22][23][24], the state equivalence function is +much more sophisticated, since it must account for (hyper)graph isomorphism, whose precise complexity +class GI remains unknown (i.e. it is not known whether GI is P, NP-complete or NP-intermediate)[35][36]. +In the analysis that follows, we shall be using a generalized version of the “uniqueness tree” isomorphism +algorithm presented in [37]. +By defining a (directed/ordered) hypergraph H = ⟨V, E⟩ in terms of a finite collection/multiset of ordered +relations (hyperedges) between elements: +E ⊆ P (V ) \ {∅} , +(70) +where P denotes the power set, we can formalize the notion of hypergraph rewriting rules H1 → H2 in terms +of a span of monomorphisms[38][39] of the form: +L +K +R, +l +r +(71) +in some category C (whose objects are hypergraphs, and whose morphisms represent subhypergraph inclusion +maps), where L is a hypergraph pattern designating the left-hand side of the rule, R is a hypergraph +pattern designating the right-hand side of the rule, and K is a pattern designating the subhypergraph that +remains invariant when the left-hand side is “extracted” and the right-hand side is “injected”. (Note that +hypergraph categories, in the terminology of Kissinger[40] and Fong[41][42], namely symmetric monoidal +categories equipped with a Frobenius algebra structure which ensures that all string diagrams correspond +to hypergraphs, constitute a particularly natural categorical setting in which to construct such a rewriting +system). In the above, the morphisms l : K → L and r : K → R form a span in the sense that they constitute +a pair of morphisms with a common domain, and they are monomorphisms in the sense that they are +29 + +injective/left-cancellative: for any pairs of morphisms f1 : X → K and f2 : X → K that make either of the +following diagrams commute: +∀X +K +L +∀f1 +∀f2 +l +, +or +∀X +K +R, +∀f1 +∀f2 +r +(72) +one necessarily has (f1 : X → K) = (f2 : X → K), i.e: +∀X ∈ ob (C) , +∀ (f1 : X → K) , (f2 : X → K) ∈ hom (C) , +(l ◦ f1 : X → L) = (l ◦ f2 : X → L) +=⇒ +(f1 : X → K) = (f2 : X → K) , +(73) +and: +∀X ∈ ob (C) , +∀ (f1 : X → K) , (f2 : X → K) ∈ hom (C) , +(r ◦ f1 : X → R) = (r ◦ f2 : X → R) +=⇒ +(f1 : X → K) = (f2 : X → K) . +(74) +A graphical representation of such a hypergraph rewriting rule with relations/hyperedges of arity-2 (corre- +sponding to the set substitution rule {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}} - since any hyper- +graph rewriting rule of this form can always be reformulated as a symbolic set substitution rule acting on +multisets of ordered relations between vertices) is shown in Figure 12. +Figure 12: A graphical representation of the (hyper)graph transformation rule corresponding to the set +substitution system {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}}. +A hypergraph rewriting rule of the form presented above can then be said to match a given hypergraph +G if there exists a morphism (m : L → G) ∈ hom (C), and the resulting hypergraph H obtained by applying +the rewriting rule at that match can be computed by means of the following double-pushout diagram[43]: +30 + +L +K +R +G +D +H +∀G∗ +∀H∗. +m +∀m∗ +l +r +n +p +∀p∗ +∃!u1 +g +∀h∗ +∀g∗ +h +∃!u2 +(75) +More concretely, D ∈ ob (C) is a hypergraph and (n : K → D) , (g : D → G) ∈ hom (C) are hypergraph in- +clusions such that the leftmost square commutes: +(g ◦ n : K → G) = (m ◦ l : K → G) , +(76) +and are universal in the sense that, for any hypergraph G∗ ∈ ob (C) equipped with inclusions: +(m∗ : L → G∗) , (g∗ : D → G∗) ∈ hom (C) , +(77) +there exists a unique inclusion (u1 : G → G∗) ∈ hom (C) such that: +(m∗ : L → G∗) = (u1 ◦ m : L → G∗) , +and +(g∗ : D → G∗) = (u1 ◦ g : D → G∗) . +(78) +This allows one to compute the “residual” hypergraph D obtained by extracting out a subhypergraph +isomorphic to L from G at the position defined by the rule match m : L → G. Moreover, D, H ∈ ob (C) +are hypergraphs and (p : R → H) , (h : D → H) ∈ hom (C) are inclusions such that the rightmost square +commutes: +(h ◦ n : K → H) = (p ◦ r : K → H) , +(79) +and are universal in the sense that, for any hypergraph H∗ ∈ ob (C) equipped with inclusions: +(p∗ : R → H∗) , (h∗D → H∗) ∈ hom (C) , +(80) +there exists a unique inclusion (u2 : H → H∗) ∈ hom (C) such that: +(p∗ : R → H∗) = (u2 ◦ p : R → H∗) , +and +(h∗ : D → H∗) = (u2 ◦ h : D → H∗) . +(81) +This, in turn, allows one to compute the resulting hypergraph H obtained by gluing a subhypergraph +31 + +isomorphic to R into the residual hypergraph D at the position defined by the injection map m : K → D +(from the invariant subhypergraph to the residual hypergraph). This double-pushout construction yields +the class of possible hypergraph transitions from which we are able to construct a multiway evolution +graph (whose vertices represent hypergraphs and whose edges represent hypergraph rewrites), by a process +that is directly analogous to the Turing machine case considered previously: an explicit example for the +hypergraph rewriting rule presented above is shown in Figure 13, and the category that is freely generated +by this multiway evolution graph (considered as a quiver) is shown in Figure 14. Figure 15 shows each +morphism/edge tagged with the number of hypergraph transitions necessary to perform the corresponding +computation, illustrating that the composition ◦ and tensor product ⊗ operations are not purely additive +(although they are both somewhat close), thus indicating that the system is largely (multi)computationally +irreducible, but not entirely so. The construction with each vertex/object tagged with its step number and +each edge/morphism tagged with a list of step numbers traversed throughout the course of its corresponding +computation is shown in Figure 16. As expected, we see that the hypergraph rewriting system is more +computationally reducible than multicomputationally so (i.e. the composition operation ◦ is distorted more +by the map Z′ than the tensor product operation ⊗): a consequence of the fact that the state equivalence +function (based on hypergraph isomorphism) is more computationally complex than it was for the non- +deterministic Turing machine case previously. +Figure 13: A multiway evolution graph corresponding to the non-deterministic evolution of the set substi- +tution system {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}}, starting from a “double self-loop” initial +condition, for 3 steps; each edge represents a single application of the corresponding (hyper)graph rewriting +rule. +The usual setting in which double-pushout rewriting takes place is within an adhesive category[44], and +thus adhesivity is the usual condition that one would impose on the category C described above; loosely +speaking, adhesivity provides sufficient conditions for pushouts to be “glued” along monomorphisms in the +necessary manner for double-pushout diagrams to be constructed. More formally, a category is adhesive if +it has pullbacks, and all pushouts along monomorphisms satisfy the van-Kampen square condition. Having +32 + +Figure 14: +A graph-theoretic representation of the category that is freely generated by the multi- +way evolution graph corresponding to the non-deterministic evolution of the set substitution system +{{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}} (considered as a quiver), starting from a “double self- +loop” initial condition, for 3 steps. +pullbacks simply entails that, for some cospan (i.e. a pair of morphisms with a common codomain): +Y +X +Z +f +g +(82) +in C, there exists an object P ∈ ob (C) and a pair of morphisms (f ′ : P → Z) , (g′ : P → Y ) ∈ hom (C) such +that the following square commutes: +P +Z +Y +X, +f ′ +g′ +g +f +(83) +i.e: +(f ◦ g′ : P → X) = (g ◦ f ′ : P → X) , +(84) +and that are universal in the sense that, for any object P ∗ ∈ ob (C) equipped with morphisms: +(f ∗ : P ∗ → Z) , (g∗ : P ∗ → Y ) ∈ hom (C) , +(85) +there exists a unique morphism (u : P ∗ → P) ∈ hom (C) such that: +(f ∗ : P ∗ → Z) = (f ′ ◦ u : P ∗ → Z) , +and +(g∗ : P ∗ → Y ) = (g′ ◦ u : P ∗ → Y ) , +(86) +33 + +区Figure 15: +A graph-theoretic representation of the category that is freely generated by the multi- +way evolution graph corresponding to the non-deterministic evolution of the set substitution system +{{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}} (considered as a quiver), with edges/morphisms tagged +with additional metadata corresponding to the number of “steps” (i.e. rewriting rule applications) required +to perform the requisite computation. +i.e. the following diagram commutes: +∀P ∗ +P +Z +Y +X. +∀f ∗ +∀g∗ +∃!u +f ′ +g′ +g +f +(87) +Though the condition of having pullbacks along cospans is relatively straightforward to state and understand, +the van-Kampen square condition on pushouts along monomorphisms is, on the other hand, far more opaque +and technical. It asserts that, if the object W ∈ ob (C) and the morphisms (f ′ : Y → W) , (g′ : Z → W) ∈ hom (C) +in the following diagram: +X +Z +Y +W +∀W ∗, +f +g +g′ +∀g∗ +f ′ +∀f ∗ +∃!u +(88) +constitute a pushout of the span (f : X → Z) , (g : X → Y ) ∈ hom (C), i.e: +(f ′ ◦ g : X → W) = (g′ ◦ f : X → W) , +(89) +34 + +3 +2 +3 +3 +3 +3 +3 +33 +3 +3 +2 +2 +2 +N +2222 +2 +2 +2 +2 +2 +2 +11 +11111 +区 +0 +0X00X +000000Figure 16: +A graph-theoretic representation of the category that is freely generated by the multi- +way evolution graph corresponding to the non-deterministic evolution of the set substitution system +{{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}} (considered as a quiver), with vertices/objects tagged with +additional metadata corresponding to the step number on which they occur (shown in blue) and with +edges/morphisms tagged with additional metadata corresponding to all intermediate step numbers traversed +as part of the requisite computation. +and: +∀W ∗ ∈ ob (C) , +∀ (f ∗ : Y → W ∗) , (g∗ : Z → W ∗) ∈ hom (C) , +such that +(f ∗ ◦ g : X → W ∗) = (g∗ ◦ f : X → W ∗) , +∃! (u : W → W ∗) ∈ hom (C) , +(90) +such that: +(f ∗ : Y → W ∗) = (u ◦ f ′ : Y → W ∗) , +and +(g∗ : Z → W ∗) = (u ◦ g′ : Z → W ∗) , +(91) +then that pushout is a van-Kampen square if and only if, for every commutative cube of the form: +X′ +Z′ +X +Z +Y +W +Y ′ +W ′, +fh +gh +hx +hZ +g′ +h +f +g +g′ +f ′ +hY +f ′ +h +hW +(92) +35 + +[2] +[1444412.3] +[1.2.3] +[1,2.3] +2.31 +3 +[2(2222:4] +3.47 +[2. 42424243 0 +[344] +4 +4 +A +区 +区 +X +(4 4/4/444 4] +[4] +[4] +[4]4] [4][4] [4][4] [4] [4] [4]for which the top and left faces are pullback squares, in other words if and only if the object X′ ∈ ob (C) +together with the morphisms (hX : X′ → X) , (gh : X′ → Y ′) ∈ hom (C), and the object X′ ∈ ob (C) together +with the morphisms (fh : X′ → Z′) , (hX : X′ → X) ∈ hom (C) in the following pair of diagrams: +∀X∗ +Y ′ +X′ +Y +X +, +∀g∗ +h +∀h∗ +X +∃!u +hY +gh +hX +g +and +∀X∗ +X′ +Z′ +X +Z, +∀f ∗ +h +∀h∗ +X +∃!u +fh +hX +hZ +f +(93) +constitute pullbacks of the cospans (g : X → Y ) , (hY : Y ′ → Y ) ∈ hom (C) and (f : X → Z) , (hZ : Z′ → Z) ∈ hom (C), +respectively, i.e: +(g ◦ hX : X′ → Y ) = (hY ◦ gh : X′ → Y ) , +and +(f ◦ hX : X′ → Z) = (hZ ◦ fh : X′ → Z) , +(94) +and, moreover: +∀X′ ∈ ob (C) , +∀ (g∗ +h : X∗ → Y ′) , (h∗ +X : X∗ → X) ∈ hom (C) , +such that +(g ◦ h∗ +X : X∗ → Y ) = (hY ◦ g∗ +h : X∗ → Y ) , +∃! (u : X∗ → X′) ∈ hom (C) , +(95) +such that: +(h∗ +X : X∗ → X) = (hX ◦ u : X∗ → X) , +and +(g∗ +h : X∗ → Y ′) = (gh ◦ u : X∗ → Y ′) , +(96) +for the case of the first (leftmost) diagram, and: +∀X′ ∈ ob (C) , +∀ (f ∗ +h : X∗ → Z′) , (h∗ +X : X∗ → X) ∈ hom (C) , +such that +(f ◦ h∗ +X : X∗ → Z) = (hZ ◦ f ∗ +h : X∗ → Z) , +∃! (u : X∗ → X′) ∈ hom (C) , +(97) +36 + +such that: +(h∗ +X : X∗ → X) = (hX ◦ u : X∗ → X) , +and +(f ∗ +h : X∗ → Z′) = (fh ◦ u : X∗ → Z′) , +(98) +for the case of the second (rightmost) diagram, then a certain compatibility condition is satisfied between the +pushout and pullback squares. In detail, this compatibility condition states that the rear face is a pushout +square, in other words the object W ′ ∈ ob (C) and the morphisms (f ′ +h : Y ′ → W ′) , (g′ +h : Z′ → W ′) ∈ hom (C) +in the following diagram: +X′ +Z′ +Y ′ +W ′ +∀W ∗, +fh +gh +g′ +h +∀g∗ +h +f ′ +h +∀f ∗ +h +∃!u +(99) +constitute a pushout of the span (fh : X′ → Z′) , (gh : X′ → Y ′) ∈ hom (C), i.e: +(f ′ +h ◦ gh : X′ → W ′) = (g′ +h ◦ fh : X′ → W ′) , +(100) +and: +∀W ∗ ∈ ob (C) , +∀ (f ∗ +h : Y ′ → W ∗) , (g∗ +h : Z′ → W ∗) ∈ hom (C) , +such that +(f ∗ +h ◦ gh : X′ → W ∗) = (g∗ +h ◦ fh : X′ → W ∗) , +∃! (u : W ′ → W ∗) , +(101) +such that: +(f ∗ +h : Y ′ → W ∗) = (u ◦ f ′ +h :′→ W ∗) , +and +(g∗ +h : Z′ → W ∗) = (u ◦ g′ +h : Z′ → W ∗) , +(102) +if and only if the bottom and right faces are pullback squares, in other words if and only if the ob- +ject Y ′ ∈ ob (C) together with the morphisms (f ′ +h : Y ′ → W ′) , (hY : Y ′ → Y ) ∈ hom (C), and the object +Z′ ∈ ob (C) together with the morphisms (g′ +h : Z′ → W ′) , (hZ : Z′ → Z) ∈ hom (C) in the following pair of +37 + +diagrams: +Y +W +Y ′ +W ′ +∀Y ∗ +f ′ +hY +f ′ +h +hW +∀h∗ +Y +∀f ∗ +h +∃!u +and +W +Z +W ′ +Z′ +∀Z∗, +g′ +hW +hZ +g′ +h +∀g∗ +h +∀h∗ +Z +∃!u +(103) +constitute pullbacks of the cospans (f ′ : Y → W) , (hW : W ′ → W) ∈ hom (C) and (g′ : Z → W) , (hW : W ′ → W) ∈ hom (C), +respectively, i.e: +(hW ◦ f ′ +h : Y ′ → W) = (f ′ ◦ hY : Y ′ → W) , +and +(hW ◦ g′ +h : Z′ → W) = (g′ ◦ hZ : Z′ → W) , +(104) +and, moreover: +∀Y ∗ ∈ ob (C) , +∀ (f ∗ +h : Y ∗ → W ′) , (h∗ +Y : Y ∗ → Y ) ∈ hom (C) , +such that +(hW ◦ f ∗ +h : Y ∗ → W) = (f ′ ◦ h∗ +Y : Y ∗ → W) , +∃! (u : Y ∗ → Y ′) ∈ hom (C) , +(105) +such that: +(f ∗ +h : Y ∗ → W ′) = (f ′ +h ◦ u : Y ∗ → W ′) , +and +(h∗ +Y : Y ∗ → Y ) = (hY ◦ u : Y ∗ → Y ) , +(106) +for the case of the first (leftmost) diagram, and: +∀Z∗ ∈ ob (C) , +∀ (g∗ +h : Z∗ → W ′) , (h∗ +Z : Z∗ → Z) ∈ hom (C) , +such that +(hW ◦ g∗ +h : Z∗ → W) = (g′ ◦ h∗ +Z : Z∗ → W) , +∃! (u : Z∗ → Z′) ∈ hom (C) , +(107) +such that: +38 + +(g∗ +h : Z∗ → W ′) = (g′ +h ◦ u : Z∗ → W ′) , +and +(h∗ +Z : Z∗ → Z) = (hZ ◦ u : Z∗ → Z) . +(108) +for the case of the second (rightmost) diagram. Although the category of hypergraphs and subhypergraph +inclusion maps considered in the case of Wolfram model evolution is not strictly an adhesive category +(due to the arbitrary connectivity of vertices within each hyperedge, implying that not all pushouts along +monomorphisms are guaranteed to exist), as noted by Kissinger[45][46][47] it constitutes a full subcategory +of an adhesive category, and can be “embedded” in the ambient adhesive category in such a way as to inherit +sufficient “adhesivity” to allow double-pushout rewriting to be performed (through the mechanisms of either +selective adhesivity or partial adhesivity - essentially by allowing the functor that embeds the subcategory +into the adhesive category to preserve monomorphisms). +As detailed in [48], one can construct both the ordinary composition structure and the symmetric +monoidal structure of the resulting category T of hypergraphs and hypergraph rewritings in a very explicit +way using a combination of the concurrency and parallelism theorems from algebraic graph transformation +theory. Specifically, if one has a pair of hypergraph productions p1 and p2 (i.e. two spans of monomor- +phisms corresponding to two hypergraph rewrites) rewriting hypergraph G to H and hypergraph H to G′ +respectively (known as an E-related transformation sequence), then the concurrency theorem allows one to +compose the productions to obtain an E-concurrent hypergraph production p1 ∗E p2 of the form: +H +G +G′, +p2 +p1 +p1∗Ep2 +(109) +thus giving rise to the ordinary (sequential) composition of morphisms ◦ in T . On the other hand, if one has +a pair of hypergraph productions p1 and p2 yielding two different, sequentially-independent transformation +sequences G to H1 to G′ and G to H2 to G′ (obtained by applying p1 then p2 vs. p2 then p1), then the +parallelism theorem allows one to compose the productions to obtain a parallel hypergraph production p1 + p2 +of the form: +39 + +G +H1 +H2 +G′ +, +p1 +p2 +p1+p2 +p2 +p1 +(110) +thus giving rise to the tensor product (parallel) composition of morphisms ⊗ in T , as required. +4 +Correspondence with Categorical Quantum Mechanics and Func- +torial Quantum Field Theory +In conventional (non-relativistic) quantum mechanics, one typically associates to each moment of time t ∈ R +a corresponding (Hilbert) space of states Vt for the system, and to every interval of time [t1, t2], where +t1, t2 ∈ R such that t1 ≤ t2, a corresponding linear (unitary) time evolution operator ˆU (t1, t2) : Vt1 → Vt2. +If the Hamiltonian ˆH (t) is a Hermitian/self-adjoint operator that depends smoothly on t ∈ R, then we can +express ˆU (t1, t2) explicitly in terms of the following Dyson formula (otherwise known more generally as an +iterated integral expansion for parallel transport) for the time-dependent Schr¨odinger equation: +ˆU (t1, t2) = P exp +� i +ℏ +� t2 +t1 +ˆH (t) dt +� +, +(111) +where P exp denotes the path-ordered exponential operator for non-commutative algebras; for the time- +independent case in which the Hamiltonian ˆH is fixed, this simplifies to just an ordinary exponential: +∀t ∈ R, +ˆH (t) = ˆH, +=⇒ +ˆU (t1, t2) = exp +� i +ℏ (t2 − t1) ˆH +� +. +(112) +This explicit representation of ˆU (t1, t2) makes manifest one of the fundamental features of quantum mechan- +ical time evolution: that it is necessarily local in time. In other words, due to the linearity of integration, the +“global” time interval [t1, t2] can always be subdivided into many “local” subintervals, in such a way that +the single global time evolution is obtained by integrating up the effects of the many local time evolutions. +More formally, we have: +∀t1, t2, t3 ∈ R, +such that +t1 ≤ t2 ≤ t3, +ˆU (t1, t3) = ˆU (t2, t3) ◦ ˆU (t1, t2) , +(113) +40 + +in other words, time evolution has a natural composition structure ◦ such that, when combined with the +(mostly, though not entirely, innocent) condition that: +∀t ∈ R, +ˆU (t, t) = idVt, +(114) +we see that time evolution in quantum mechanics satisfies the requisite axioms of a category (with associativ- +ity inherited from the associativity of products of linear operators); this was ultimately the insight underlying +Abramsky and Coecke’s formulation of categorical quantum mechanics[49][50]. Thus, if BordRiem +1 +refers to +the 1-dimensional category of (Riemannian) manifolds and their cobordisms, and Vect refers to the category +of vector spaces and their linear isomorphisms, then the statement of locality of time evolution in quantum +mechanics simply becomes a statement that the map: +Z : BordRiem +1 +→ Vect, +(115) +is a functor, i.e. (at least in this context) locality is functoriality. This functoriality between the categories +BordRiem +1 +and Vect can be illustrated diagrammatically as follows: +t2 +Vt2 +t1 +t3 +Vt1 +Vt3.. +[t2,t3] +ˆU(t2,t3) +[t1,t2] +[t2,t3]∪[t1,t2] +=[t1,t3] +ˆU(t1,t2) +ˆU(t2,t3)◦ ˆU(t1,t2) += ˆU(t1,t3) +Z +(116) +If we think of the (Hilbert) spaces of states Vt as being data structures, and the unitary time evolution +operators ˆU (t1, t2) as being elementary computations (as is the case in, for instance, quantum information +theory, where they represent the actions of compositions of quantum gates), then this looks structurally very +similar to the continuous case of the functor Z′ : T → B from a category of data structures and computations +to a category of manifolds and cobordisms, namely: +Y +t2 +X +Z +t1 +t3, +g +[t2,t3] +f +g◦f +[t1,t2] +[t2,t3]∪[t1,t2] +=[t1,t3] +Z′ +(117) +41 + +considered in the preceding sections, albeit with the domain and the codomain categories swapped around. +In the most general case of a functorial quantum mechanics theory, defined by some functor from cobor- +disms to computations Z : B → T , although it is not necessarily the case that Z will always be a strict +inverse of Z′ : T → B, the two functors will at least be adjoint to one another. Formally, the statement +that Z′ : T → B is left adjoint to Z : B → T corresponds to the assertion that, for every object (manifold) +X ∈ ob (B), there exists a universal morphism (cobordism) (εX : Z′ (Z (X)) → X) ∈ hom (B) from Z′ to X +for some object (data structure) Z (X) ∈ ob (T ), where the universality property necessitates that, for any +object (data structure) Y ∈ ob (T ) and any morphism (cobordism) (f : Z′ (Y ) → X) ∈ hom (B), there exists +a unique morphism (computation) (g : Y → Z (X)) ∈ hom (T ) such that: +(εX ◦ Z′ (g) : Z′ (Y ) → X) = (f : Z′ (Y ) → X) . +(118) +This condition may be restated succinctly via the following commutative diagram: +Z′ (∀Y ) +Z′ (∃Z (X)) +∀X. +Z′(∃!g) +∀f +∃εX +(119) +On the other hand, the statement that Z : B → T is right adjoint to Z′ : T → B corresponds to the asser- +tion that, for every object (data structure) Y ∈ ob (T ), there exists a universal morphism (computation) +(ηY : Y → Z (Z′ (Y ))) ∈ hom (T ) from Y to Z for some object (manifold) Z′ (Y ) ∈ ob (B), where the uni- +versality property necessitates that, for any object (manifold) X ∈ ob (B) and any morphism (computation) +(g : Y → Z (X)) ∈ hom (T ), there exists a unique morphism (cobordism) (f : Z′ (Y ) → X) ∈ hom (B) such +that: +(Z (f) ◦ ηY : Y → Z (X)) = (g : Y → Z (X)) . +(120) +This condition may be restated succinctly via the following commutative diagram: +∀Y +Z (∃Z′ (Y )) +Z (∀X) . +∃ηY +∀g +Z(∃!f) +(121) +42 + +It is in this rather precise sense that we are able to claim that the irreducibility of computations in computa- +tional complexity theory is dual/adjoint to the locality of time evolution in categorical quantum mechanics: +for any functor Z′ : T → B describing an irreducible computation, we can uniquely construct a corresponding +functor Z : B → T describing a local quantum time evolution such that: +∀X, X′ ∈ ob (B) , +∀ (f : X′ → X) ∈ hom (B) , +(εX ◦ Z′ (Z (f)) : Z′ (Z (X′)) → X) = (f ◦ εX′ : Z′ (Z (X′)) → X) , +(122) +and, conversely, for any functor Z : B → T describing a local quantum time evolution, we can uniquely +construct a corresponding functor Z′ : T → B describing an irreducible computation such that: +∀Y, Y ′ ∈ ob (T ) , +∀ (g : Y → Y ′) ∈ hom (T ) , +(Z (Z′ (g)) ◦ ηY : Y → Z (Z′ (Y ′))) = (ηY ′ ◦ g : Y → Z (Z′ (Y ′))) , +(123) +as described by the following pair of commutative diagrams: +Z′ (Z (X′)) +Z′ (Z (X)) +X′ +X, +Z′(Z(f)) +εX′ +εX +f +and +Y +Z (Z′ (Y )) +Y ′ +Z (Z′ (Y ′)) . +ηY +g +Z(Z′(g)) +ηY ′ +(124) +Somewhat more cryptically, this enables us to make the claim that, in this very restricted sense, computa- +tional complexity theory is dual/adjoint to (non-relativistic) quantum mechanics. +In the non-deterministic/multicomputational case, in which categories T and B are both equipped with +a (symmetric) monoidal structure, and thus in which the adjoint functors Z′ and Z are now (symmetric) +monoidal functors: +Z′ : ⟨T, ⊗, I⟩ → ⟨B, ⊕, ∅⟩ , +and +Z : ⟨B, ⊕, ∅⟩ → ⟨T , ⊗, I⟩ , +(125) +we obtain a higher-dimensional analog of the time evolution functor for non-relativistic quantum mechanics +on the right-hand side of the adjunction. For the example considered initially, in which the category T +43 + +of data structures and computations is abstractly represented as a category of vector spaces and linear +isomorphisms (with the tensor product operation given by the usual tensor product of vector spaces), this +functor consequently takes the form: +Z : BordRiem +d +→ Vect, +(126) +where d > 1. +In the context of functorial approaches to quantum field theory (e.g. +in topological field +theories or 2-dimensional conformal field theories)[51][52], such a functor plays the role of a propagator, +i.e. the Lorentz-invariant analog of the time evolution functor from non-relativistic quantum mechanics: to +every codimension-1 spacelike hypersurface Md−1 ∈ ob +� +BordRiem +d +� +, this functor assigns a corresponding +vector space Z (Md−1) ∈ ob (Vect) designating the space of states over that hypersurface, and to every +cobordism/spacetime/worldvolume M ∈ hom +� +BordRiem +d +� +with boundaries ∂ (M), i.e.: +(M : ∂in (M) → ∂out (M)) ∈ hom +� +BordRiem +d +� +, +(127) +this functor assigns a corresponding linear isomorphism: +(Z (M) : Z (∂in (M)) → Z (∂out (M))) ∈ hom (Vect) , +(128) +designating the propagator/scattering amplitude/S-matrix for a process of shape M mapping from hyper- +surface ∂in (M) to hypersurface ∂out (M). Hence, just as computational irreducibility may be said to be +formally dual/adjoint to locality of time evolution in quantum mechanics, multicomputational irreducibility +may be said to be formally dual/adjoint to adherence to the Atiyah-Segal sewing laws[15][16][17] in func- +torial quantum field theory, under which the path integral over a domain Σ which can be decomposed into +subdomains Σ1 and Σ2 (i.e. Σ = Σ1 ⊔ Σ2) must be equal to the path integral over subdomain Σ1 composed +with the path integral over subdomain Σ2. +Conventionally, the symmetric monoidal functors Z : BordRiem +1 +→ Vect and Z : BordRiem +d +→ Vect that +define time evolution in category quantum mechanics and functorial quantum field theory are assumed to +be strong (in the sense described previously that the coherence maps ϵ and µX,Y are isomorphisms for all +X, Y ∈ ob (Vect)). However, simply preserving the tensor product structure of cobordisms/vector spaces +is not sufficient to define a truly physical theory of quantum mechanics or quantum fields: at the very +least, one must also preserve the dagger structure of time evolution (which generalizes the Hermitian adjoint +operation on linear transformations, otherwise known as the conjugate transpose in the finite-dimensional +44 + +case), as well as the compact structure of vector spaces (which generalizes the operation of taking duals of a +finite-dimensional vector space). In this setting, we say that Vect is a dagger category[53][54] to mean that +it is equipped with an involutive contravariant endofunctor † : Vect → Vect, i.e. a functor from Vect to +itself which has the effect of swapping the sources and targets of each morphism: +∀ (f : X → Y ) ∈ hom (Vect) , +∃ +� +f † : Y → X +� +∈ hom (Vect) , +(129) +and reversing the direction of composition: +∀ (f : X → Y ) , (g : Y → Z) ∈ hom (Vect) , +� +(g ◦ f)† : Z → X +� += +� +f † ◦ g† : Z → X +� +, +(130) +which acts as the identity on objects: +∀X ∈ ob (Vect) , +X† = X, +and +� +id† +X : X → X +� += (idX : X → X) , +(131) +and which is involutive in the sense that it acts as its own inverse functor: +∀ (f : X → Y ) ∈ hom (Vect) , +�� +f †�† : X → Y +� += (f : X → Y ) . +(132) +Since Vect is also a symmetric monoidal category, we would ideally like for the dagger structure † to be +compatible with the tensor product structure ⊗, meaning that: +∀ (f : X → Y ) , (g : Z → W) ∈ hom (Vect) , +� +(f ⊗ g)† : Y ⊗ W → X ⊗ Z +� += +� +f † ⊗ g† : Y ⊗ W → X ⊗ Z +� +, +(133) +in such a way that one maintains compatibility with the defining natural isomorphisms of the symmetric +monoidal structure, namely the associator isomorphism α: +45 + +∀X, Y, Z ∈ ob (Vect) , +� +α† +X,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z) +� += +� +α−1 +X,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z) +� +, +(134) +the left and right unitor isomorphisms λ: +∀X ∈ ob (Vect) , +� +λ† +X : X → I ⊗ X +� += +� +λ−1 +X : X → I ⊗ X +� +, +(135) +and ρ: +∀X ∈ ob (Vect) , +� +ρ† +X : X → X ⊗ I +� += +� +ρ−1 +X : X → X ⊗ I +� +, +(136) +and the symmetry/braiding isomorphism σ: +∀X, Y ∈ ob (Vect) , +� +σ† +X,Y : Y ⊗ X → X ⊗ Y +� += +� +σ−1 +X,Y : Y ⊗ X → X ⊗ Y +� +. +(137) +For the case of vector spaces (respectively finite-dimensional vector spaces), the Hermitian adjoint operation +(respectively the conjugate transpose operation) clearly furnishes Vect/FdVect with a canonical dagger +structure. +For the case of the cobordism categories BordRiem +1 +and BordRiem +d +, the operation of “time +reversal” (i.e. inversion of the orientation of cobordisms) yields a compatible dagger structure. Likewise, +for the case of the category T of data structures and computations, obvious dagger structures exist for the +cases of both Turing machine evolution and hypergraph rewriting considered previously (since for any given +Turing machine rule, it is always possible to find a Turing machine rule of the same signature that reverses +its evolution, and for any given hypergraph rewriting system, one can always swap the L and R objects +within the span of monomorphisms that defines the rewriting rule in order to obtain a time-reversed version +of the same evolution). +On the other hand, we also say that FdVect (the category of finite-dimensional vector spaces) is a compact +closed[55][56] symmetric monoidal category as a shorthand for saying that every object X ∈ ob (FdVect) +has a corresponding dual object X∗ ∈ ob (FdVect) that is unique up to canonical isomorphism, which is +equipped with a pair of morphisms ηX and εX of the form: +(ηX : I → A∗ ⊗ A) , (εA : A ⊗ A∗ → I) ∈ hom (FdVect) , +(138) +46 + +known as the unit and counit morphisms, respectively, satisfying the following pair of coherence conditions +(sometimes known as the yanking conditions): +∀X ∈ ob (FdVect) , +� +λX ◦ +� +(εX ⊗ idX) ◦ +� +αX,X∗,X ◦ +� +(idX ⊗ ηX) ◦ ρ−1 +X +��� +: X → X +� += (idX : X → X) , +(139) +and: +∀X ∈ ob (FdVect) , +� +ρX∗ ◦ +� +(idX∗ ⊗ εX) ◦ +� +α−1 +X∗,X,X∗ ◦ +� +(ηX ⊗ idX∗) ◦ λ−1 +X∗ +��� +: X∗ → X∗� += (idX∗ : X∗ → X∗) . +(140) +We can reformulate these two yanking conditions diagrammatically as the statement that the following pair +of diagrams commute for all X ∈ ob (FdVect): +X +X ⊗ I +X ⊗ (X∗ ⊗ X) +(X ⊗ X∗) ⊗ X +I ⊗ X +X, +ρ−1 +X +idX +idX⊗ηX +αX,X∗,X +εX⊗idX +λX +(141) +and: +X∗ +I ⊗ X∗ +(X∗ ⊗ X) ⊗ X∗ +X∗ ⊗ (X ⊗ X∗) +X∗ ⊗ I +X∗. +λ−1 +X∗ +idX∗ +ηX⊗idX∗ +α−1 +X∗,X,X∗ +idX∗⊗εX +ρX∗ +(142) +If, moreover, there exists a dagger structure † that is compatible with the compact structure in such a way +47 + +that the unit and counit morphisms η and ε can be related by means of the dagger operation (as indeed is +the case for the category of finite-dimensional vector spaces and their linear isomorphisms), in other words +if the following diagram commutes for all objects X ∈ ob (FdVect): +I +X ⊗ X∗ +X∗ ⊗ X, +ε† +X +ηX +σX,X∗ +(143) +i.e: +∀X ∈ ob (FdVect) , +� +σX,X∗ ◦ ε† +X : I → X∗ ⊗ X +� += (ηX : I → X∗ ⊗ X) , +(144) +then we describe the symmetric monoidal category as being dagger compact. For the case of finite-dimensional +vector spaces, the passage to the dual vector space (i.e. the of vector space of linear forms under pointwise +addition and scalar multiplication) furnishes FdVect with a canonical compact structure[57][58]; for the +case of the infinite-dimensional vector spaces in Vect, dual spaces also exist, although they are inherently +less “well-behaved” (since the axiom of choice here implies that the dual space is always of strictly larger +dimension than the original space) and satisfy fewer compatibility conditions. In topological quantum field +theories, in which the number of degrees of freedom (and therefore the dimensionality of the spaces of states) +is always finite, the propagator for processes of shape M from hypersurface ∂in (M) to hypersurface ∂out (M), +namely: +Z (M) : Z (∂in (M)) → Z (∂out (M)) , +(145) +generated by the functor Z : BordRiem +d +→ Vect can be formulated in terms of dual spaces as: +Z (M) : C → Z (∂ (M)) = Z (∂out (M)) ⊗ Z (∂in (M))∗ , +(146) +i.e. in the form of a correlator or n-point function. For the category of hypergraphs and subhypergraph +inclusion maps, every hypergraph has a dual (that is trivially compatible with the dagger structure) given +by the interchange of its vertex set V and its hyperedge set E, i.e if: +H = ⟨V = {vi |i ∈ Iv } , E = {ei |i ∈ Ie, ei ⊆ V, ei ̸= ∅}⟩ , +(147) +48 + +then: +H∗ = ⟨V ∗ = E, E∗ = {{ei |vm ∈ ei } |m ∈ Iv }⟩ , +(148) +where Iv and Ie are index sets for the vertices and for the hyperedges, respectively. Corresponding dual +structures may well exist for other categories of data structures and computations too (such as the category +of Turing machine states and transitions), but, if they do, then their explicit forms remain unknown at +present. +Just as we have shown that deformation of the sequential composition structure ◦ in an ordinary category +can be used to characterize and quantify computational reducibility, and that deformation of the parallel +composition structure/tensor product ⊗ in a symmetric monoidal category can be used to characterize and +quantify multicomputational reducibility, it is entirely conceivable that, as a consequence of this formal du- +ality/adjunction relating computational complexity theory to quantum mechanics and multicomputational +complexity theory to quantum field theory, these various other algebraic properties and structures inher- +ent to the physical theories will end up having rather natural, and purely complexity-theoretic, analogs in +the abstract theory of computation. For instance, it is plausible that deformation of the dagger structure +† in a dagger symmetric monoidal category may provide some means of characterizing the irreversibil- +ity (meaning the pragmatic computational difficulty of reversing a computation that is, in principle, fully +reversible) of certain classes of (multi)computations - a concept which is, of course, deeply conceptually +related to (multi)computational irreducibility itself - while deformation of the compact structure η, ε in a +dagger compact category may be useful for quantifying the computational difficulty involved in exchanging +outputs/values with inputs/arguments in a multicomputation consisting of one or more multi-argument, +multi-valued functions (e.g. as represented by a tensor network or a monoidal string diagram). These ex- +citing possibilities remain topics for future investigation, as further detailed within the concluding remarks +below. +5 +Concluding Remarks +Throughout the course of this article, we have sought to develop a systematic procedure by which the +abstract syntax of a category may be endowed with a concrete computational semantics, in which all objects +are interpreted as data structures and all morphisms are interpreted as computations, in such a way that +each morphism carries with it certain metadata corresponding to the time complexity of its underlying +49 + +computation, along with an algebra that defines how these time complexities behave under composition. We +have shown that this can be achieved by defining a map from the category of data structures and computations +to a discrete cobordism category (of 0-dimensional manifolds and discrete cobordisms/intervals), in such +a way that the irreducibility of a given computation is characterized by the extent to which this map +preserves additivity of time complexities under sequential composition (i.e. +its functoriality), and such +that the multicomputational irreducibility of a given multicomputation (described by the case of higher- +dimensional manifolds and discrete cobordisms, in which both categories carry an additional symmetric +monoidal structure) is characterized by the extent to which this map preserves additivity of time complexities +under parallel/tensor product composition (i.e. its symmetric monoidal functoriality). This latter extension +is achieved by exploiting the fact that symmetric monoidal categories provide a convenient compositional +semantics for describing multiway systems, branchial graphs and the general algebraic structure of non- +deterministic computations. In so doing, we have effectively defined the outlines of a potential novel extension +to the standard techniques of category theory and monoidal category theory, in which the compositions of +morphisms (both in sequence and in parallel) carry with them additional algebraic structure resulting from +the constraints of computational complexity theory. This new formalism brings with it many compelling +directions for future research and exploration, a few of which we shall indicate below. +Firstly, although traditional computational complexity theory has tended to restrict itself to the investi- +gation of complexity classes with very simply and well-defined algebraic constraints (e.g. P, EXP, NP, etc.), +this category-theoretic formalism, along with the explicit computational tools developed for the purposes of +the present article, potentially enables one to conduct a systematic and empirical investigation of computa- +tional complexity classes exhibiting far less a priori algebraic structure. Further, by considering the extension +to monoidal categories, it also enables the rigorous investigation of multicomputational complexity theory, +which differs fundamentally from ordinary non-deterministic complexity theory in that, within a standard +non-deterministic complexity class (such as NP), although the non-deterministic nature of the computation +implies that one is inevitably forced to consider a multiway system of different singleway/deterministic com- +putations, the complexity class itself is ultimately concerned only with the time complexity along a single +branch of that system (albeit a branch whose identity may not necessarily be known in advance). Multicom- +putational complexity theory, on the other hand, investigates the computational complexity of the multiway +system itself, and specifically of the tensor product structure of its branchial graphs (encoding, as it does, the +complex branching and merging behavior of the entire multiway system), featuring, as discussed previously, +contributions from the complexity-theoretic properties of both the state evolution function and the state +50 + +equivalence function of the multiway system, and, most crucially, the properties of the interactions between +the two. To the best of the author’s knowledge, the space of possible multicomputational complexity classes +remains essentially unexplored, and constitutes a major topic for planned future examination. +Next, all of the analysis presented within this article has focused exclusively on singleway/multiway evo- +lution structure, and has neglected any consideration of causality; however, a natural causal semantics does +exist for the case of hypergraph rewriting[59][60] (and, indeed, for non-deterministic Turing machine evolu- +tion, as depicted as part of the multiway evolution causal graph shown in Figure 17 for the non-deterministic +2-state, 2-color Turing machine analyzed previously), allowing one to encode formally, by means of an +explicit partial order, the notion of one transition/rewriting event only being applicable if another transi- +tion/rewriting event had previously occurred. Just as the multiway evolution graph can be interpreted as +freely generating a symmetric monoidal category, the multiway evolution causal graph (featuring a combi- +nation of both evolution edges, indicating transitions/rewriting events, and causal edges, indicating causal +relationships between those transitions/rewriting events) can be interpreted as freely generating a weak 2- +category[61], with the 2-cells representing causal relationships. These 2-cells are also equipped with their +own tensor product operation, usually denoted ⊗C, that satisfies the axioms of a partial monoidal structure +in the sense defined by Coecke and Lal in the context of causal categories[62], and allows one to compose +causal relationships between spacelike-separated (causally-independent) events in parallel, but not between +timelike-separated (causally-dependent) events, which may only be composed in sequence. This leads to +the intriguing possibility that one may be able to equip the discrete cobordism category B with a second +(partial) monoidal structure and, with it, to examine the 2-functoriality of the resulting map Z′ : T → B +as a proxy for extending the definition of multicomputational irreducibility so as also to incorporate some +kind of inherent complexity of causality, and not simply of evolution (essentially by examining the extent to +which Z′ also distorts the sequential and parallel composition of the 2-cells representing the causal structure +of the system). The aim of such an undertaking would be to formalize a notion of causal irreducibility, in +which the complete causal structure of a causally irreducible multiway system cannot be preempted without +effectively tracing the explicit causal relationships between all transitions/rewriting events, spanning across +all branches of the multiway evolution graph. +As discussed within the preceding section, there are also various features of the formal duality/adjunction +relationship between (multi)computational complexity theory and categorical quantum mechanics/functorial +quantum field theory that might potentially yield additional, purely complexity-theoretic, extensions to this +formalism. For instance, it has often been assumed that the practical irreversibility of computations which +51 + +Figure 17: A multiway evolution causal graph corresponding to the non-deterministic evolution of a 2- +state, 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition +functions for rules 2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 3 steps; the gray edges are +the usual evolution edges, while each orange edge represents a causal relationship between two transitions. +are in principle reversible (e.g. in the case of one-way functions in cryptography) is merely a byproduct of +computational irreducibility, but this assumption tacitly presupposes a compatibility condition between the +irreducibilities of forward and backward evolution that may or may not hold for certain classes of systems. +By equipping our category T with an involutive dagger structure † and examining the effects of the map Z′ +on that structure using methods from categorical quantum mechanics, it is conceivable that we may be able +to disentangle the irreducibility of evolution from the irreducibility of reversal in a relatively fine-grained way. +Deeply related to this is the irreducibility of swapping arguments/inputs with values/outputs in a composition +of several multi-argument, multi-valued functions; in the case of a tensor network or monoidal string diagram +(where this swapping operation is encoded as the raising and lowering of contravariant/covariant indices), +such operations are usually assumed to be computationally trivial, but for the case of computations whose +reversal operation is irreducible this need not be the case. +Equipping the category T with a compact +structure, with unit/counit η/ε, and observing how the compact structure gets distorted under the action +of Z′ may permit one to quantify the complexities of these operations in a more meaningful way. +52 + +However, the conventional multiway system formalism does not make full use of the compact structure +that is available within dagger compact categories, since transitions/events in a multiway system are tradi- +tionally single-argument but potentially multi-valued (that is, a state evolution function conventionally takes +in a single state as input and produces a list of possible successor states as output). On the other hand, a +glocal multiway system promotes the transitions/events to being true multi-argument functions (and, hence, +to true symbolic tensors in a tensor network/string diagram representation of the multiway system) by ef- +fectively “shattering” the states into their constituent “tokens” (for instance, into individual hyperedges for +the case of hypergraph rewriting systems, or individual tape positions for the case of Turing machines) and +then reassembling them on-demand for the application of particular transitions/events; the term “glocal” +refers here to the fact that the tokens are local but the events are global, and so in particular the multiway +equivalence function acts at the level of events rather than at the level of states. An example of a glocal +multiway evolution causal graph, for the same non-deterministic 2-state, 2-color Turing machine as before, +is shown in Figure 18, with its associated glocal branchial graph shown in Figure 19. Note that, on a given +glocal branchial graph, some of the tokens are separated spatially (i.e. they correspond to different regions +of the tape), whilst some of the tokens are separated “branchially” (i.e. they correspond to the same region +of the tape, but on two different branches of the multiway system), and so, unlike the branchial graphs +considered previously, glocal branchial graphs actually encode two different tensor product structures: the +standard symmetric monoidal structure inherited from the multiway system, and a “spatial” tensor product +structure (which is really identical to the causal tensor product structure ⊗C described above). The com- +patibility conditions between these two tensor product structures are not known (for instance, it is unknown +whether they form a rig category in special cases, although this possibility is unlikely) and remain a topic +for future research. Note that the question of whether the ordering of tokens matters (as for Turing machine +tape states) or does not (as for hyperedges in a hypergraph) corresponds to the question of whether the +“spatial” part of the category is symmetric monoidal or simply monoidal. Rather excitingly, just as the +multiway tensor product structure enabled us to quantify multicomputational complexity, it is conceivable +that the spatial tensor product structure (once it is better understood) may enable us to quantify space +complexity, and thus to examine questions surrounding (e.g.) the trade-offs between space complexity and +time complexity in arbitrary (multi)computations. A meticulous treatment of the relationship between mul- +tiway systems, the notion of “spatiality”, higher categories and type theory was previously conducted by +Arsiwalla[64][65] within the context of Shulman’s cohesive homotopy type theory[66]. +Finally, there are several conceptual consequences of the realization of the orthogonality (independence) +53 + +Figure 18: A “glocal” multiway evolution causal graph corresponding to the non-deterministic evolution of a +2-state, 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition +functions for rules 2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 3 steps; the orange edges +represent causal relationships between transitions, while each gray edge represents either the “ingestion” or +the “egestion” of a single “token” (tape position) into or out of a single transition. +that exists between the complexity of state evolution and state equivalence (and thus between computational +and multicomputational irreducibility). For instance, the computational interpretation of the second law of +thermodynamics advocated by Wolfram[2] implies that entropy increase is a consequence of computational +irreducibility, wherein the progression of a reversible computation can have the effect of “encrypting” the +details of its initial conditions (such that, even if the computation is in principle reversible, in practice it +can represent an arbitrarily hard problem of cryptanalysis to enact that reversal). However, synthesizing +this idea with the “orthogonality” principle indicates that there should exist at least two distinct concepts of +entropy at play within any given multicomputation: one essentially computational, and the other essentially +multicomputational, in origin. The standard thermodynamic description of entropy is essentially a measure +of non-injectivity of coarse-graining (i.e. how many distinct microstates get mapped to the same macrostate +under the action of the coarse-graining function), and, due to Liouville’s theorem, and thus the one-to-one +correspondence that exists between position/momentum values and possible evolution histories, in classical +mechanics this definition is provably equivalent to a definition in terms of possible evolution trajectories +(i.e. how many distinct branches of history would have resulted in the same coarse-grained macrostate). +However, for more general multiway systems with more complex branching and merging structure, this +correspondence does not necessarily hold, and so the two definitions of entropy diverge (although many +computations of e.g. entanglement entropies in the context of quantum gravity may implicitly assume that +they are equivalent[67][68][69][70]). The manifold implications of this disambiguation remain another worthy +54 + +Figure 19: The corresponding “glocal” branchial graph associated to the default “foliation” of the glocal +multiway evolution causal graph for the non-deterministic evolution of the 2-state, 2-color Turing machine +constructed from parallel composition of the transition functions for rules 2506 and 3506, after 3 steps, +showing a mixture of both “spatial” and “branchial” tensor product structure. +topic for future study. +Acknowledgments +The author would like to thank Mohamed Barakat, Nicolas Behr, Matteo Capucci, Bob Coecke, Fabrizio +Genovese, Manojna Namuduri, Stephen Wolfram and Yorick Zeschke for various stimulating and insightful +discussions, enjoyed at a variety of different gestational stages of the ideas presented within this work. The +author would also like to acknowledge James Boyd for his christening of the term “multicomputational +irreducibility” in an earlier blog post, and Juan Arturo Silva-Ordaz for forcing the author to think about the +complexity of equivalence functions to a far greater extent than he would ever have chosen to do voluntarily. +55 + +. +88 +日 +EReferences +[1] S. Wolfram (1985), “Undecidability and Intractability in Theoretical Physics”, Physical Review Let- +ters 54 (8): 735–738. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.54.735. +[2] S. Wolfram (2002), A New Kind of Science. Champaign, IL: Wolfram Media, Inc. https://www. +wolframscience.com. +[3] J. Gorard (2018), “The Slowdown Theorem: A Lower Bound for Computational Irreducibility in Phys- +ical Systems”, Complex Systems 27 (2): 177–185. https://www.complex-systems.com/abstracts/ +v27_i02_a05/. +[4] A. M. 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Zhao (2017), “Quantum Complexity and Negative Curvature”, +Physical Review D 95 (4): 045010. https://arxiv.org/abs/1608.02612. +62 + diff --git a/JtE3T4oBgHgl3EQfvAvH/content/tmp_files/load_file.txt b/JtE3T4oBgHgl3EQfvAvH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88ddfde8c9aeaf3b8f7717b47a674dc7c3e84fd1 --- /dev/null +++ b/JtE3T4oBgHgl3EQfvAvH/content/tmp_files/load_file.txt @@ -0,0 +1,1327 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf,len=1326 +page_content='A Functorial Perspective on (Multi)computational Irreducibility Jonathan Gorard1,2 1Cardiff University, Cardiff, UK∗ 2University of Cambridge, Cambridge, UK† January 13, 2023 Abstract This article aims to provide a novel formalization of the concept of computational irreducibility in terms of the exactness of functorial correspondence between a category of data structures and elementary computations and a corresponding category of (1-dimensional) cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We proceed to demonstrate that, by equipping both categories with a symmetric monoidal structure and considering the case of higher-dimensional cobordism categories, we obtain a natural extension of this formalism that serves also to encompass non-deterministic or “multiway” computations, in which one quantifies not only the irreducibility in the behavior of a single (deterministic) computation path, but in the branching and merging behavior of an entire “multiway system” of such paths too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We finally outline how, in the most general case, the resulting symmetric monoidal functor may be considered to be adjoint to the functor characterizing the Atiyah-Segal axiomatization of a functorial quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Thus, we conclude by arguing that the irreducibility of (multi)computations may be thought of as being dual to the locality of time evolution in functorial approaches to quantum mechanics and quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In the process, we propose an extension of the methods of standard (monoidal) category theory, in which morphisms are effectively equipped with intrinsic computational complexity data, together with an algebra for how those complexities compose (both in sequence and in parallel, subject to the monoidal structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Some possible extensions of this formalism (for instance to encompass notions of causality, space complexity, irreversibility, complexity of index manipulation in tensor networks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ), as well as potential implications for physics (for instance by providing an ability to distinguish formally between certain computational and multicomputational definitions of entropy) are briefly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∗gorardj@cardiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='uk †jg865@cantab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='uk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='04690v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='CC] 13 Oct 2022 1 Introduction Computational irreducibility, as first proposed by Stephen Wolfram in the 1980s[1], explored empirically within A New Kind of Science[2] (NKS) and subsequently refined and formalized by the author in later work[3], refers to the general phenomenon in which the outcome of any sufficiently sophisticated computa- tional process cannot be predicted (or “shortcut”) using any less computational effort than the system itself requires for its own explicit evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In this way, computational irreducibility may be considered to be an extension of the standard recursion-theoretic concepts of universality and undecidability[4][5][6] (indeed, it is straightforward to prove by diagonalization that any Turing-complete computation must be irreducible, and undecidability may be thought of as corresponding to a limiting case of irreducibility in which certain computations become infinite in length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Within the NKS paradigm (in which all natural processes are con- ceived of as corresponding to computations), traditional methods of theoretical science thus correspond to those cases in which computations are reducible, and can therefore be preempted by means of sufficiently so- phisticated tools of scientific and mathematical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The abstract phenomenon of complexity in natural systems may, in turn, be thought of as corresponding to those cases in which computations are fundamentally irreducible, and therefore in which the only reasonable course of action available to the theoretical scientist is to simulate the behavior of the system explicitly[7][8][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The present article seeks to recast the author’s previous formal definition of computational irreducibility within the broader conceptual framework of functoriality[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In Section 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' by considering a category whose objects are data structures and whose morphisms are elementary computations (in the first instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the article employs Turing machines as its chosen formal model of computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' whose corresponding category has as its objects configurations of the Turing machine’s tape/head and head position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and as its morphisms applications of the Turing machine’s partial transition function),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we are able to reformulate the definition of reducibility/irreducibility in terms of the compositional properties of a certain map from this category to a category whose objects are time steps/step numbers and whose morphisms are discrete intervals between these step numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This has the effect of equipping the morphisms of our original category of Turing machine states and computations with intrinsic computational complexity data, along with a natural algebra (encoded by the action of the map on the composition operation) describing how the corresponding time complexities compose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Computational irreducibility then corresponds to the case where these complexities compose purely additively, and therefore in which this map is a pure functor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' this allows us to say, in a rather precise sense, that irreducibility is functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Computational reducibility is then a measure of the extent to which this map distorts pure additive composition of time complexities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' it is a measure of deviation away from 2 pure functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The codomain of this map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the category of step numbers and discrete intervals) may be interpreted as being a certain discretization of a (1-dimensional) category of cobordisms/“gluings of boundaries” between (0-dimensional) manifolds[11][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In Section 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we proceed to demonstrate that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' when our category of data structures and elementary com- putations is also equipped with a symmetric tensor product structure (thereby promoting it to a symmetric monoidal category[13]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' which allows us to describe a multiway system of many different branching and merging (singleway) computation paths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with the branchial graphs of that multiway system describing the tensor product structure of the corresponding monoidal category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' then there exists a very natural extension of the previous irreducibility formalism for ordinary categories and singleway computations to encompass the multicomputational case too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Just as (singleway) computational reducibility may be formulated as a measure of distortion of the additivity of time complexity under ordinary (sequential) composition of com- putations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' multicomputational reducibility may therefore be formulated as a measure of distortion of the additivity of time complexity under parallel (tensor product) composition of computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a multicomputationally irreducible computation is one under which the map from the (monoidal) category of data structures and elementary computations to the (monoidal) category consisting of parallel compositions of step numbers and parallel compositions of discrete intervals (effectively describing the coordinatization of branchial graphs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' is a pure symmetric monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This idea formally encodes and concretizes the intuition that a singleway computation is irreducible if one must explicitly trace every intermediate step in the composition in order to determine the final result, whereas a multiway computation is irreducible if one must explicitly trace every singleway computation path in order to determine the system’s overall branchial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The codomain of this map between symmetric monoidal categories is now interpretable as a certain discretization of a higher-dimensional category of cobordisms between higher-dimensional man- ifolds (equipped with a Riemannian structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We discuss how ordinary computational irreducibility is a byproduct of the complexity of the state evolution function used in the specification of a multiway system, whilst multicomputational irreducibility is a byproduct of the complexity of its state equivalence function, and emphasize that the two concepts are therefore essentially orthogonal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' one can easily build mul- ticomputationally irreducible systems out of tensor products of many computationally reducible singleway paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' To illustrate this fact, we also analyze the case of multiway hypergraph rewriting systems, in which the state evolution function has comparable computational complexity to that of Turing machine evolution, but the state equivalence function (based on hypergraph isomorphism) is now far more non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In Section 4, we illustrate how, in a certain general sense, the functor from data structures and compu- 3 tations to step numbers and intervals that characterizes an irreducible computation may be considered to be the left adjoint of the functor encoding the passage from moments of time and intervals to vector spaces and linear isomorphisms in the context of categorical quantum mechanics[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Thus, the same functoriality that characterizes the irreducibility of a computation may be thought of as being formally dual/adjoint to the functoriality that characterizes the locality of time evolution in non-relativistic quantum mechanics (in which any global unitary evolution over an interval may be decomposed into several local unitary evolutions over subintervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In a fairly natural extension of this idea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we show how the symmetric monoidal func- tor encoding an irreducible multicomputation can be viewed as the left adjoint of the symmetric monoidal functor from manifolds and cobordisms to vector spaces and linear isomorphisms that defines a functorial quantum field theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with the functoriality of multicomputational irreducibility now being dual/adjoint to the Atiyah-Segal sewing laws[15][16][17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in which the path integral over any global interval may be decom- posed into local path integrals over subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Hence, we argue for the existence of a general adjunction relationship between computational complexity theory and categorical quantum mechanics, and between multicomputational complexity theory and functorial quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We discuss how some of the other algebraic structure inherent to categorical quantum mechanics and functorial quantum field theory models (in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the involutive dagger and compact structures of the corresponding categories) might therefore have potential complexity-theoretic interpretations (for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in terms of the irreducibility of reversing computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' or of swapping arguments and values in a composition of multi-argument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' multi- valued functions as described by a tensor network or monoidal string diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ), although a complete analysis of these interpretations lies beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Finally, in Section 5, we outline some of the broader implications of the formalism developed herein, including the generalization of techniques of ordinary (monoidal) category theory to accommodate the case where morphisms designate explicit computations with associated computational complexity classes, and thus in which composition operations must respect the underlying algebra for how the complexities of those various computations interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We also discuss the possibility of using these new algebraic techniques to conduct a more systematic investigation of computational complexity classes and (multi)computational complexity classes that are in some way “wilder” or less structured (in the sense that their algebra of composition is less well-behaved) than those studied within the setting of traditional computational complexity theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we, moreover, indicate the various ways in which multicomputational complexity classes differ from traditional non-deterministic complexity classes (namely by considering the inherent computational complexity of the branching and merging operations of the multiway system, rather than solely considering the result yielded 4 non-deterministically by a single multiway branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We summarize some of the potential implications for physics, including a plausible disambiguation of two different computational definitions of entropy - one in which microstates are treated as being possible instantaneous states of a system (and thus based on computational irreducibility), and one in which microstates are treated as being possible paths of evolution history for the system (and thus based on multicomputational irreducibility) - that are often otherwise conflated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We propose some possible future directions of investigation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' including the extension of the relevant maps/functors to also encompass preservation of dagger structure and/or compact structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the extension of the overall formalism to encompass the additional information encoded within the structure of causal relations via certain higher categories/higher functors and the extension to multi-argument computations as encoded through glocal multiway systems and their description as tensor networks/string diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note that the majority (though by no means all) of the categories considered within this article are small (and all of the ones which are not are sufficiently widely studied that their behavior is known to be at least somewhat well-behaved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For this reason, we shall henceforth neglect all considerations of set-theoretic issues, and shall use terms such as “object set” and “hom set” irrespective of whether the structures involved are sets or proper classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' note that the code necessary to reproduce the results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' proofs and figures from this article is open source and freely exposed through the Wolfram Function Repository,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for instance via MultiwayTuringMachine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' TuringMachineGlocalMultiwaySystem and MultiwaySystem for system evo- lution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' AbstractCategory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' AbstractFunctor and AbstractStrictMonoidalCategory for representation of the underlying category-theoretic structures (and for automating the process of theorem-proving over them),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 2 (Singleway) Irreducibility as Functoriality We begin by adopting the formal definition of computational irreducibility proposed by the author in [3]: Definition 1 If f : N → N is a function on natural numbers and T is a Turing machine that computes the value of f (i) for some fixed input i in n steps, then T’s computation is reducible if and only if there exists a Turing machine T ∗ that computes f (i) in m steps, where m < n[5][6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The “degree” of reducibility of the computation may thus be quantified in terms of the discrepancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the value of n − m (the converse value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' m − n for the case of irreducible computations in which m ≥ n, is known as the slowdown of T ∗’s simulation of T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note that we are here and henceforth assuming that all Turing machines are 1-tape Turing machines,[7] which we can do without loss of generality since any k-tape 5 Turing machine M operating in time f (n) may be simulated by a 1-tape Turing machine M ′ operating in time O � [f (n)]2� , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for any input x, M ′ (x) = M (x)[18][19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We have also (implicitly) adopted the Hopcroft-Ullman formalization[6] of a 1-tape Turing machine T as a 7-tuple T = ⟨Q, Γ, b, Σ, δ, q0, F⟩, with finite alphabet set Γ ̸= ∅, blank symbol b ∈ Γ, input symbols Σ ⊆ Γ \\ {b}, finite state set Q ̸= ∅, initial state q0 ∈ Q, accepting states F ⊆ Q and (partial) transition function: δ : (Q \\ F) × Γ ↛ Q × Γ × {L, F} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (1) We are now in a position to be able to construct a category[10] representing a particular formal (abstract) model of computation, whose objects are data structures and whose morphisms are elementary/primitive computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For the specific case considered here of the category generated by the Turing machine T, which we shall denote T , we choose as its object set ob (T ) the set Γℵ0 × Q × N of possible ordered triples consisting of tape state, head state and head position (assuming an infinite-length tape), and we wish for its morphism set hom (T ) to consist of all possible valid transitions of the Turing machine T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In order to construct this morphism set, we therefore begin by constructing a quiver (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a directed multigraph) whose arrows/edges correspond to applications of the (partial) transition function δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for our present purposes, it suffices to think of the set of all possible such arrows as being N × � Γℵ0 × Q × N �2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' each arrow is treated as an ordered triple (f, X, Y ), denoted f : X → Y , consisting of a name f ∈ N, and a pair of elements X, Y ∈ ob (T ) (X being the arrow’s source/domain and Y being its target/codomain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This quiver freely generates a category if we populate the morphism set hom (T ) initially with the set of arrows/transitions, and proceed to introduce a binary composition operator ◦ such that: ∀ (f : X → Y ) , (g : Y → Z) ∈ hom (T ) , (g ◦ f : X → Z) ∈ hom (T ) , (2) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' any pair of morphisms with matching codomain and domain can be composed by means of the operator .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' If X, Y and Z represent possible Turing machine states (including tape state, head state and head position), and f and g represent possible Turing machine transitions, this procedure of generating a category from the underlying quiver can be illustrated diagrammatically as follows: Y Y X Z X Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' g g f f g◦f (3) The resulting algebraic structure is indeed a category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' since the operator ◦ inherits associativity: 6 ∀ (f : X → Y ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (g : Y → Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (h : Z → W) ∈ hom (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ((h ◦ g) ◦ f : X → W) = (h ◦ (g ◦ f) : X → W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (4) from the fact that (partial) function composition is associative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and the identity axiom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' namely: ∀X ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∃ (idX : X → X) ∈ hom (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (5) such that: ∀ (f : X → Y ) ∈ hom (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (f ◦ idX : X → Y ) = (f : X → Y ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (6) and: ∀ (g : Y → X) ∈ hom (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (idX ◦ g : Y → X) = (g : Y → X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (7) holds by virtue of the fact that the (partial) transition function may be augmented by a neutral (“no shift”) operation id as follows: δ : (Q \\ F) × Γ ↛ Q × Γ × {L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' id} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (8) Note that, since the Turing machines considered here are all deterministic/classical, meaning that the (par- tial) transition function is always single-valued, it follows that the resulting quiver must take the form of either a path graph, a cycle graph or a union of path and cycle graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' An explicit example of this construc- tion, for the 2-state, 2-color Turing machine shown in Figure 1 (known as rule number 2506 in the canonical Turing machine enumeration) is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 1: A graphical representation of the 2-state, 2-color Turing machine rule number 2506, with the black icon representing the location and state of the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The process of obtaining the full category T representing the Turing machine T may consequently be 7 Figure 2: On the left, a graph-theoretic representation of the evolution of the 2-state, 2-color Turing machine rule 2506, starting from the tape state {0, 1, 0, 0}, for 4 evolution steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' each arrow/edge of the quiver represents a single application of the Turing machine’s transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' On the right, a graph-theoretic representation of the category that is freely generated from this quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' thought of graph-theoretically as taking a reflexive transitive closure of the underlying transition quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' However, this operation of taking transitive closures appears, at least conceptually, to be very much against the spirit of computational irreducibility - it seems intuitively to imply that whenever there exists a com- putation from X to Y , and another computation from Y to Z, then one can necessarily always “jump ahead” to get directly from X to Z with the same amount of computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Ideally, we would like to consider some form of “decorated” category in which morphisms are tagged with some additional metadata corresponding to the computational complexity of the underlying computation that the morphism signifies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we could then proceed to define an algebra to describe formally how these complexities behave under the action of the composition operator ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Under such an algebra, irreducible computations would correspond to those computations whose complexities behave purely additively under composition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' if the computa- tion underlying morphism f : X → Y requires at least n steps to execute, and the computation underlying morphism g : Y → Z requires at least m steps to execute, then the composite computation g ◦ f : X → Z is irreducible (under the definition given above) if and only if it cannot be executed in fewer than n + m steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Conversely, if the complexities of the computations behave strictly subadditively under composition, then 8 the composite computation is reducible (with the “degree” of subadditivity corresponding to the “degree” of reducibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This intuition is illustrated in Figure 3 for the case of the 2-state, 2-color Turing machine rule considered above, with each edge/morphism tagged with the number of transitions necessary to per- form the corresponding computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the composition operation here is purely additive, indicating that all computations shown are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 3: A graph-theoretic representation of the category that is freely generated from the quiver repre- senting the evolution of the 2-state, 2-color Turing machine rule 2506, with edges/morphisms tagged with additional metadata corresponding to the number of “steps” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' transitions) required to perform the req- uisite computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In order to formalize this intuition, let us proceed on the assumption that all computations are fully irre- ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We can now consider a function Z′ mapping from our category of data structures and computations (for the case of Turing machines, this is the category T defined above) to a category B whose objects are natural numbers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ob (B) = N) corresponding to step numbers/moments of time, and whose morphisms are discrete intervals between these step numbers/moments of time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: hom (B) = {[n, m] ∩ N |n, m ∈ N and n ≤ m} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (9) If we now equip B with a composition operation given by the union of discrete (contiguous) intervals ∪, then it 9 trivially forms a category (with [n, n] ∩ N = {n} being the identity morphism for any n ∈ N), and, moreover, since all computations are irreducible (and therefore computational complexities are purely additive under composition) by hypothesis, the function Z′ : T → B trivially forms a functor[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' More precisely, Z′ is a map between categories T and B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∃Z′ (X) ∈ ob (B) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (10) and: ∀ (f : X → Y ) ∈ hom (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∃ (Z′ (f) : Z′ (X) → Z′ (Y )) ∈ hom (B) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with (Z′ (f) : Z′ (X) → Z′ (Y )) = [Z′ (X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z′ (Y )] ∩ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (11) such that the structure of composition is preserved: ∀ (f : X → Y ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (g : Y → Z) ∈ hom (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (Z′ (g ◦ f) : Z′ (X) → Z′ (Z)) = (Z′ (g) ∪ Z′ (f) : Z′ (X) → Z′ (Z)) = [Z′ (X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z′ (Z)] ∩ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (12) and all identity morphisms are preserved: ∀X ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (Z′ (idX) : Z′ (X) → Z′ (X)) = � idZ′(X) : Z′ (X) → Z′ (X) � = [Z′ (X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z′ (X)] ∩ N = {Z′ (X)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (13) If X, Y and Z represent Turing machine states reached at step numbers t1, t2 and t3 respectively, and f : X → Y and g : Y → Z represent the transitions between states X and Y and states Y and Z respectively, then this functoriality between categories T and B can be illustrated diagrammatically as follows: 10 Y t2 X Z t1 t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' g [t2,t3]∩N f g◦f [t1,t2]∩N ([t2,t3]∪[t1,t2])∩N =[t1,t3]∩N Z′ (14) In this way, the cardinality of any morphism in B (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' |[t1, t2] ∩ N|) represents, up to an additive constant, the time complexity of the corresponding computation in T (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' f : X → Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Since this functorial relationship holds only in the case where all computations in T are irreducible, we are consequently able to conclude that, in some precise sense, computational irreducibility is functoriality, and, moreover, that computational reducibility corresponds precisely to a deformation of the map Z′ away from being a pure functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This construction is demonstrated in Figure 4 for the 2-state, 2-color Turing machine described previously, with each vertex/object tagged with its step number and each edge/morphism tagged with a list of step numbers traversed throughout the course of its corresponding computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 4: A graph-theoretic representation of the category that is freely generated from the quiver repre- senting the evolution of the 2-state, 2-color Turing machine rule 2506, with vertices/objects tagged with additional metadata corresponding to the step number on which they occur (shown in blue) and with edges/morphisms tagged with additional metadata corresponding to all intermediate step numbers traversed as part of the requisite computation (shown in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note that the axioms obeyed by this “algebra of complexity” are formally almost identical to those 11 1, 2,3, 4 3 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 3, 4] 3,4,5 [1, 2, 3, 4, 5] [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 4, 5]of a metric space: the complexity of applying the identity transition id mapping from a state to itself is always 0 (or possibly 1, depending upon precisely how the intervals are defined), the complexity of applying any transition between two distinct states is always strictly positive, and the complexities always obey the triangle inequality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' non-strict subadditivity) under composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the only metric space axiom for which there exists no immediate analog is the symmetry axiom, since transitions need not necessarily be reversible, and the complexity of reversing a transition (when such a reversal exists) need not necessarily be equal to the complexity of performing the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note, moreover, that, if we modify the definition of the category B such that its objects are not merely natural numbers but real ones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ob (B) = R), and its morphisms are not merely discrete intervals but continuous ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: hom (B) = {[x, y] |x, y ∈ R and x ≤ y } , (15) with the composition operation still being the standard union of contiguous intervals ∪, then nothing in the above argument is substantively modified: the category T is still in functorial correspondence with the new version of category B if and only if it was in functorial correspondence with the old category B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The only material difference that this modification makes is that it guarantees that the functor/map Z′ (functor in the irreducible case, map in the reducible one) cannot be surjective on objects, but there was no requirement for it to be so in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' However, this slightly generalized definition of the category B does carry the definite advantage of making the underlying topological intuition of this construction manifest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the objects of R (real numbers, representing moments of time) can be thought of as being 0-dimensional manifolds, and its morphisms can be thought of as being 1-dimensional cobordisms between those manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Thus, B is really a cobordism category[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Here and henceforth, in relation to category B and its descendants, we employ the formal definition of a cobordism category as an ordered triple ⟨B, ∂, i⟩, where B has all finite coproducts and is equipped with an initial object ∅, ∂ : B → B is an additive endofunctor that preserves all coproducts and i : ∂ ⇒ IdB (with IdB denoting the identity functor on B that sends every object/morphism to itself) is a natural transformation satisfying: ∀M ∈ ob (B) , ∂ (∂ (M)) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (16) In the above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the coproduct of a pair of objects M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' M2 ∈ ob (B) is taken to be an object M1 ⊕ M2 ∈ ob (B) equipped with a pair of canonical injection morphisms: 12 (i1 : M1 → M1 ⊕ M2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (i2 : M2 → M1 ⊕ M2) ∈ hom (B) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (17) that are universal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in the sense that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for any object M ∗ ∈ ob (B) equipped with morphisms: (i∗ 1 : M1 → M ∗) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (i∗ 2 : M2 → M ∗) ∈ hom (B) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (18) there exists a unique morphism (u : M1 ⊕ M2 → M ∗) ∈ hom (B) such that: (i∗ 1 : M1 → M ∗) = (u ◦ i1 : M1 → M ∗) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and (i∗ 2 : M2 → M ∗) = (u ◦ i2 : M2 → M ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (19) This condition may be restated concisely by means of the following commutative diagram: ∀M ∗ M1 M1 ⊕ M2 M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∀i∗ 1 i1 ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u ∀i∗ 2 i2 (20) This definition can be extended in the obvious way to any finite collection of objects Mj ∈ ob (B) to yield a finite coproduct � j Mj ∈ ob (B) equipped with a finite collection of (universal) injection morphisms: � ij : Mj → � k Mk � ∈ hom (B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (21) An initial object ∅ ∈ ob (B) is a distinguished object such that, for any object M ∈ ob (B), there exists a unique morphism (u : ∅ → M) ∈ hom (B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∅ ∀M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u (22) An endofunctor is any functor from a category to itself, and an additive functor is any functor that preserves finite coproducts (or, in certain contexts, finite biproducts, though this is not the case considered here);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in other words, zero objects are preserved (up to isomorphism): ∂ (∅) ∼= ∅ ∈ ob (B) , (23) and, for any pair of objects M1, M2 ∈ ob (B), there exists an isomorphism: 13 ∂ (M1 ⊕ M2) ∼= ∂ (M1) ⊕ ∂ (M2) , (24) that preserves the canonical injection morphisms of the coproduct construction: M1 ⊕ M2 ∂ (M1 ⊕ M2) ∼= ∂ (M1) ⊕ ∂ (M2) M1 M2 ∂ (M1) ∂ (M2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' i1 i2 ∂(i1) ∂(i2) ∂ (25) An isomorphism (indicated by ∼=) here refers to any morphism (f : X → Y ) ∈ hom (B) for which there exists a corresponding morphism � f −1 : Y → X � ∈ hom (B) that acts as both a left and right inverse of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: � f −1 ◦ f : X → X � = (idX : X → X) , and � f ◦ f −1 : Y → Y � = (idY : Y → Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (26) Finally, a natural transformation η : F ⇒ G between functors F : C → D and G : C → D is characterized by a family of component morphisms: ∀X ∈ ob (C) , ∃ (ηX : F (X) → G (X)) ∈ hom (D) , (27) such that one has: ∀ (f : X → Y ) ∈ hom (C) , (ηY ◦ F (f) : F (X) → G (Y )) = (G (f) ◦ ηX : F (X) → G (Y )) , (28) or, represented in the form of a commutative diagram: F (X) F (Y ) G (X) G (Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' F (f) ηX ηY G(f) (29) The intuition lying behind this formalization of cobordism categories can be articulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Every object M ∈ ob (B) is interpreted as a (generalized) manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' every morphism (f : M1 → M2) ∈ hom (B) is interpreted as a (generalized) cobordism between those manifolds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a “gluing” together of those manifolds along their boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The initial object ∅ plays the role of the empty set/empty manifold, and the 14 additive endofunctor ∂ represents the boundary relation: ∂ (M) (for some M ∈ ob (B)) corresponds to the (generalized) boundary of manifold M, with the condition that ∂ (∂ (M)) = ∅ thus designating that the boundary of a boundary is always empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Coproducts ⊕ in the cobordism category B play the role of the direct sum/disjoint union of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' A pair of objects/manifolds M1, M2 ∈ ob (B) may be said to be cobordant (in the sense of their disjoint union forming the boundary of a manifold in one dimension higher), written M1 ∼ M2, if and only if: ∃V1, V2 ∈ ob (B) , such that M1 ⊕ ∂ (V1) ∼= M2 ⊕ ∂ (V2) , (30) where ∼= is an isomorphism in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the cobordism relation ∼ is hence an equivalence relation in which all isomorphic objects/manifolds are cobordant: ∀M1, M2 ∈ ob (B) , M1 ∼= M2 =⇒ M1 ∼ M2, (31) and where: ∀M ∈ ob (B) , ∂ (M) ∼ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (32) This connection to cobordism categories may appear to be a rather trivial technical point, but it will proceed to play a crucial role in the forthcoming discussion on formal correspondences with categorical quantum mechanics and functorial quantum field theory[14], and the relationship between computational irreducibility and the locality of time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 3 Multicomputational Irreducibility as Monoidal Functoriality We now proceed to consider the case of non-deterministic (multiway) computations, in which the (partial) transition function acting on data structures/computational states is not necessarily single-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Such computations may be parameterized by means of a multiway system[22], or, more precisely, a multiway evolution graph[23][24], namely a directed acyclic graph whose vertices represent computational states and whose directed edges represent single-step transitions between those states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' An explicit example of the multiway system construction, for a pair of 2-state, 2-color Turing machine rules shown in Figure 5 (rule numbers 2506 and 3506 in the canonical Turing machine enumeration) is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' By considering the resulting multiway evolution graph as a quiver, we are able to construct the category that this quiver 15 freely generates via the same procedure outlined in the previous section, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 5: A graphical representation of two 2-state, 2-color Turing machine rules (rule numbers 2506 and 3506), with the black icons representing the locations and states of the heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 6: A multiway evolution graph corresponding to the non-deterministic evolution of a 2-state, 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 4 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' each edge represents a single application of one of the two Turing machine transition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' However, we are now able to construct a deterministic/singleway computation from this non-deterministic/multiway one by exploiting the formalism of branchial graphs, through a procedure that is more-or-less directly anal- ogous to the Rabin-Scott powerset/subset construction[25] for converting non-deterministic finite automata into deterministic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' More concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we “foliate” the multiway evolution graph into an ordered sequence of non-intersecting “branchlike hypersurfaces” Σt that cover the entire multiway evolution graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with the ordering relation defined by a universal time function: 16 Figure 7: A graph-theoretic representation of the category that is freely generated by the multiway evolution graph corresponding to the non-deterministic evolution of a 2-state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506 (considered as a quiver),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' starting from the single tape state {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for 3 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t : V → Z, such that ∆t ̸= 0 everywhere, (33) where V is the vertex set of the multiway evolution graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the set of reachable Turing machine states), and where each “branchlike hypersurface” is now a level set of this function, satisfying: ∀t1, t2 ∈ Z, Σt1 = {p ∈ V : t (p) = t1} , and Σ1 ∩ Σ2 = ∅ ⇐⇒ t1 ̸= t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (34) Branchial graphs constitute a discrete/combinatorial representation of these abstract branchlike hypersur- faces, in which the common ancestry distance between state vertices is represented for any given value of the universal time function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' vertices X and Y in the branchial graph for a given time step are connected by an undirected edge in the branchial graph if and only if they share a common ancestor state Z in the multiway evolution graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The default choice of foliation for the multiway evolution graph corresponding to the evolution of the non-deterministic 2-state, 2-color Turing machine discussed above is shown in Figure 8, with the associated sequence of branchial graphs shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We can therefore think of every multiway system as being equipped with a certain tensor product structure, in which certain pairs of states occur in parallel (as defined by the simultaneity surfaces of the universal time function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the branchlike hypersurfaces), and are therefore considered to be “tensored” together, such that the entire multiway system can be decomposed into a tensor product of single branches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' of individual deterministic/singleway com- 17 putations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The branchial graphs thus provide a combinatorial description of the tensor product structure of the multiway computation at each time step, and the overall non-deterministic/multiway computation can be recast as a deterministic/singleway computation over these tensor products/branchial graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 8: The default “foliation” of the multiway evolution graph for the non-deterministic evolution of the 2-state, 2-color Turing machine constructed from parallel composition of the transition functions for rules 2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 4 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 9: The corresponding sequence of “branchlike hypersurfaces” associated to the default “foliation” of the multiway evolution graph for the non-deterministic evolution of the 2-state, 2-color Turing machine constructed from parallel composition of the transition functions for rules 2506 and 3506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We can make this intuition mathematically rigorous by noting that our category T of Turing machine states and transitions is now (in the non-deterministic/multiway case) a monoidal category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a category equipped with a tensor product structure, as proved for the case of generic multiway systems based on symbolic rewriting in [26] and [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' concretely, this entails that T is really an ordered triple ⟨T, ⊗, I⟩ consisting of an underlying category, a tensor product operation ⊗ and a distinguished unit object I ∈ ob (T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The tensor product operation is encoded as a bifunctor of the form: ⊗ : T × T → T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (35) 18 in other words a functor whose domain is the product category T × T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' whose object set is given by ordered pairs of objects in T : ob (T × T ) = {(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y ) |X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y ∈ ob (T )} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (36) whose morphism set is given by ordered pairs of morphisms in T : hom (T × T ) = {((f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' g) : (X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' X2) → (Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y2)) |(f : X1 → Y1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (g : X2 → Y2) ∈ hom (T )} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (37) and in which composition and identity are defined component-wise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀ ((f1, g1) : (X1, X2) → (Y1, Y2)) , ((f2, g2) : (Y1, Y2) → (Z1, Z2)) ∈ hom (T × T ) , ((f2, g2) ◦ (f1, g1) : (X1, X2) → (Z1, Z2)) = ((f2 ◦ f1, g2 ◦ g1) : (X1, X2) → (Z1, Z2)) , (38) and: ∀ (X, Y ) ∈ ob (T × T ) , � id(X,Y ) : (X, Y ) → (X, Y ) � = ((idX, idY ) : (X, Y ) → (X, Y )) , (39) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In order to encode the fact that the tensor product operation should be (weakly) associa- tive, there should exist a natural isomorphism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a natural transformation whose components are all isomorphisms) α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' which we call the associator[28][29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' of the general form: α : (−) ⊗ ((−) ⊗ (−)) ∼= ((−) ⊗ (−)) ⊗ (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (40) with components: ∀X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' αX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z : X ⊗ (Y ⊗ Z) ∼= (X ⊗ Y ) ⊗ Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (41) such that the following diagram (the associator coherence) commutes for all X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' W ∈ ob (T ): 19 X ⊗ (Y ⊗ (Z ⊗ W)) (X ⊗ Y ) ⊗ (Z ⊗ W) ((X ⊗ Y ) ⊗ Z) ⊗ W X ⊗ ((Y ⊗ Z) ⊗ W) (X ⊗ (Y ⊗ Z)) ⊗ W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' αX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z⊗W idX⊗αY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='W αX⊗Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='W αX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y ⊗Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='W αX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z⊗idZ (42) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X, Y, Z, W ∈ ob (T ) , αX,Y,Z ⊗ idZ ◦ (αX,Y ⊗Z,W ◦ idX ⊗ αY,Z,W ) = αX⊗Y,Z,W ◦ αX,Y,Z⊗W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (43) Moreover, in order to encode the fact that the tensor product is (weakly) unital, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' that the distinguished unit object I acts as both a left and right identity for ⊗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' there should exist a further pair of natural isomorphisms λ and ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' which we call the left and right unitor isomorphisms respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' of the general form: λ : I ⊗ (−) ∼= (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and ρ : (−) ⊗ I ∼= (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (44) with components: ∀X ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' λX : I ⊗ X ∼= X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and ρX : X ⊗ I ∼= X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (45) such that the following diagram (the unitor coherence) commutes for all X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y ∈ ob (T ): X ⊗ (I ⊗ Y ) (X ⊗ I) ⊗ Y X ⊗ Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' αX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y idX⊗λY ρX⊗idY (46) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X, Y ∈ ob (T ) , ρX ⊗ idY ◦ αX,I,Y = idX ⊗ λY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (47) Our monoidal category ⟨T , ⊗, I⟩ inherits the associativity and unitality of its tensor product structure ⊗ from the associativity and unitality of the disjoint union operation ⊔ (with the halt state HALT of the Turing machine playing the role of the unit object I, since, by definition, parallel composition with the halt state does not substantively modify the structure of any multiway computation because the halt state 20 does not evolve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' since the disjoint union operation is also commutative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' it follows that our monoidal category is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' symmetric[30][31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in the sense that it is also equipped with an additional natural isomorphism σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' called the symmetry or braiding isomorphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' of the general form: σ : (−) ⊗ (−) ∼= (−) ⊗ (−) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (48) with components: ∀X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' σX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y : X ⊗ Y ∼= Y ⊗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (49) such that the symmetry/braiding isomorphism is compatible with the associator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' meaning that the following diagram (essentially an additional associator coherence condition) commutes for all X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z ∈ ob (T ): (X ⊗ Y ) ⊗ Z (Y ⊗ X) ⊗ Z X ⊗ (Y ⊗ Z) Y ⊗ (X ⊗ Z) (Y ⊗ Z) ⊗ X Y ⊗ (Z ⊗ X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' σX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y ⊗idZ α−1 X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z α−1 Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z σX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y ⊗Z idY ⊗σX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z α−1 Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X (50) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X, Y, Z ∈ ob (T ) , idY ⊗ σX,Z ◦ � α−1 Y,X,Z ◦ σX,Y ⊗ idZ � = α−1 Y,Z,X ◦ � σX,Y ⊗Z ◦ α−1 X,Y,Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (51) Additionally, the symmetry/braiding isomorphism should be compatible with the left and right unitor iso- morphisms, meaning that the following diagram (an additional unitor coherence condition) commutes for all X ∈ ob (T ): X ⊗ I I ⊗ X X , σX,I ρX λX (52) 21 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X ∈ ob (T ) , λX ◦ σX,I = ρX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (53) Finally, the symmetry/braiding isomorphism should be involutive/self-inverse, meaning that the following diagram commutes for all X, Y ∈ ob (T ): Y ⊗ X X ⊗ Y X ⊗ Y, σY,X idX⊗Y σX,Y (54) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X, Y ∈ ob (T ) , σY,X ◦ σX,Y = idX⊗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (55) Note that, if we relax the last condition (namely that the natural isomorphism σ should be involutive/self- inverse), then we obtain not a symmetric monoidal category but the weaker notion of a braided monoidal category[32][33], in which the action of σ on an n-fold tensor product factors not through the symmetric group but through the braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For the case of braided monoidal categories, we must impose one further associator coherence condition, wherein the following diagram commutes for all X, Y, Z ∈ ob (T ): X ⊗ (Y ⊗ Z) X ⊗ (Z ⊗ Y ) (X ⊗ Y ) ⊗ Z (X ⊗ Z) ⊗ Y Z ⊗ (X ⊗ Y ) (Z ⊗ X) ⊗ Y, idX⊗σY,Z αX,Y,Z αX,Z,Y σX⊗Y,Z σX,Z⊗idY αZ,X,Y (56) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X, Y, Z ∈ ob (T ) , σX,Z ⊗ idY ◦ (αX,Z,Y ◦ idX ⊗ σY,Z) = αZ,X,Y ◦ (σX⊗Y,Z ◦ αX,Y,Z) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (57) however, this coherence condition is unnecessary for the case of symmetric monoidal categories, since it can be derived from a combination of the first associator coherence law and the involutive/self-inverse property 22 of the symmetry/braiding isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' As before, we can now consider equipping the morphisms of the category T with additional semantic metadata specifying the computational complexities of the corresponding computations that the morphisms signify symbolically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' However, since we are now able to compose computations not only in sequence (using the standard composition operator ◦) but also in parallel (using the tensor product operation ⊗), the algebra describing how these complexities compose is now somewhat more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' As previously described, (singleway) computational irreducibility is characterized by additivity of sequential composition ◦, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' if morphism f : X → Y requires at least n computational steps to enact and morphism g : Y → Z requires at least m computational steps to enact, then their sequential composition g ◦ f : X → Z requires at least n + m computational steps to enact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' On the other hand, due to the presence of the tensor product structure, we can now characterize a form of multicomputational irreducibility in terms of additivity of parallel composition ⊗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' if morphism f : X1 → Y1 requires at least n computational steps to enact and morphism g : X2 → Y2 requires at least m computational steps to enact, then their parallel composition f ⊗ g : X1 ⊗ X2 → Y1 ⊗ Y2 (acting on the branchial graph consisting initially of states X1 and X2) is multicomputationally irreducible if and only if it cannot be enacted in fewer than n + m computational steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Likewise, if the complexi- ties of computations behave subadditively under parallel composition/tensor product, then the composite (multiway) computation is multicomputationally reducible (with the degree of subadditivity quantifying the degree of multicomputational reducibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The intuition behind this characterization is that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' much as “or- dinary” computational irreducibility is intended to describe the case in which the outcome of a deterministic computation cannot be preempted without explicitly tracing out all intermediate steps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' multicomputational irreducibility is intended to describe the case in which the outcome of a non-deterministic/multiway compu- tation cannot be preempted without explicitly tracing out all parallel deterministic/singleway computations of which it is the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This intuition is illustrated in Figure 10 for the case of the non-deterministic 2-state, 2-color Turing machine discussed above, with each edge/morphism tagged with the number of tran- sitions necessary to perform the corresponding computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' neither the sequential composition operation ◦ nor the parallel tensor product operation ⊗ is purely additive, indicating that the system is, at least in part, multicomputationally reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We can consequently extend our previous formalization of computational irreducibility in terms of the functoriality of the map Z′ : T → B to deal with the new case of multicomputational irreducibility too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' If we suppose, in the first instance, that all singleway computation paths through the multiway system for the non-deterministic Turing machine T are computationally irreducible, then Z′ is a functor (as proved above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 23 Figure 10: A graph-theoretic representation of the category that is freely generated by the multiway evolution graph corresponding to the non-deterministic evolution of a 2-state, 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506 (considered as a quiver), with edges/morphisms tagged with additional metadata corresponding to the number of “steps” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' transitions) required to perform the requisite computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Furthermore, the (discrete) cobordism category B of step numbers/moments of time and (discrete) intervals between them can trivially be equipped with a commutative tensor product structure (namely the disjoint union of sets and intervals ⊔) with a unit object given by the empty set ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' recall that, for more abstract and general cobordism categories, the coproduct ⊕ plays the role of the disjoint union ⊔ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the direct sum of manifolds), with the initial object ∅ playing the role of the empty set/empty manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The objects in the symmetric monoidal category ⟨B, ⊕, ∅⟩ therefore represent not merely the time steps associated to individual computational states (as in the singleway case considered within the previous section), but parallel compo- sitions of time steps associated to multiple states on the same “branchlike hypersurface”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we can, as such, think of the category B as providing a “coordinatization” of the time-ordered sequence of branchial graphs computed by the multiway system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Thus, both ⟨T , ⊗, I = HALT⟩, and ⟨B, ⊕, ∅⟩ are symmetric monoidal categories, and therefore, subject to the additional hypothesis that all parallel compositions of singleway computations are multicomputationally irreducible (and therefore all computational complexities behave purely additively under tensor products), the map Z′ : T → B forms a symmetric monoidal functor[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' More precisely, Z′ is a monoidal functor in the sense that it is a functor between monoidal categories: Z′ : ⟨T , ⊗, I = HALT⟩ → ⟨B, ⊕, ∅⟩ , (58) that preserves the tensor product structure, meaning concretely that Z′ is equipped with a morphism 24 20(ε : ∅ → Z′ (I)) ∈ hom (B), along with a natural transformation µ between functors from T × T to B, with components: ∀X, Y ∈ ob (T ) , µX,Y : Z′ (X) ⊕ Z′ (Y ) → Z′ (X ⊗ Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (59) Together, ε and µ are known as the coherence maps of the monoidal functor Z′, and satisfy coherence conditions with the associators αT , αB, left unitor isomorphisms λT , λB and right unitor isomorphisms ρT , ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The associator coherence condition is represented by the assertion that the following diagram commutes for all X, Y, Z ∈ ob (T ): (Z′ (X) ⊕ Z′ (Y )) ⊕ Z′ (Z) Z′ (X) ⊕ (Z′ (Y ) ⊕ Z′ (Z)) Z′ (X ⊗ Y ) ⊕ Z′ (Z) Z′ (X) ⊕ Z′ (Y ⊗ Z) Z′ ((X ⊗ Y ) ⊗ Z) Z′ (X ⊗ (Y ⊗ Z)) , αB Z′(X),Z′(Y ),Z′(Z) µX,Y ⊕idB Z′(Z) idB Z′(Z) µX⊗Y,Z µX,Y ⊗Z Z′(αT X,Y,Z) (60) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (recalling that the composition operation in the cobordism category B is given by the ordinary union of contiguous intervals/cobordisms ∪): ∀X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' µX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y ⊗Z ∪ � idZ′(Z) ∪ αB Z′(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z′(Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z′(Z) � = Z′ � αT X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z � ∪ � µX⊗Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z ∪ µX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y ⊕ idB Z′(Z) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (61) while the left and right unitor coherence conditions are represented by the assertion that the following diagrams commute for all X ∈ ob (T ): 25 Z′ (X) ⊕ ∅ Z′ (X) ⊕ Z′ (I) Z′ (X) Z′ (X ⊗ I) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' idB Z′(X)⊕ε ρB Z′(X) µX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='I Z′(ρT X) (62) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X ∈ ob (T ) , Z′ � ρT X � ∪ � µX,I ∪ idB Z′(X) ⊕ ε � = ρB Z′(X), (63) and: ∅ ⊕ Z′ (X) Z′ (I) ⊕ Z′ (X) Z′ (X) Z′ (I ⊗ X) , ϵ⊕idB Z′(X) λB Z′(X) µI,X Z′(λT X) (64) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X ∈ ob (T ) , Z′ � λT X � ∪ � µI,X ∪ ϵ ⊕ idB Z′(X) � = λB Z′(X), (65) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note that the definition presented here is for a lax monoidal functor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' if the coherence maps ε and µX,Y were, additionally, either isomorphisms or identities for all X, Y ∈ ob (T ), then one would obtain a strong or a strict monoidal functor, respectively, instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The monoidal functor Z′ : ⟨T , ⊗, I⟩ → ⟨B, ⊕, ∅⟩ is also symmetric, in the sense that the coherence maps ε and µ also satisfy a further coherence condition with the braiding/symmetry isomorphisms σT , σB, represented by the assertion that the following diagram commutes for all X, Y ∈ ob (T ): Z′ (X) ⊕ Z′ (Y ) Z′ (X) ⊕ Z′ (Y ) Z′ (X ⊗ Y ) Z′ (Y ⊗ X) , σB Z′(X),Z′(Y ) µX,Y µY,X Z′(σT X,Y ) (66) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: 26 ∀X, Y ∈ ob (T ) , µY,X ∪ σB Z′(X),Z′(Y ) = Z′ � σT X,Y � ∪ µX,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (67) If ⟨T , ⊗, I⟩ and ⟨B, ⊕, ∅⟩ were merely braided monoidal rather than symmetric monoidal categories, then a monoidal functor Z′ satisfying the above coherence condition would instead be a braided monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Thus, if X1, Y1, X2 and Y2 represent Turing machine states reached at step numbers t1, t2, s1 and s2 respectively, and f : X1 → Y1 and g : X2 → Y2 represent the transitions between states X1 and Y1 and states X2 and Y2 respectively, then the (symmetric) monoidal functor Z′: X1 X2 t1 s1 Y1 Y2 t2 s2, f g [t1,t2]∩N [s1,s2]∩N Z′ (68) preserves the (symmetric) tensor product structure in the sense depicted by the following diagram: X1 ⊗ X2 t1 ⊕ s1 Y1 ⊗ Y2 t2 ⊕ s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' f⊗g ([t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t2]∩N)⊕([s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='s2]∩N) Z′ (69) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' by equipping B with a tensor product structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the resulting symmetric monoidal category ⟨B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ⊕,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∅⟩ is actually a higher-dimensional cobordism category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in which the objects are potentially higher-dimensional manifolds (consisting of direct sums/disjoint unions of several 0-dimensional manifolds) and the morphisms are potentially higher-dimensional cobordisms (consisting of direct sums/disjoint unions of several 1-dimensional cobordisms),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' since there now exist two distinct directions in which cobordisms can be “glued” (either se- quentially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' via the standard union of contiguous intervals ∪,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' or in parallel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' via the tensor product/direct sum of intervals ⊔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For any given morphism in B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the cardinalities of the various individual (1-dimensional) intervals of which it is composed represent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' up to an additive constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the time complexities of the cor- responding deterministic/singleway computations in T (as in the singleway case analyzed in the previous section),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' whereas the number of distinct (1-dimensional) intervals appearing in the tensor product/direct sum within that morphism represents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' up to an additive constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the time complexity of the resulting composite non-deterministic/multiway computation in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' As a consequence, we can conclude that, just as (singleway) computational irreducibility is reflected in the functoriality of Z′, multicomputational irreducibility is re- flected in the symmetric monoidal functoriality of Z′, and just as computational reducibility corresponds 27 precisely to a deformation of Z′ away from a pure functor, multicomputational reducibility corresponds to a deformation of Z′ from being symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Another, perhaps cleaner, way to articulate this would be to say that computational reducibility corresponds to how much the map Z′ distorts the sequential composition (◦) of computations in T , while multicomputational reducibility corresponds to how much the map Z′ distorts the parallel composition (⊗) of computations in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This construction is demonstrated in Figure 11 for the case of the non-deterministic 2-state, 2-color Turing machine discussed previously, with each vertex/object tagged with its step number and each edge/morphism tagged with a list of step numbers traversed throughout the course of its corresponding computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note that the same algebraic axioms that describe how the time complexities attached to morphisms compose (namely identity, strict positivity and subadditivity/triangle inequality) in the sequential case under ◦ also hold in the parallel case under ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 11: A graph-theoretic representation of the category that is freely generated by the multiway evolution graph corresponding to the non-deterministic evolution of a 2-state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506 (considered as a quiver),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with vertices/objects tagged with additional metadata corresponding to the step number on which they occur (shown in blue) and with edges/morphisms tagged with additional metadata corresponding to all intermediate step numbers traversed as part of the requisite computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' From this rather abstract description, it should be apparent that computational irreducibility and mul- ticomputational irreducibility are essentially orthogonal concepts: the map Z′ can distort sequential com- positions to an arbitrary extent whilst keeping the tensor product structure entirely intact, or vice versa (or anything in between).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This is a fairly intuitive consequence of the fact that computational irreducibility is a byproduct of the state evolution function of a given multiway system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the function that speci- fies which states are obtainable in a single step from which other states, and which therefore provides the rules for constructing morphisms in T ), while multicomputational irreducibility is a byproduct of the state equivalence function of a given multiway system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the function that specifies which pairs of states are 28 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 2] 1, 2,3, 4 [1,2,3] {1, 2, 3, 4 [1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 3] 51, 2,3, 4) [1, 2, 3, 4] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=',2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 4) 2,3 [2, 3] [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 4] [2, 3,4 2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 4 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='4]to be considered equivalent, and which therefore provides the rules for merging/equating objects in T ), and the evolution and equivalence functions have entirely independent definitions: one can define a state evolution function with an arbitrarily high computational complexity whilst keeping the state equivalence function essentially trivial, or vice versa (or, once again, any reasonable intermediate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For instance, in the case of non-deterministic/multiway Turing machines considered thus far, the state equivalence function is elementary (two Turing machine states are considered equivalent if their tape states, head states and head positions are both identical, which is trivial to determine algorithmically), with all of the computational complexity originating from the state evolution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' On the other hand, in the case of (hyper)graph rewriting, as considered in the context of the Wolfram model[22][23][24], the state equivalence function is much more sophisticated, since it must account for (hyper)graph isomorphism, whose precise complexity class GI remains unknown (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' it is not known whether GI is P, NP-complete or NP-intermediate)[35][36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In the analysis that follows, we shall be using a generalized version of the “uniqueness tree” isomorphism algorithm presented in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' By defining a (directed/ordered) hypergraph H = ⟨V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' E⟩ in terms of a finite collection/multiset of ordered relations (hyperedges) between elements: E ⊆ P (V ) \\ {∅} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (70) where P denotes the power set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we can formalize the notion of hypergraph rewriting rules H1 → H2 in terms of a span of monomorphisms[38][39] of the form: L K R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' l r (71) in some category C (whose objects are hypergraphs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and whose morphisms represent subhypergraph inclusion maps),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' where L is a hypergraph pattern designating the left-hand side of the rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' R is a hypergraph pattern designating the right-hand side of the rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and K is a pattern designating the subhypergraph that remains invariant when the left-hand side is “extracted” and the right-hand side is “injected”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (Note that hypergraph categories, in the terminology of Kissinger[40] and Fong[41][42], namely symmetric monoidal categories equipped with a Frobenius algebra structure which ensures that all string diagrams correspond to hypergraphs, constitute a particularly natural categorical setting in which to construct such a rewriting system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In the above, the morphisms l : K → L and r : K → R form a span in the sense that they constitute a pair of morphisms with a common domain, and they are monomorphisms in the sense that they are 29 injective/left-cancellative: for any pairs of morphisms f1 : X → K and f2 : X → K that make either of the following diagrams commute: ∀X K L ∀f1 ∀f2 l , or ∀X K R, ∀f1 ∀f2 r (72) one necessarily has (f1 : X → K) = (f2 : X → K), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X ∈ ob (C) , ∀ (f1 : X → K) , (f2 : X → K) ∈ hom (C) , (l ◦ f1 : X → L) = (l ◦ f2 : X → L) =⇒ (f1 : X → K) = (f2 : X → K) , (73) and: ∀X ∈ ob (C) , ∀ (f1 : X → K) , (f2 : X → K) ∈ hom (C) , (r ◦ f1 : X → R) = (r ◦ f2 : X → R) =⇒ (f1 : X → K) = (f2 : X → K) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (74) A graphical representation of such a hypergraph rewriting rule with relations/hyperedges of arity-2 (corre- sponding to the set substitution rule {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}} - since any hyper- graph rewriting rule of this form can always be reformulated as a symbolic set substitution rule acting on multisets of ordered relations between vertices) is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 12: A graphical representation of the (hyper)graph transformation rule corresponding to the set substitution system {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' A hypergraph rewriting rule of the form presented above can then be said to match a given hypergraph G if there exists a morphism (m : L → G) ∈ hom (C), and the resulting hypergraph H obtained by applying the rewriting rule at that match can be computed by means of the following double-pushout diagram[43]: 30 L K R G D H ∀G∗ ∀H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' m ∀m∗ l r n p ∀p∗ ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u1 g ∀h∗ ∀g∗ h ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u2 (75) More concretely, D ∈ ob (C) is a hypergraph and (n : K → D) , (g : D → G) ∈ hom (C) are hypergraph in- clusions such that the leftmost square commutes: (g ◦ n : K → G) = (m ◦ l : K → G) , (76) and are universal in the sense that, for any hypergraph G∗ ∈ ob (C) equipped with inclusions: (m∗ : L → G∗) , (g∗ : D → G∗) ∈ hom (C) , (77) there exists a unique inclusion (u1 : G → G∗) ∈ hom (C) such that: (m∗ : L → G∗) = (u1 ◦ m : L → G∗) , and (g∗ : D → G∗) = (u1 ◦ g : D → G∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (78) This allows one to compute the “residual” hypergraph D obtained by extracting out a subhypergraph isomorphic to L from G at the position defined by the rule match m : L → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Moreover, D, H ∈ ob (C) are hypergraphs and (p : R → H) , (h : D → H) ∈ hom (C) are inclusions such that the rightmost square commutes: (h ◦ n : K → H) = (p ◦ r : K → H) , (79) and are universal in the sense that, for any hypergraph H∗ ∈ ob (C) equipped with inclusions: (p∗ : R → H∗) , (h∗D → H∗) ∈ hom (C) , (80) there exists a unique inclusion (u2 : H → H∗) ∈ hom (C) such that: (p∗ : R → H∗) = (u2 ◦ p : R → H∗) , and (h∗ : D → H∗) = (u2 ◦ h : D → H∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (81) This, in turn, allows one to compute the resulting hypergraph H obtained by gluing a subhypergraph 31 isomorphic to R into the residual hypergraph D at the position defined by the injection map m : K → D (from the invariant subhypergraph to the residual hypergraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This double-pushout construction yields the class of possible hypergraph transitions from which we are able to construct a multiway evolution graph (whose vertices represent hypergraphs and whose edges represent hypergraph rewrites),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' by a process that is directly analogous to the Turing machine case considered previously: an explicit example for the hypergraph rewriting rule presented above is shown in Figure 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and the category that is freely generated by this multiway evolution graph (considered as a quiver) is shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 15 shows each morphism/edge tagged with the number of hypergraph transitions necessary to perform the corresponding computation, illustrating that the composition ◦ and tensor product ⊗ operations are not purely additive (although they are both somewhat close), thus indicating that the system is largely (multi)computationally irreducible, but not entirely so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The construction with each vertex/object tagged with its step number and each edge/morphism tagged with a list of step numbers traversed throughout the course of its corresponding computation is shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' As expected, we see that the hypergraph rewriting system is more computationally reducible than multicomputationally so (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the composition operation ◦ is distorted more by the map Z′ than the tensor product operation ⊗): a consequence of the fact that the state equivalence function (based on hypergraph isomorphism) is more computationally complex than it was for the non- deterministic Turing machine case previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Figure 13: A multiway evolution graph corresponding to the non-deterministic evolution of the set substi- tution system {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}}, starting from a “double self-loop” initial condition, for 3 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' each edge represents a single application of the corresponding (hyper)graph rewriting rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The usual setting in which double-pushout rewriting takes place is within an adhesive category[44], and thus adhesivity is the usual condition that one would impose on the category C described above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' loosely speaking, adhesivity provides sufficient conditions for pushouts to be “glued” along monomorphisms in the necessary manner for double-pushout diagrams to be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' More formally, a category is adhesive if it has pullbacks, and all pushouts along monomorphisms satisfy the van-Kampen square condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Having 32 Figure 14: A graph-theoretic representation of the category that is freely generated by the multi- way evolution graph corresponding to the non-deterministic evolution of the set substitution system {{x, y} , {x, z}} → {{x, z} , {x, w} , {y, w} , {z, w}} (considered as a quiver), starting from a “double self- loop” initial condition, for 3 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' pullbacks simply entails that, for some cospan (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a pair of morphisms with a common codomain): Y X Z f g (82) in C, there exists an object P ∈ ob (C) and a pair of morphisms (f ′ : P → Z) , (g′ : P → Y ) ∈ hom (C) such that the following square commutes: P Z Y X, f ′ g′ g f (83) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: (f ◦ g′ : P → X) = (g ◦ f ′ : P → X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (84) and that are universal in the sense that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for any object P ∗ ∈ ob (C) equipped with morphisms: (f ∗ : P ∗ → Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (g∗ : P ∗ → Y ) ∈ hom (C) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (85) there exists a unique morphism (u : P ∗ → P) ∈ hom (C) such that: (f ∗ : P ∗ → Z) = (f ′ ◦ u : P ∗ → Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and (g∗ : P ∗ → Y ) = (g′ ◦ u : P ∗ → Y ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (86) 33 区Figure 15: A graph-theoretic representation of the category that is freely generated by the multi- way evolution graph corresponding to the non-deterministic evolution of the set substitution system {{x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' y} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' z}} → {{x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' z} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' w} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' w} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' w}} (considered as a quiver),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with edges/morphisms tagged with additional metadata corresponding to the number of “steps” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' rewriting rule applications) required to perform the requisite computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the following diagram commutes: ∀P ∗ P Z Y X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∀f ∗ ∀g∗ ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u f ′ g′ g f (87) Though the condition of having pullbacks along cospans is relatively straightforward to state and understand, the van-Kampen square condition on pushouts along monomorphisms is, on the other hand, far more opaque and technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' It asserts that, if the object W ∈ ob (C) and the morphisms (f ′ : Y → W) , (g′ : Z → W) ∈ hom (C) in the following diagram: X Z Y W ∀W ∗, f g g′ ∀g∗ f ′ ∀f ∗ ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u (88) constitute a pushout of the span (f : X → Z) , (g : X → Y ) ∈ hom (C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: (f ′ ◦ g : X → W) = (g′ ◦ f : X → W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (89) 34 3 2 3 3 3 3 3 33 3 3 2 2 2 N 2222 2 2 2 2 2 2 11 11111 区 0 0X00X 000000Figure 16: A graph-theoretic representation of the category that is freely generated by the multi- way evolution graph corresponding to the non-deterministic evolution of the set substitution system {{x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' y} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' z}} → {{x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' z} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' w} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' w} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' {z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' w}} (considered as a quiver),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with vertices/objects tagged with additional metadata corresponding to the step number on which they occur (shown in blue) and with edges/morphisms tagged with additional metadata corresponding to all intermediate step numbers traversed as part of the requisite computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and: ∀W ∗ ∈ ob (C) , ∀ (f ∗ : Y → W ∗) , (g∗ : Z → W ∗) ∈ hom (C) , such that (f ∗ ◦ g : X → W ∗) = (g∗ ◦ f : X → W ∗) , ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (u : W → W ∗) ∈ hom (C) , (90) such that: (f ∗ : Y → W ∗) = (u ◦ f ′ : Y → W ∗) , and (g∗ : Z → W ∗) = (u ◦ g′ : Z → W ∗) , (91) then that pushout is a van-Kampen square if and only if, for every commutative cube of the form: X′ Z′ X Z Y W Y ′ W ′, fh gh hx hZ g′ h f g g′ f ′ hY f ′ h hW (92) 35 [2] [1444412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='3] [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='3] [1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='3] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='31 3 [2(2222:4] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='47 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 42424243 0 [344] 4 4 A 区 区 X (4 4/4/444 4] [4] [4] [4]4] [4][4] [4][4] [4] [4] [4]for which the top and left faces are pullback squares, in other words if and only if the object X′ ∈ ob (C) together with the morphisms (hX : X′ → X) , (gh : X′ → Y ′) ∈ hom (C), and the object X′ ∈ ob (C) together with the morphisms (fh : X′ → Z′) , (hX : X′ → X) ∈ hom (C) in the following pair of diagrams: ∀X∗ Y ′ X′ Y X , ∀g∗ h ∀h∗ X ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u hY gh hX g and ∀X∗ X′ Z′ X Z, ∀f ∗ h ∀h∗ X ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u fh hX hZ f (93) constitute pullbacks of the cospans (g : X → Y ) , (hY : Y ′ → Y ) ∈ hom (C) and (f : X → Z) , (hZ : Z′ → Z) ∈ hom (C), respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: (g ◦ hX : X′ → Y ) = (hY ◦ gh : X′ → Y ) , and (f ◦ hX : X′ → Z) = (hZ ◦ fh : X′ → Z) , (94) and, moreover: ∀X′ ∈ ob (C) , ∀ (g∗ h : X∗ → Y ′) , (h∗ X : X∗ → X) ∈ hom (C) , such that (g ◦ h∗ X : X∗ → Y ) = (hY ◦ g∗ h : X∗ → Y ) , ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (u : X∗ → X′) ∈ hom (C) , (95) such that: (h∗ X : X∗ → X) = (hX ◦ u : X∗ → X) , and (g∗ h : X∗ → Y ′) = (gh ◦ u : X∗ → Y ′) , (96) for the case of the first (leftmost) diagram, and: ∀X′ ∈ ob (C) , ∀ (f ∗ h : X∗ → Z′) , (h∗ X : X∗ → X) ∈ hom (C) , such that (f ◦ h∗ X : X∗ → Z) = (hZ ◦ f ∗ h : X∗ → Z) , ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (u : X∗ → X′) ∈ hom (C) , (97) 36 such that: (h∗ X : X∗ → X) = (hX ◦ u : X∗ → X) , and (f ∗ h : X∗ → Z′) = (fh ◦ u : X∗ → Z′) , (98) for the case of the second (rightmost) diagram, then a certain compatibility condition is satisfied between the pushout and pullback squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In detail, this compatibility condition states that the rear face is a pushout square, in other words the object W ′ ∈ ob (C) and the morphisms (f ′ h : Y ′ → W ′) , (g′ h : Z′ → W ′) ∈ hom (C) in the following diagram: X′ Z′ Y ′ W ′ ∀W ∗, fh gh g′ h ∀g∗ h f ′ h ∀f ∗ h ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u (99) constitute a pushout of the span (fh : X′ → Z′) , (gh : X′ → Y ′) ∈ hom (C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: (f ′ h ◦ gh : X′ → W ′) = (g′ h ◦ fh : X′ → W ′) , (100) and: ∀W ∗ ∈ ob (C) , ∀ (f ∗ h : Y ′ → W ∗) , (g∗ h : Z′ → W ∗) ∈ hom (C) , such that (f ∗ h ◦ gh : X′ → W ∗) = (g∗ h ◦ fh : X′ → W ∗) , ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (u : W ′ → W ∗) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (101) such that: (f ∗ h : Y ′ → W ∗) = (u ◦ f ′ h :′→ W ∗) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and (g∗ h : Z′ → W ∗) = (u ◦ g′ h : Z′ → W ∗) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (102) if and only if the bottom and right faces are pullback squares,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in other words if and only if the ob- ject Y ′ ∈ ob (C) together with the morphisms (f ′ h : Y ′ → W ′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (hY : Y ′ → Y ) ∈ hom (C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and the object Z′ ∈ ob (C) together with the morphisms (g′ h : Z′ → W ′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (hZ : Z′ → Z) ∈ hom (C) in the following pair of 37 diagrams: Y W Y ′ W ′ ∀Y ∗ f ′ hY f ′ h hW ∀h∗ Y ∀f ∗ h ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u and W Z W ′ Z′ ∀Z∗, g′ hW hZ g′ h ∀g∗ h ∀h∗ Z ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='u (103) constitute pullbacks of the cospans (f ′ : Y → W) , (hW : W ′ → W) ∈ hom (C) and (g′ : Z → W) , (hW : W ′ → W) ∈ hom (C), respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: (hW ◦ f ′ h : Y ′ → W) = (f ′ ◦ hY : Y ′ → W) , and (hW ◦ g′ h : Z′ → W) = (g′ ◦ hZ : Z′ → W) , (104) and, moreover: ∀Y ∗ ∈ ob (C) , ∀ (f ∗ h : Y ∗ → W ′) , (h∗ Y : Y ∗ → Y ) ∈ hom (C) , such that (hW ◦ f ∗ h : Y ∗ → W) = (f ′ ◦ h∗ Y : Y ∗ → W) , ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (u : Y ∗ → Y ′) ∈ hom (C) , (105) such that: (f ∗ h : Y ∗ → W ′) = (f ′ h ◦ u : Y ∗ → W ′) , and (h∗ Y : Y ∗ → Y ) = (hY ◦ u : Y ∗ → Y ) , (106) for the case of the first (leftmost) diagram, and: ∀Z∗ ∈ ob (C) , ∀ (g∗ h : Z∗ → W ′) , (h∗ Z : Z∗ → Z) ∈ hom (C) , such that (hW ◦ g∗ h : Z∗ → W) = (g′ ◦ h∗ Z : Z∗ → W) , ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (u : Z∗ → Z′) ∈ hom (C) , (107) such that: 38 (g∗ h : Z∗ → W ′) = (g′ h ◦ u : Z∗ → W ′) , and (h∗ Z : Z∗ → Z) = (hZ ◦ u : Z∗ → Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (108) for the case of the second (rightmost) diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Although the category of hypergraphs and subhypergraph inclusion maps considered in the case of Wolfram model evolution is not strictly an adhesive category (due to the arbitrary connectivity of vertices within each hyperedge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' implying that not all pushouts along monomorphisms are guaranteed to exist),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' as noted by Kissinger[45][46][47] it constitutes a full subcategory of an adhesive category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and can be “embedded” in the ambient adhesive category in such a way as to inherit sufficient “adhesivity” to allow double-pushout rewriting to be performed (through the mechanisms of either selective adhesivity or partial adhesivity - essentially by allowing the functor that embeds the subcategory into the adhesive category to preserve monomorphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' As detailed in [48], one can construct both the ordinary composition structure and the symmetric monoidal structure of the resulting category T of hypergraphs and hypergraph rewritings in a very explicit way using a combination of the concurrency and parallelism theorems from algebraic graph transformation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Specifically, if one has a pair of hypergraph productions p1 and p2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' two spans of monomor- phisms corresponding to two hypergraph rewrites) rewriting hypergraph G to H and hypergraph H to G′ respectively (known as an E-related transformation sequence), then the concurrency theorem allows one to compose the productions to obtain an E-concurrent hypergraph production p1 ∗E p2 of the form: H G G′, p2 p1 p1∗Ep2 (109) thus giving rise to the ordinary (sequential) composition of morphisms ◦ in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' On the other hand, if one has a pair of hypergraph productions p1 and p2 yielding two different, sequentially-independent transformation sequences G to H1 to G′ and G to H2 to G′ (obtained by applying p1 then p2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' p2 then p1), then the parallelism theorem allows one to compose the productions to obtain a parallel hypergraph production p1 + p2 of the form: 39 G H1 H2 G′ , p1 p2 p1+p2 p2 p1 (110) thus giving rise to the tensor product (parallel) composition of morphisms ⊗ in T , as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 4 Correspondence with Categorical Quantum Mechanics and Func- torial Quantum Field Theory In conventional (non-relativistic) quantum mechanics, one typically associates to each moment of time t ∈ R a corresponding (Hilbert) space of states Vt for the system, and to every interval of time [t1, t2], where t1, t2 ∈ R such that t1 ≤ t2, a corresponding linear (unitary) time evolution operator ˆU (t1, t2) : Vt1 → Vt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' If the Hamiltonian ˆH (t) is a Hermitian/self-adjoint operator that depends smoothly on t ∈ R, then we can express ˆU (t1, t2) explicitly in terms of the following Dyson formula (otherwise known more generally as an iterated integral expansion for parallel transport) for the time-dependent Schr¨odinger equation: ˆU (t1, t2) = P exp � i ℏ � t2 t1 ˆH (t) dt � , (111) where P exp denotes the path-ordered exponential operator for non-commutative algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for the time- independent case in which the Hamiltonian ˆH is fixed, this simplifies to just an ordinary exponential: ∀t ∈ R, ˆH (t) = ˆH, =⇒ ˆU (t1, t2) = exp � i ℏ (t2 − t1) ˆH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (112) This explicit representation of ˆU (t1, t2) makes manifest one of the fundamental features of quantum mechan- ical time evolution: that it is necessarily local in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In other words, due to the linearity of integration, the “global” time interval [t1, t2] can always be subdivided into many “local” subintervals, in such a way that the single global time evolution is obtained by integrating up the effects of the many local time evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' More formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we have: ∀t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t3 ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' such that t1 ≤ t2 ≤ t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ˆU (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t3) = ˆU (t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t3) ◦ ˆU (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (113) 40 in other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' time evolution has a natural composition structure ◦ such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' when combined with the (mostly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' though not entirely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' innocent) condition that: ∀t ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ˆU (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t) = idVt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (114) we see that time evolution in quantum mechanics satisfies the requisite axioms of a category (with associativ- ity inherited from the associativity of products of linear operators);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' this was ultimately the insight underlying Abramsky and Coecke’s formulation of categorical quantum mechanics[49][50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Thus, if BordRiem 1 refers to the 1-dimensional category of (Riemannian) manifolds and their cobordisms, and Vect refers to the category of vector spaces and their linear isomorphisms, then the statement of locality of time evolution in quantum mechanics simply becomes a statement that the map: Z : BordRiem 1 → Vect, (115) is a functor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (at least in this context) locality is functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This functoriality between the categories BordRiem 1 and Vect can be illustrated diagrammatically as follows: t2 Vt2 t1 t3 Vt1 Vt3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' [t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3] ˆU(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3) [t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t2] [t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3]∪[t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t2] =[t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3] ˆU(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t2) ˆU(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3)◦ ˆU(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t2) = ˆU(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3) Z (116) If we think of the (Hilbert) spaces of states Vt as being data structures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and the unitary time evolution operators ˆU (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' t2) as being elementary computations (as is the case in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' quantum information theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' where they represent the actions of compositions of quantum gates),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' then this looks structurally very similar to the continuous case of the functor Z′ : T → B from a category of data structures and computations to a category of manifolds and cobordisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' namely: Y t2 X Z t1 t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' g [t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3] f g◦f [t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t2] [t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3]∪[t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t2] =[t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='t3] Z′ (117) 41 considered in the preceding sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' albeit with the domain and the codomain categories swapped around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In the most general case of a functorial quantum mechanics theory, defined by some functor from cobor- disms to computations Z : B → T , although it is not necessarily the case that Z will always be a strict inverse of Z′ : T → B, the two functors will at least be adjoint to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the statement that Z′ : T → B is left adjoint to Z : B → T corresponds to the assertion that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for every object (manifold) X ∈ ob (B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' there exists a universal morphism (cobordism) (εX : Z′ (Z (X)) → X) ∈ hom (B) from Z′ to X for some object (data structure) Z (X) ∈ ob (T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' where the universality property necessitates that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for any object (data structure) Y ∈ ob (T ) and any morphism (cobordism) (f : Z′ (Y ) → X) ∈ hom (B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' there exists a unique morphism (computation) (g : Y → Z (X)) ∈ hom (T ) such that: (εX ◦ Z′ (g) : Z′ (Y ) → X) = (f : Z′ (Y ) → X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (118) This condition may be restated succinctly via the following commutative diagram: Z′ (∀Y ) Z′ (∃Z (X)) ∀X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z′(∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g) ∀f ∃εX (119) On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the statement that Z : B → T is right adjoint to Z′ : T → B corresponds to the asser- tion that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for every object (data structure) Y ∈ ob (T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' there exists a universal morphism (computation) (ηY : Y → Z (Z′ (Y ))) ∈ hom (T ) from Y to Z for some object (manifold) Z′ (Y ) ∈ ob (B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' where the uni- versality property necessitates that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for any object (manifold) X ∈ ob (B) and any morphism (computation) (g : Y → Z (X)) ∈ hom (T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' there exists a unique morphism (cobordism) (f : Z′ (Y ) → X) ∈ hom (B) such that: (Z (f) ◦ ηY : Y → Z (X)) = (g : Y → Z (X)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (120) This condition may be restated succinctly via the following commutative diagram: ∀Y Z (∃Z′ (Y )) Z (∀X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∃ηY ∀g Z(∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='f) (121) 42 It is in this rather precise sense that we are able to claim that the irreducibility of computations in computa- tional complexity theory is dual/adjoint to the locality of time evolution in categorical quantum mechanics: for any functor Z′ : T → B describing an irreducible computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we can uniquely construct a corresponding functor Z : B → T describing a local quantum time evolution such that: ∀X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' X′ ∈ ob (B) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∀ (f : X′ → X) ∈ hom (B) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (εX ◦ Z′ (Z (f)) : Z′ (Z (X′)) → X) = (f ◦ εX′ : Z′ (Z (X′)) → X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (122) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' conversely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for any functor Z : B → T describing a local quantum time evolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we can uniquely construct a corresponding functor Z′ : T → B describing an irreducible computation such that: ∀Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y ′ ∈ ob (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∀ (g : Y → Y ′) ∈ hom (T ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (Z (Z′ (g)) ◦ ηY : Y → Z (Z′ (Y ′))) = (ηY ′ ◦ g : Y → Z (Z′ (Y ′))) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (123) as described by the following pair of commutative diagrams: Z′ (Z (X′)) Z′ (Z (X)) X′ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z′(Z(f)) εX′ εX f and Y Z (Z′ (Y )) Y ′ Z (Z′ (Y ′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ηY g Z(Z′(g)) ηY ′ (124) Somewhat more cryptically, this enables us to make the claim that, in this very restricted sense, computa- tional complexity theory is dual/adjoint to (non-relativistic) quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In the non-deterministic/multicomputational case, in which categories T and B are both equipped with a (symmetric) monoidal structure, and thus in which the adjoint functors Z′ and Z are now (symmetric) monoidal functors: Z′ : ⟨T, ⊗, I⟩ → ⟨B, ⊕, ∅⟩ , and Z : ⟨B, ⊕, ∅⟩ → ⟨T , ⊗, I⟩ , (125) we obtain a higher-dimensional analog of the time evolution functor for non-relativistic quantum mechanics on the right-hand side of the adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For the example considered initially, in which the category T 43 of data structures and computations is abstractly represented as a category of vector spaces and linear isomorphisms (with the tensor product operation given by the usual tensor product of vector spaces), this functor consequently takes the form: Z : BordRiem d → Vect, (126) where d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In the context of functorial approaches to quantum field theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in topological field theories or 2-dimensional conformal field theories)[51][52], such a functor plays the role of a propagator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the Lorentz-invariant analog of the time evolution functor from non-relativistic quantum mechanics: to every codimension-1 spacelike hypersurface Md−1 ∈ ob � BordRiem d � , this functor assigns a corresponding vector space Z (Md−1) ∈ ob (Vect) designating the space of states over that hypersurface, and to every cobordism/spacetime/worldvolume M ∈ hom � BordRiem d � with boundaries ∂ (M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' : (M : ∂in (M) → ∂out (M)) ∈ hom � BordRiem d � , (127) this functor assigns a corresponding linear isomorphism: (Z (M) : Z (∂in (M)) → Z (∂out (M))) ∈ hom (Vect) , (128) designating the propagator/scattering amplitude/S-matrix for a process of shape M mapping from hyper- surface ∂in (M) to hypersurface ∂out (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Hence, just as computational irreducibility may be said to be formally dual/adjoint to locality of time evolution in quantum mechanics, multicomputational irreducibility may be said to be formally dual/adjoint to adherence to the Atiyah-Segal sewing laws[15][16][17] in func- torial quantum field theory, under which the path integral over a domain Σ which can be decomposed into subdomains Σ1 and Σ2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Σ = Σ1 ⊔ Σ2) must be equal to the path integral over subdomain Σ1 composed with the path integral over subdomain Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Conventionally, the symmetric monoidal functors Z : BordRiem 1 → Vect and Z : BordRiem d → Vect that define time evolution in category quantum mechanics and functorial quantum field theory are assumed to be strong (in the sense described previously that the coherence maps ϵ and µX,Y are isomorphisms for all X, Y ∈ ob (Vect)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' simply preserving the tensor product structure of cobordisms/vector spaces is not sufficient to define a truly physical theory of quantum mechanics or quantum fields: at the very least,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' one must also preserve the dagger structure of time evolution (which generalizes the Hermitian adjoint operation on linear transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' otherwise known as the conjugate transpose in the finite-dimensional 44 case),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' as well as the compact structure of vector spaces (which generalizes the operation of taking duals of a finite-dimensional vector space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In this setting, we say that Vect is a dagger category[53][54] to mean that it is equipped with an involutive contravariant endofunctor † : Vect → Vect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a functor from Vect to itself which has the effect of swapping the sources and targets of each morphism: ∀ (f : X → Y ) ∈ hom (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ∃ � f † : Y → X � ∈ hom (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (129) and reversing the direction of composition: ∀ (f : X → Y ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (g : Y → Z) ∈ hom (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � (g ◦ f)† : Z → X � = � f † ◦ g† : Z → X � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (130) which acts as the identity on objects: ∀X ∈ ob (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' X† = X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and � id† X : X → X � = (idX : X → X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (131) and which is involutive in the sense that it acts as its own inverse functor: ∀ (f : X → Y ) ∈ hom (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' �� f †�† : X → Y � = (f : X → Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (132) Since Vect is also a symmetric monoidal category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we would ideally like for the dagger structure † to be compatible with the tensor product structure ⊗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' meaning that: ∀ (f : X → Y ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (g : Z → W) ∈ hom (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � (f ⊗ g)† : Y ⊗ W → X ⊗ Z � = � f † ⊗ g† : Y ⊗ W → X ⊗ Z � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (133) in such a way that one maintains compatibility with the defining natural isomorphisms of the symmetric monoidal structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' namely the associator isomorphism α: 45 ∀X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Z ∈ ob (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � α† X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z) � = � α−1 X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (134) the left and right unitor isomorphisms λ: ∀X ∈ ob (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � λ† X : X → I ⊗ X � = � λ−1 X : X → I ⊗ X � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (135) and ρ: ∀X ∈ ob (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � ρ† X : X → X ⊗ I � = � ρ−1 X : X → X ⊗ I � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (136) and the symmetry/braiding isomorphism σ: ∀X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Y ∈ ob (Vect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � σ† X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y : Y ⊗ X → X ⊗ Y � = � σ−1 X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='Y : Y ⊗ X → X ⊗ Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (137) For the case of vector spaces (respectively finite-dimensional vector spaces), the Hermitian adjoint operation (respectively the conjugate transpose operation) clearly furnishes Vect/FdVect with a canonical dagger structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For the case of the cobordism categories BordRiem 1 and BordRiem d , the operation of “time reversal” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' inversion of the orientation of cobordisms) yields a compatible dagger structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Likewise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for the case of the category T of data structures and computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' obvious dagger structures exist for the cases of both Turing machine evolution and hypergraph rewriting considered previously (since for any given Turing machine rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' it is always possible to find a Turing machine rule of the same signature that reverses its evolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and for any given hypergraph rewriting system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' one can always swap the L and R objects within the span of monomorphisms that defines the rewriting rule in order to obtain a time-reversed version of the same evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we also say that FdVect (the category of finite-dimensional vector spaces) is a compact closed[55][56] symmetric monoidal category as a shorthand for saying that every object X ∈ ob (FdVect) has a corresponding dual object X∗ ∈ ob (FdVect) that is unique up to canonical isomorphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' which is equipped with a pair of morphisms ηX and εX of the form: (ηX : I → A∗ ⊗ A) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (εA : A ⊗ A∗ → I) ∈ hom (FdVect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (138) 46 known as the unit and counit morphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' satisfying the following pair of coherence conditions (sometimes known as the yanking conditions): ∀X ∈ ob (FdVect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � λX ◦ � (εX ⊗ idX) ◦ � αX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X ◦ � (idX ⊗ ηX) ◦ ρ−1 X ��� : X → X � = (idX : X → X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (139) and: ∀X ∈ ob (FdVect) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' � ρX∗ ◦ � (idX∗ ⊗ εX) ◦ � α−1 X∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X∗ ◦ � (ηX ⊗ idX∗) ◦ λ−1 X∗ ��� : X∗ → X∗� = (idX∗ : X∗ → X∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' (140) We can reformulate these two yanking conditions diagrammatically as the statement that the following pair of diagrams commute for all X ∈ ob (FdVect): X X ⊗ I X ⊗ (X∗ ⊗ X) (X ⊗ X∗) ⊗ X I ⊗ X X, ρ−1 X idX idX⊗ηX αX,X∗,X εX⊗idX λX (141) and: X∗ I ⊗ X∗ (X∗ ⊗ X) ⊗ X∗ X∗ ⊗ (X ⊗ X∗) X∗ ⊗ I X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' λ−1 X∗ idX∗ ηX⊗idX∗ α−1 X∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X∗ idX∗⊗εX ρX∗ (142) If,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' there exists a dagger structure † that is compatible with the compact structure in such a way 47 that the unit and counit morphisms η and ε can be related by means of the dagger operation (as indeed is the case for the category of finite-dimensional vector spaces and their linear isomorphisms),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in other words if the following diagram commutes for all objects X ∈ ob (FdVect): I X ⊗ X∗ X∗ ⊗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ε† X ηX σX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='X∗ (143) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e: ∀X ∈ ob (FdVect) , � σX,X∗ ◦ ε† X : I → X∗ ⊗ X � = (ηX : I → X∗ ⊗ X) , (144) then we describe the symmetric monoidal category as being dagger compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For the case of finite-dimensional vector spaces, the passage to the dual vector space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the of vector space of linear forms under pointwise addition and scalar multiplication) furnishes FdVect with a canonical compact structure[57][58];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for the case of the infinite-dimensional vector spaces in Vect, dual spaces also exist, although they are inherently less “well-behaved” (since the axiom of choice here implies that the dual space is always of strictly larger dimension than the original space) and satisfy fewer compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In topological quantum field theories, in which the number of degrees of freedom (and therefore the dimensionality of the spaces of states) is always finite, the propagator for processes of shape M from hypersurface ∂in (M) to hypersurface ∂out (M), namely: Z (M) : Z (∂in (M)) → Z (∂out (M)) , (145) generated by the functor Z : BordRiem d → Vect can be formulated in terms of dual spaces as: Z (M) : C → Z (∂ (M)) = Z (∂out (M)) ⊗ Z (∂in (M))∗ , (146) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in the form of a correlator or n-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For the category of hypergraphs and subhypergraph inclusion maps, every hypergraph has a dual (that is trivially compatible with the dagger structure) given by the interchange of its vertex set V and its hyperedge set E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e if: H = ⟨V = {vi |i ∈ Iv } , E = {ei |i ∈ Ie, ei ⊆ V, ei ̸= ∅}⟩ , (147) 48 then: H∗ = ⟨V ∗ = E, E∗ = {{ei |vm ∈ ei } |m ∈ Iv }⟩ , (148) where Iv and Ie are index sets for the vertices and for the hyperedges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Corresponding dual structures may well exist for other categories of data structures and computations too (such as the category of Turing machine states and transitions), but, if they do, then their explicit forms remain unknown at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Just as we have shown that deformation of the sequential composition structure ◦ in an ordinary category can be used to characterize and quantify computational reducibility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and that deformation of the parallel composition structure/tensor product ⊗ in a symmetric monoidal category can be used to characterize and quantify multicomputational reducibility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' it is entirely conceivable that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' as a consequence of this formal du- ality/adjunction relating computational complexity theory to quantum mechanics and multicomputational complexity theory to quantum field theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' these various other algebraic properties and structures inher- ent to the physical theories will end up having rather natural,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and purely complexity-theoretic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' analogs in the abstract theory of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' it is plausible that deformation of the dagger structure † in a dagger symmetric monoidal category may provide some means of characterizing the irreversibil- ity (meaning the pragmatic computational difficulty of reversing a computation that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in principle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' fully reversible) of certain classes of (multi)computations - a concept which is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' of course,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' deeply conceptually related to (multi)computational irreducibility itself - while deformation of the compact structure η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ε in a dagger compact category may be useful for quantifying the computational difficulty involved in exchanging outputs/values with inputs/arguments in a multicomputation consisting of one or more multi-argument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' multi-valued functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' as represented by a tensor network or a monoidal string diagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' These ex- citing possibilities remain topics for future investigation, as further detailed within the concluding remarks below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 5 Concluding Remarks Throughout the course of this article,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' we have sought to develop a systematic procedure by which the abstract syntax of a category may be endowed with a concrete computational semantics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in which all objects are interpreted as data structures and all morphisms are interpreted as computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in such a way that each morphism carries with it certain metadata corresponding to the time complexity of its underlying 49 computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' along with an algebra that defines how these time complexities behave under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' We have shown that this can be achieved by defining a map from the category of data structures and computations to a discrete cobordism category (of 0-dimensional manifolds and discrete cobordisms/intervals), in such a way that the irreducibility of a given computation is characterized by the extent to which this map preserves additivity of time complexities under sequential composition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' its functoriality), and such that the multicomputational irreducibility of a given multicomputation (described by the case of higher- dimensional manifolds and discrete cobordisms, in which both categories carry an additional symmetric monoidal structure) is characterized by the extent to which this map preserves additivity of time complexities under parallel/tensor product composition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' its symmetric monoidal functoriality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This latter extension is achieved by exploiting the fact that symmetric monoidal categories provide a convenient compositional semantics for describing multiway systems, branchial graphs and the general algebraic structure of non- deterministic computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' In so doing, we have effectively defined the outlines of a potential novel extension to the standard techniques of category theory and monoidal category theory, in which the compositions of morphisms (both in sequence and in parallel) carry with them additional algebraic structure resulting from the constraints of computational complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This new formalism brings with it many compelling directions for future research and exploration, a few of which we shall indicate below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Firstly, although traditional computational complexity theory has tended to restrict itself to the investi- gation of complexity classes with very simply and well-defined algebraic constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' P, EXP, NP, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' ), this category-theoretic formalism, along with the explicit computational tools developed for the purposes of the present article, potentially enables one to conduct a systematic and empirical investigation of computa- tional complexity classes exhibiting far less a priori algebraic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' by considering the extension to monoidal categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' it also enables the rigorous investigation of multicomputational complexity theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' which differs fundamentally from ordinary non-deterministic complexity theory in that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' within a standard non-deterministic complexity class (such as NP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' although the non-deterministic nature of the computation implies that one is inevitably forced to consider a multiway system of different singleway/deterministic com- putations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the complexity class itself is ultimately concerned only with the time complexity along a single branch of that system (albeit a branch whose identity may not necessarily be known in advance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Multicom- putational complexity theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' on the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' investigates the computational complexity of the multiway system itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and specifically of the tensor product structure of its branchial graphs (encoding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' as it does,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the complex branching and merging behavior of the entire multiway system),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' featuring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' as discussed previously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' contributions from the complexity-theoretic properties of both the state evolution function and the state 50 equivalence function of the multiway system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' most crucially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the properties of the interactions between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' To the best of the author’s knowledge, the space of possible multicomputational complexity classes remains essentially unexplored, and constitutes a major topic for planned future examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Next, all of the analysis presented within this article has focused exclusively on singleway/multiway evo- lution structure, and has neglected any consideration of causality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a natural causal semantics does exist for the case of hypergraph rewriting[59][60] (and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' for non-deterministic Turing machine evolu- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' as depicted as part of the multiway evolution causal graph shown in Figure 17 for the non-deterministic 2-state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 2-color Turing machine analyzed previously),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' allowing one to encode formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' by means of an explicit partial order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the notion of one transition/rewriting event only being applicable if another transi- tion/rewriting event had previously occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Just as the multiway evolution graph can be interpreted as freely generating a symmetric monoidal category, the multiway evolution causal graph (featuring a combi- nation of both evolution edges, indicating transitions/rewriting events, and causal edges, indicating causal relationships between those transitions/rewriting events) can be interpreted as freely generating a weak 2- category[61], with the 2-cells representing causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' These 2-cells are also equipped with their own tensor product operation, usually denoted ⊗C, that satisfies the axioms of a partial monoidal structure in the sense defined by Coecke and Lal in the context of causal categories[62], and allows one to compose causal relationships between spacelike-separated (causally-independent) events in parallel, but not between timelike-separated (causally-dependent) events, which may only be composed in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' This leads to the intriguing possibility that one may be able to equip the discrete cobordism category B with a second (partial) monoidal structure and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' with it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' to examine the 2-functoriality of the resulting map Z′ : T → B as a proxy for extending the definition of multicomputational irreducibility so as also to incorporate some kind of inherent complexity of causality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' and not simply of evolution (essentially by examining the extent to which Z′ also distorts the sequential and parallel composition of the 2-cells representing the causal structure of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The aim of such an undertaking would be to formalize a notion of causal irreducibility, in which the complete causal structure of a causally irreducible multiway system cannot be preempted without effectively tracing the explicit causal relationships between all transitions/rewriting events, spanning across all branches of the multiway evolution graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' As discussed within the preceding section, there are also various features of the formal duality/adjunction relationship between (multi)computational complexity theory and categorical quantum mechanics/functorial quantum field theory that might potentially yield additional, purely complexity-theoretic, extensions to this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For instance, it has often been assumed that the practical irreversibility of computations which 51 Figure 17: A multiway evolution causal graph corresponding to the non-deterministic evolution of a 2- state, 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 3 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the gray edges are the usual evolution edges, while each orange edge represents a causal relationship between two transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' are in principle reversible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in the case of one-way functions in cryptography) is merely a byproduct of computational irreducibility, but this assumption tacitly presupposes a compatibility condition between the irreducibilities of forward and backward evolution that may or may not hold for certain classes of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' By equipping our category T with an involutive dagger structure † and examining the effects of the map Z′ on that structure using methods from categorical quantum mechanics, it is conceivable that we may be able to disentangle the irreducibility of evolution from the irreducibility of reversal in a relatively fine-grained way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Deeply related to this is the irreducibility of swapping arguments/inputs with values/outputs in a composition of several multi-argument, multi-valued functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' in the case of a tensor network or monoidal string diagram (where this swapping operation is encoded as the raising and lowering of contravariant/covariant indices), such operations are usually assumed to be computationally trivial, but for the case of computations whose reversal operation is irreducible this need not be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Equipping the category T with a compact structure, with unit/counit η/ε, and observing how the compact structure gets distorted under the action of Z′ may permit one to quantify the complexities of these operations in a more meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 52 However, the conventional multiway system formalism does not make full use of the compact structure that is available within dagger compact categories, since transitions/events in a multiway system are tradi- tionally single-argument but potentially multi-valued (that is, a state evolution function conventionally takes in a single state as input and produces a list of possible successor states as output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' a glocal multiway system promotes the transitions/events to being true multi-argument functions (and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' to true symbolic tensors in a tensor network/string diagram representation of the multiway system) by ef- fectively “shattering” the states into their constituent “tokens” (for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' into individual hyperedges for the case of hypergraph rewriting systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' or individual tape positions for the case of Turing machines) and then reassembling them on-demand for the application of particular transitions/events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the term “glocal” refers here to the fact that the tokens are local but the events are global, and so in particular the multiway equivalence function acts at the level of events rather than at the level of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' An example of a glocal multiway evolution causal graph, for the same non-deterministic 2-state, 2-color Turing machine as before, is shown in Figure 18, with its associated glocal branchial graph shown in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note that, on a given glocal branchial graph, some of the tokens are separated spatially (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' they correspond to different regions of the tape), whilst some of the tokens are separated “branchially” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' they correspond to the same region of the tape, but on two different branches of the multiway system), and so, unlike the branchial graphs considered previously, glocal branchial graphs actually encode two different tensor product structures: the standard symmetric monoidal structure inherited from the multiway system, and a “spatial” tensor product structure (which is really identical to the causal tensor product structure ⊗C described above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The com- patibility conditions between these two tensor product structures are not known (for instance, it is unknown whether they form a rig category in special cases, although this possibility is unlikely) and remain a topic for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Note that the question of whether the ordering of tokens matters (as for Turing machine tape states) or does not (as for hyperedges in a hypergraph) corresponds to the question of whether the “spatial” part of the category is symmetric monoidal or simply monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Rather excitingly, just as the multiway tensor product structure enabled us to quantify multicomputational complexity, it is conceivable that the spatial tensor product structure (once it is better understood) may enable us to quantify space complexity, and thus to examine questions surrounding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=') the trade-offs between space complexity and time complexity in arbitrary (multi)computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' A meticulous treatment of the relationship between mul- tiway systems, the notion of “spatiality”, higher categories and type theory was previously conducted by Arsiwalla[64][65] within the context of Shulman’s cohesive homotopy type theory[66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Finally, there are several conceptual consequences of the realization of the orthogonality (independence) 53 Figure 18: A “glocal” multiway evolution causal graph corresponding to the non-deterministic evolution of a 2-state, 2-color Turing machine constructed from the (parallel) composition of the Turing machine transition functions for rules 2506 and 3506, starting from the single tape state {0, 1, 0, 0}, for 3 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' the orange edges represent causal relationships between transitions, while each gray edge represents either the “ingestion” or the “egestion” of a single “token” (tape position) into or out of a single transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' that exists between the complexity of state evolution and state equivalence (and thus between computational and multicomputational irreducibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' For instance, the computational interpretation of the second law of thermodynamics advocated by Wolfram[2] implies that entropy increase is a consequence of computational irreducibility, wherein the progression of a reversible computation can have the effect of “encrypting” the details of its initial conditions (such that, even if the computation is in principle reversible, in practice it can represent an arbitrarily hard problem of cryptanalysis to enact that reversal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' However, synthesizing this idea with the “orthogonality” principle indicates that there should exist at least two distinct concepts of entropy at play within any given multicomputation: one essentially computational, and the other essentially multicomputational, in origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The standard thermodynamic description of entropy is essentially a measure of non-injectivity of coarse-graining (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' how many distinct microstates get mapped to the same macrostate under the action of the coarse-graining function), and, due to Liouville’s theorem, and thus the one-to-one correspondence that exists between position/momentum values and possible evolution histories, in classical mechanics this definition is provably equivalent to a definition in terms of possible evolution trajectories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' how many distinct branches of history would have resulted in the same coarse-grained macrostate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' However, for more general multiway systems with more complex branching and merging structure, this correspondence does not necessarily hold, and so the two definitions of entropy diverge (although many computations of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' entanglement entropies in the context of quantum gravity may implicitly assume that they are equivalent[67][68][69][70]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The manifold implications of this disambiguation remain another worthy 54 Figure 19: The corresponding “glocal” branchial graph associated to the default “foliation” of the glocal multiway evolution causal graph for the non-deterministic evolution of the 2-state, 2-color Turing machine constructed from parallel composition of the transition functions for rules 2506 and 3506, after 3 steps, showing a mixture of both “spatial” and “branchial” tensor product structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' topic for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' Acknowledgments The author would like to thank Mohamed Barakat, Nicolas Behr, Matteo Capucci, Bob Coecke, Fabrizio Genovese, Manojna Namuduri, Stephen Wolfram and Yorick Zeschke for various stimulating and insightful discussions, enjoyed at a variety of different gestational stages of the ideas presented within this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' The author would also like to acknowledge James Boyd for his christening of the term “multicomputational irreducibility” in an earlier blog post, and Juan Arturo Silva-Ordaz for forcing the author to think about the complexity of equivalence functions to a far greater extent than he would ever have chosen to do voluntarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfvAvH/content/2301.04690v1.pdf'} +page_content=' 55 .' metadata={'source': 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Xue1, Sameer Antani1 1 Computational Health Research Branch, National Library of Medicine, National Institutes of Health, Bethesda, Maryland, USA Correspondence: sivaramakrishnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='rajaraman@nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='gov Abstract: Deep learning (DL) models are becoming state‐of‐the‐art in segmenting anatomical and disease regions of interest (ROIs) in medical images, particularly chest X‐rays (CXRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' However, these models are reportedly trained on reduced image resolutions citing reasons for the lack of com‐ putational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Literature is sparse considering identifying the optimal image resolution to train these models for the task under study, particularly considering segmentation of Tuberculosis (TB)‐consistent lesions in CXRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' In this study, we used the (i) Shenzhen TB CXR dataset, investigated performance gains achieved through training an Inception‐V3‐based UNet model using various im‐ age/mask resolutions with/without lung ROI cropping and aspect ratio adjustments, and (ii) identi‐ fied the optimal image resolution through extensive empirical evaluations to improve TB‐consistent lesion segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We proposed a combinatorial approach consisting of storing model snapshots, optimizing test‐time augmentation (TTA) methods, and selecting the optimal seg‐ mentation threshold to further improve performance at the optimal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We emphasize that (i) higher image resolutions are not always necessary and (ii) identifying the optimal image resolu‐ tion is indispensable to achieve superior performance for the task under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Keywords: aspect ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' chest X‐ray;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' image resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' tuberculosis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' test‐time augmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' threshold selection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Introduction Mycobacterium tuberculosis (MTB) infects the lungs, thereby causing pulmonary tu‐ berculosis (TB) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The infection can also spread to other body organs including the brain, spine, and kidneys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' TB infection can further be categorized into latent and active types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Latent TB refers to cases where the MTB remains inactive and causes no symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Active TB is contagious ad can spread to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Latent TB can turn into active TB, so, immediate treatment becomes indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The Centers for Disease Control and Prevention recom‐ mends those having an increased risk of acquiring TB infection including HIV/AIDS, us‐ ing intravenous drugs, and from countries with a high prevalence of TB, among others, be screened for TB infection [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Chest X‐ray (CXR) is the commonly used radiographic technique to screen for cardi‐ opulmonary abnormalities, particularly TB [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Some of the TB‐consistent abnormal man‐ ifestations in the lungs include apical thickening, calcified, non‐calcified, and clustered nodules, infiltrates, cavities, linear densities, adenopathy, miliary patterns, and retraction, among others [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' These manifestations can be observed anywhere in the lungs and may vary in size, shape, and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' While CXRs are widely adopted to screen for TB infection, there is an increasing scar‐ city of human experts, particularly in low and middle‐income countries, to interpret CXRs and make decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Such scarcity necessitates the development of artificial intelligence (AI)‐based machine learning (ML) tools that could automate the process of segmenting disease‐consistent regions of interest (ROIs) in medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Currently, deep learning MDPI 2 (DL) models, a subset of ML algorithms, are observed to perform on par with human ex‐ perts in segmenting body organs like the lungs, heart, clavicles [4,5], and other cardiopul‐ monary disease manifestations including COVID‐19 [6], pneumonia [7], and TB [8] in CXRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' A study of the literature reveals that the CXRs and disease ROI masks are down‐ sampled to either 224×224 or 256×256 pixel resolution citing reasons for reducing the com‐ putational overhead due to GPU constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' However, extensive reduction in image res‐ olution may eliminate information, particularly when every pixel in an image is critical, as in the case of a segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The important information may be hidden in small details, like the surface and contour of the lesion, and other patterns in findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The de‐ tails preserved in the visual information can drastically vary with the changes in image resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The choice of image resolution depends not on the computational hardware availability but the characteristics of the data under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' A study of the literature reveals that changes in endoscopy image resolution impact classification performance [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Another study [10] discussed that the disease classification performance improved at lower CXR image resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The authors observed that the overfitting issues were resolved at lower input image resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Identifying the optimal image resolution for the task under study remains an open avenue for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Until the writing of writing this manuscript, we observed no available literature that discussed the impact of image resolution on a CXR‐based segmentation task, particularly considering segmenting TB‐consistent lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The primary goal of this study is to study the impact of training a UNet model on varying image resolutions with/without lung ROI cropping and aspect ratio adjustments and find the optimal resolution that improved fine‐grained TB‐ consistent lesion segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We further improved performance at the optimal resolu‐ tion through a combinatorial approach consisting of storing model snapshots, optimizing the test‐time augmentation (TTA) methods, optimizing the segmentation threshold, and averaging the predictions of the model snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Section 2 discusses the materials and methods, Section 3 elaborates on the results and Section 4 discusses and concludes this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Materials and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Data Characteristics This retrospective study uses the Shenzhen TB CXR dataset [11] collected from the Shenzhen No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 3 hospital, Longgang, Shenzhen, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The CXRs were deidentified and published by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' National Library of Medicine (NLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The dataset comprises 336 CXRs collected from microbiologically‐confirmed TB cases and 326 CXRs showing normal lungs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 1 shows the dataset characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Dataset characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The age of the men and women population, image width, and image height are given in terms of mean± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' # TB CXRs # Men # Women Men age (in years) Women age (in years) # Lung masks # TB Masks Image width (in pixels) Image height (in pixels) 336 228 108 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='29±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='12 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='5±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='75 287 336 2644±253 2799±206 The CXRs manifesting TB were annotated using the Firefly annotation tool [1] by two radiologists from the Chinese University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The annotations were stored as both grayscale masks and in JSON format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The authors of [12] manually segmented the lung regions and made them available as grayscale masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' These masks are available for 287 CXRs manifesting TB‐consistent abnormalities and 279 CXRs showing normal lungs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We used these 287 TB CXRs out of 336 TB CXRs that have both lung masks and TB lesion‐ consistent masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 1 shows the following: (a) The heatmaps were created by resizing the TB masks of men and women to 1024×1024 to maintain uniformity in scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The gray‐ scale intensities were then added and shown using the “hot” colormap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (b) Pie chart 3 showing the proportion and distribution of TB in men and women, and (c) Age‐wise dis‐ tribution of normal and TB‐infected men and women population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Data characteristics are shown as a proportion of men and women in the Shen‐ zhen TB CXR collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (a) Heatmaps showing regions of TB infestation in men and TBmanifestationsinMen TBmanifestationsinWomen 23458 10025 No Finding No Finding (a) TB women 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='3% women 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='29 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4% TB TB No Finding 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 9% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8% No Finding Men 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2% No Finding Men 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4% (b) Men Women 25 20 15 10 5 0 0 5 10 15 20 25 30 No Finding No Finding 95 TB TB 90 85 80 75 70 65 6055 4 40 35 30 25 20 15 10 c 4 women (all images are resized to 1024×1024 to maintain uniformity in scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (b) Pie chart showing the proportion and distribution of TB in men and women, and (c) Age‐wise dis‐ tribution of normal and TB‐infected population in men and women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The 287 CXRs were further divided at the patient level into 70% for training (n = 201), 10% for validation (n = 29), and 20% for hold‐out testing (n= 57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The masks were thresholded and binarized to separate the foreground lung/TB‐lesion pixels from the background pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Model architecture We used the Inception‐V3 UNet model architecture that delivered superior TB‐con‐ sistent lesion segmentation performance from our previous study [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The Inception‐V3‐ based encoder [13] was initialized with ImageNet weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The model was trained for 128 epochs at various image resolutions discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We used an Adam optimizer with an initial learning rate of 1 × 10‐3 to minimize the boundary‐uncertainty augmented focal Tversky loss [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The learning rate was reduced if the validation loss ceased to im‐ prove after 5 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We stored the model weights whenever the validation loss de‐ creased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The best‐performing model with the validation data was used to predict the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The modes were trained using Keras with Tensorflow backend (ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='7) using a single NVIDIA GTX 1080 Ti GPU and CUDA dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Image resolution We empirically identified the optimal image resolution at which the Inception‐V3 UNet model delivered superior performance toward the TB‐consistent lesion segmenta‐ tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The model was trained using various image/mask resolutions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', 32×32, 64×64, 128×128, 256×256, 512×512, 768×768, and 1024×1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We used a batch size of 128, 64, 32, 16, 8, 4, and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The CXR images in the Shenzhen TB CXR dataset are of high resolution (2644 pixel width × 2799 pixel height average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We used bicubic interpolation to downsample the 287 CXR images and their associated TB masks to the aforementioned resolutions as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed that the visual details improved with in‐ creasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' CXRs and their corresponding TB‐consistent lesion masks at various image res‐ olutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (a) 32×32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (b) 64×64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (c) 128×128;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (d) 256×256;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (e) 512×512;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (f) 768×768, and (g) (a) (b) (c)20 4c0 (d) (c) (0) (g) 5 1024×1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' All images and masks are rescaled to 256×256 to compare quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The red con‐ tours indicate ground truth annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We evaluated model performance under the following conditions: (i) 287 CXRs and their associated TB masks were directly downsampled to the afore‐ mentioned resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (ii) The lung masks were overlaid on the CXRs and their associated TB masks to de‐ lineate the lung boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The lung ROI was cropped to the size of a bounding box and downsampled to the aforementioned resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (iii) Based on performance, the data from step (i) or step (ii) is corrected for aspect ratio, the details are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The aspect ratio‐corrected CXRs/masks were further downsampled to the aforementioned resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Aspect ratio correction The aspect ratio can be defined as the ratio of width to height [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The mean and standard deviation of the widths and heights of the CXRs manifesting TB‐consistent ab‐ normalities in the Shenzhen TB CXR dataset was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' For original CXRs, we ob‐ served the width and height are 2644±253 pixels and 2799±206 pixels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' For lung‐cropped CXRs, we observed the width and height are 1929±151 pixels and 1999±231 pixels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' For original CXRs, we observed the aspect ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='945:1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', the width is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='945 times the height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' For lung‐cropped CXRs, the aspect ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='965:1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', the width is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='965 times the height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed that the height is larger than the width for both the original and lung‐cropped CXRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We maintained the larger dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', height) as constant at various image resolutions and modified the smaller dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', width) to correct the aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We were, however, constrained by the fact that the width and height of the images/masks should be divisible by 32 to be compatible with the UNet architecture [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We, therefore, rounded the widths to the nearest lower value that is di‐ visible by 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Performance evaluation The trained models were evaluated using (i) pixel‐wise metrics consisting of the in‐ tersection of union (IoU) and Dice score [16] and (ii) image‐wise metrics consisting of structural similarity index measure (SSIM) [17,18] and signal‐to‐reconstruction error ratio (SRE) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' While IoU and Dice are the most commonly used metrics to evaluate segmen‐ tation performance, literature studies reveal that pixel‐wise metrics would ignore explor‐ ing the dependencies among the neighboring pixels [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This might not help to optimally explore the model’s segmentation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The authors of [21] minimized a loss function derived from SSIM to segment ROIs in the Cityscapes and PASCAL VOC 2012 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' It was observed that the masks predicted by the model that was trained to minimize the SSIM loss were more structurally similar to the ground truth masks compared to the model trained using the conventional cross‐entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Motivated by this study, we used the SSIM metric to evaluate the structural similarity between the ground truth and predicted TB masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The SSIM of a pair of images (𝑎, 𝑏) is given by a multiplicative combination of the structure (s), contrast (c), and luminance (l) factors, as shown in equations (1 – 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 𝑆𝑆𝐼𝑀 �𝑎, 𝑏� � �𝑙�𝑎, 𝑏���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' �𝑐�𝑎, 𝑏���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' �𝑠�𝑎, 𝑏�� � (1) 𝑙�𝑎, 𝑏� � 2µ�µ� � 𝐶� µ��� µ� � � 𝐶� (2) 𝑐�𝑎, 𝑏� � 2𝜎�𝜎� � 𝐶� 𝜎��� 𝜎� � � 𝐶� (3) 6 𝑠�𝑎, 𝑏� � 𝜎�� � 𝐶� 𝜎�𝜎� � 𝐶� (4) Here, µ�, µ�, 𝜎�, 𝜎�, 𝜎�� denote the mean, standard deviation, and cross‐covariance, re‐ spectively, for the images (𝑎, 𝑏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The value of IoU, Dice, and SSIM range from [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We visualized the SSIM quality map (using “jet” colormap) to interpret the quality of the pre‐ dicted masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The quality map is identical in size to the corresponding scaled version of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Small values of SSIM appear as dark blue activations, denoting regions of poor similarity to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Large values of SSIM appear as dark red activations, de‐ noting regions of high similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The authors of [19] proposed a metric called signal‐to‐reconstruction error ratio (SRE) that measures the error relative to the mean image intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The authors discussed that the SRE metric is robust to brightness changes while measuring the similarity between the predicted image and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The SRE metric is measured in decibels (DB) and is given by equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 𝑆𝑅𝐸 � 10 𝑙𝑜𝑔�� ⎝ ⎜ ⎛ µ� � �|𝑎� � 𝑎|� � 𝑛 ⎠ ⎟ ⎞ (5) Here, µ� denotes the average value of the image 𝑎 and 𝑛 denotes the number of pixels in image 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Optimizing the segmentation threshold Literature studies reveal that the threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='5 is routinely used in segmentation tasks [22–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' However, the process of selecting the segmentation threshold should be driven by the data under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' An out‐of‐the‐box threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='5 is not guaranteed to be optimal, particularly considering an imbalanced segmentation task, as in our case, where the number of foreground TB‐consistent lesion pixels is considerably smaller compared to the background pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' It is therefore indispensable to iterate among different segmen‐ tation threshold values in the range of [0, 1] and find the optimal threshold that would maximize performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This is called threshold tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' In our case, we generated 200 equally spaced samples in the closed interval [0, 1] and used a looping mechanism to find the optimal segmentation threshold that maximized the IoU metric for the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This threshold was used to binarize the predicted masks using the test data and the per‐ formance was measured in terms of the evaluation metrics discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Storing model snapshots at the optimal resolution After we empirically identified the optimal resolution, we further improved perfor‐ mance at this resolution as discussed in the following steps: (i) We adopted the method called “snapshot ensembling” discussed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The process involves using an aggressive cyclic learning rate to train and store diversified model snapshots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', the model weights) during a single training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (ii) We initialized the training process with a high learning rate of 1 × 10‐2, defined the number of training epochs as 320, and the number of training cycles as 8 so that each training cycle is composed of 40 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (iii) The learning rate was rapidly decreased to the minimum value of 1 × 10‐8 at the end of each training cycle before being drastically increased during the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This acts like a simulated restart, result‐ ing in using good weights as the initialization for the subsequent cycle, thereby allowing the model to converge to different local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (iv) The weights at the bottom of each cycle are stored as snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' With 8 training cycles, we stored 8 model snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (v) we evaluated the validation performance of each of these snapshots at their optimal segmen‐ tation threshold identified as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This threshold was further used to binarize the predicted test data and the performance was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We ranked the model snapshots based on their test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Test‐time augmentation (TTA) Literature on medical image segmentation reveals the use of augmentation strategies during model training to tackle data scarcity, increase data diversity, reduce model over‐ fitting, and improve convergence [15,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Some of the commonly used augmentation strat‐ egies include width and height shifting, horizontal and vertical flipping, random rota‐ tions, center cropping, and elastic distortions, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Test‐time augmentation (TTA) refers to the process of augmenting the test set [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' That is, the trained model predicts the original and transformed versions of the test set, and the predictions are aggregated to produce the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' An advantage of using TTA is that no changes are required to be made to the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' TTA ensures diversifica‐ tion and helps the model with improved chances of better capturing the target shape, thereby improving model performance and eliminating overconfident predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The benefits of TTA are discussed in the literature [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' However, these studies are observed to perform multiple random image augmenta‐ tions without identifying the optimal augmentation method(s) that would help improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' A possible negative effect of destroying/degrading visual information with non‐optimal augmentation(s) might outweigh the benefit of augmentation while also re‐ sulting in increased computational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' After storing the model snapshots as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='7, we performed TTA with the validation data using each model snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' In addition to the original input, we used the augmentation methods consisting of horizontal flipping, pixel‐wise width and height shifting (‐5, 5), and rotation in degrees (‐5, 5) individually and in combination as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' TTA combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Method TTA combinations M1 Original + horizontal flipping M2 Original + width shifting M3 Original + height shifting M4 Original + width shifting + height shifting M5 Original + horizontal flipping + width shifting + height shifting M6 Original + rotation M7 Original + width shifting + height shifting + rotation M8 Original + horizontal flipping + width shifting + height shifting + rotation For each TTA combination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' an aggregation function takes the set of predictions and averages them to produce the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We identified the optimal segmentation threshold that maximized the IoU for each model snapshot and every TTA combination shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' With the identified optimal TTA augmentation combination and the segmentation threshold, we augmented the test data, recorded the predictions, binarized them, and evaluated performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This process is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We ranked the models based on the test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We then constructed an ensemble of the top‐K (K = 2, 3, …, 6) by averaging their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We call this “snapshot averaging”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Statistical analysis We measured the 95% binomial Clopper‐Pearson confidence intervals (CIs) for the IoU metric obtained at various stages of our empirical analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 8 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' A combinatorial workflow showing the storage of model snapshots and identi‐ fying the optimal TTA combination at the optimal segmentation threshold for each snap‐ shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Results Table 2 shows the performance achieved through training the Inception‐V3 UNet model using the CXRs/TB masks of varying image resolutions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', 32×32, 64×64, 128×128, 256×256, 512×512, 768×768, and 1024×1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 4 shows the sample predictions at these resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The performances are reported for each image resolution at its optimal seg‐ mentation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The term “O” and “CR” denote the original and lung‐cropped CXRs/masks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed poor performance at 32×32 resolution with both original and lung‐cropped data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Performance achieved by the Inception‐V3‐UNet model with original and lung‐ cropped CXRs and TB‐lesion‐consistent masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The term SRE, O, CR, and Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' T denotes signal‐to‐reconstruction error ratio, original CXRs and TB‐lesion‐consistent masks, lung‐ ROI‐cropped CXRs and TB‐lesion‐consistent masks, and the optimal segmentation thresh‐ old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Values in parenthesis denote the 95% CIs as the Exact measure of the Clopper Pearson interval for the IoU metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The bold numerical values denote superior performance for the respective columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Resolution IoU Dice SSIM SRE Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' T 32× 32 (O) 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Visualizing and comparing the segmentation predictions of the Inception‐V3 UNet model trained at various image resolutions, using a sample original, and its corre‐ sponding lung‐cropped CXR/ mask from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The red and blue contours denote Ground truth and predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The performance kept improving until 256×256 pixel resolution where the model achieved the best IoU of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4859 (95% CI: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='3561, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6157)) and superior values for Dice, SSIM, and SRE metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The performance then kept decreasing from 256×256 to 1024×1024 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The performance achieved with the lung‐cropped data is superior compared to the origi‐ nal counterparts at all resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' These observations highlighted that 256×256 is the op‐ timal resolution and using lung‐cropped CXRs/masks gave a superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 5 shows the SSIM quality maps achieved by the Inception‐V3 UNet model for a sample test CXR at varying image resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The quality maps are identical in size to the corresponding scaled version of the images/masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed high activations, shown as red pixels, in regions where the predicted masks were highly similar to the ground truth masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Blue pixel activations denote regions of poor similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We ob‐ served the following: (i) The predicted masks exhibited poor similarity to the ground truth masks along the mask edges for all image resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' (ii) The SSIM value obtained with the lung‐cropped data was superior compared to the original counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 3 shows the performance achieved by the Inception‐V3 UNet model with aspect‐ ratio corrected (AR‐CR) lung‐cropped CXRs/masks for varying image resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We ob‐ served no improvement in performance with aspect‐ratio corrected data at any given im‐ age resolution compared to the results reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Performance achieved by the Inception‐V3‐UNet model with the aspect‐ratio cor‐ rected lung‐cropped (AR‐CR) CXRs and TB‐lesion‐consistent masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The image resolu‐ tions are given in terms of height×width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Resolution (AR‐CR) IoU Dice SSIM SRE Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' T 64×32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='1583 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='0635,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2531) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 4 shows the optimal TTA combina‐ tions that delivered superior performance for each model snapshot at its optimal segmen‐ tation threshold identified from the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 11 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' SSIM quality maps shown for the predictions achieved by the Inception‐V3 UNet model trained on various image/mask resolutions using a sample original and its corre‐ sponding lung‐cropped CXR mask from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0 0 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9453 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9827 1 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0 0 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8869 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 128 × 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0 0 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9298 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 256 × 256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='959 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 512 × 512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 12 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Optimal test‐time augmentation combination for each model snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Snapshot Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' TTA combination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original+ width shifting + height shifting + rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original + height shifting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original+ horizontal flipping + width shifting + height shifting + rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original+ horizontal flipping + width shifting + height shifting + rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original+ horizontal flipping + width shifting + height shifting + rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original+ width shifting + height shifting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original+ horizontal flipping + width shifting + height shifting + rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='S8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='Original+ horizontal flipping + width shifting + height shifting + rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='The terms S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' S3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' S4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' S5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' and S6 denote the 1st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2nd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 3rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 4th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 5th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 6th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 7th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' and 8th model snapshot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The TTA combination that aggregates (averages) the predictions of the original test data with those obtained from other augmentations consisting of hori‐ zontal flipping, width shifting, height shifting, and rotation, delivered superior perfor‐ mance for the S3, S4, S5, S7, and S8 model snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The aggregation of the original pre‐ dictions with height‐shifting augmentation delivered superior performance for the S2 snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The S6 snapshot delivered superior performance while aggregating the original predictions with those obtained from the width and height‐shifted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Aggregating the predictions of the original test data with those augmented by width, height shiting, and rotation, delivered superior test performance while using the S1 model snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The first row of Table 5 shows the performance achieved by the model trained with the 256×256 lung‐cropped CXRs/masks (from Table 2), denoted as “CR‐baseline”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Performance achieved by each model snapshot before and after applying the op‐ timal TTA and averaging the snapshots after TTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Bold numerical values denote superior performance in respective columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Model IoU Dice SSIM SRE Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' T 256×256 (CR‐Baseline) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4859 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='3561,0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='3895,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6491) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8009 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4925 Rows 2 – 9 denote the performance achieved by the model snapshots S1 – S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Rows 10 – 17 show the performances achieved by the model snapshots at their optimal TTA combi‐ nation (shown in Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed that TTA improved segmentation performance for the recorded model snapshot in terms of all metrics compared to the model snapshots without TTA and the “CR baseline”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We ranked the model snapshots S1 – S8 in terms of their IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed the S2 snapshot delivered the best IoU, followed by S3, S5, S7, S6, and S4 model snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We constructed an ensemble of the top‐K snapshots (K = 2, 3, …, 6) as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 by averaging their predictions obtained using their optimal TTA combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Rows 18 – 22 show the performances achieved by the ensemble of the top‐2, top‐3, top‐4, top‐5, and top‐6 model snapshots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed that the snapshot averaging ensemble constructed using the top‐4 and top‐5 model snapshots delivered superior performance in terms of the IoU and Dice metrics while the top‐5 snapshot ensemble delivered superior values also in terms of the SSIM and SRE metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The segmentation performance im‐ proved in terms of all evaluation metrics at the optimal 256×256 resolution by constructing an averaging ensemble of the top‐5 model snapshots compared to the “CR‐baseline”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 6 shows the predictions achieved by the baseline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', the Inception‐V3 UNet model trained with the lung‐cropped CXRs/masks at the 256×256 resolution) and snap‐ shot averaging of the top‐5 model snapshots with TTA for a couple of CXRs from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' In the first row, we could observe that snapshot averaging removed the false positives (predictions shown with blue contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' In the second row, we could observe that the predicted masks were increasingly similar to the ground truth masks (shown with red contours), compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Visualizing and comparing the segmentation predictions of the baseline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=', Inception‐V3 UNet model trained with lung‐cropped CXRs/masks at the 256×256 resolu‐ tion) and the snapshot averaging of the top‐5 model snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The red and blue contours denote ground truth and predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Baseline Snapshot averaging 0 0 50 50 Img-1 100 100 150 150 200 200 250 250 0 100 200 0 100 200 0 0 50 50 100 100 Img-2 150 150 200 200 250 250 0 100 200 0 100 200 14 Figure 7 shows the SSIM quality maps achieved with the baseline and snapshot av‐ eraging for a couple of CXR instances from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' SSIM quality maps shown for the predictions achieved for a couple of test CXRs, by the baseline (Inception‐V3 UNet model trained with lung‐cropped CXRs/masks at the 256×256 resolution) and the snapshot averaging of the top‐5 model snapshots with their optimal TTA combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed higher values for the SSIM using the snapshot averaged predictions com‐ pared to the baseline, signifying that the predicted masks were increasingly similar to the ground truth masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Snapshot averaging removed the false positives, and also demon‐ strated improved prediction similarity to the ground truth, with a higher SSIM value, compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Discussion and Conclusion We observed that the segmentation performance improved with increasing image resolution from 32×32 up to 256×256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The performance achieved with the lung‐cropped CXRs/TB‐lesion masks was superior compared to their original counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' These find‐ ings are consistent with [26,29–32] in which lung cropping was reported to improve per‐ formance in medical image segmentation and classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We observed that in‐ creasing the resolution beyond 256×256 decreased segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This can be attributed to the fact that (i) increasing resolution also increased the feature space to be learned by the models, (ii) increased parameter count might have led to model overfitting to the training data because of limited data availability, and (iii) increased complexity of the optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We did not observe a considerable performance improvement with aspect ratio cor‐ rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We were constrained by the UNet architecture [15] that requires that the length and width of the images/masks should be divisible by 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' This limitation did not allow us to make precise aspect ratio corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' However, the study of literature [14] reveals that DL models trained on medical images are robust to changes in the aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Abnor‐ malities manifesting TB do not have a precise shape and they exhibit a high degree of variabilities like nodules, effusions, infiltrations, cavitations, miliary patterns, and consol‐ idations, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' These manifestations would appear as high‐frequency signals with their inherent characteristics that provide diversified features to learn for a segmen‐ tation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We identified the optimal image resolution and further improved performance at that resolution through a combinatorial approach consisting of storing model snapshots, optimizing the TTA and segmentation threshold, and averaging the snapshot predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Baseline Snapshot averaging SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='944 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 Img-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='9358 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='6 Img-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='2 0 15 These findings are consistent with the literature where storing model snapshots and per‐ forming TTA considerably improved performance in natural and medical computer vi‐ sion tasks [27,33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We further emphasize that identifying the optimal TTA method(s) is indispensable to achieve superior performance compared to randomly augmenting the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' We underscore the importance of using the optimal segmentation threshold compared to the conventional threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='5 as widely discussed in the literature [22,37– 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Due to GPU constraints, we were not able to train high‐resolution models at larger batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' However, with the advent of high‐performance computing, this can be made feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' High‐resolution datasets might require newer model architecture and hardware advancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Nevertheless, although the full potential of high‐resolution datasets is not explored yet, it is indispensable to collect data at the highest resolution possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Additionally, irrespective of the image resolution, adding more experts to the anno‐ tation process may reduce the variation in the ground truth which may improve segmen‐ tation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Despite these limitations, we show that segmenting TB‐consistent lesions in CXRs using an UNet model trained on lung‐cropped CXRs/masks delivers optimal performance at the 256×256 image resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Through these extensive empirical evaluations and dis‐ cussions, we emphasize that the characteristics of the data under study, the model perfor‐ mances at varying image resolutions with/without ROI cropping, and aspect ratio correc‐ tions, should be discussed in all research papers to report realistic predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Author Contributions: Conceptualization, Sivaramakrishnan Rajaraman, Feng Yang, Ghada Zam‐ zmi, Zhiyun Xue, and Sameer Antani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Data curation, Sivaramakrishnan Rajaraman and Feng Yang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Formal analysis, Sivaramakrishnan Rajaraman and Feng Yang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Funding acquisition, Sameer Antani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Investigation, Sameer Antani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Methodology, Sivaramakrishnan Rajaraman and Feng Yang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Project administration, Sameer Antani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Resources, Sameer Antani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Software, Sivaramakrishnan Rajaraman and Feng Yang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Supervision, Sameer Antani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Validation, Sivaramakrishnan Rajaraman, Feng Yang, and Sameer Antani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Visualization, Sivaramakrishnan Rajaraman;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Writing – original draft, Sivara‐ makrishnan Rajaraman;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Writing – review & editing, Sivaramakrishnan Rajaraman, Feng Yang, Ghada Zamzmi, Zhiyun Xue, and Sameer Antani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Funding: This research was supported by the Intramural Research Program of the National Library of Medicine, National Institutes of Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' The funders had no role in the study design, data collec‐ tion, analysis, decision to publish, or preparation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Institutional Review Board Statement: Ethical review and approval were waived for this study because of the retrospective nature of the study and the use of anonymized patient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Informed Consent Statement: Patient consent was waived by the IRBs because of the retrospective nature of this investigation and the use of anonymized patient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Data Availability Statement: The data required to reproduce this study is publicly available and cited in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' References 1.' metadata={'source': 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Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content=' 2017, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} +page_content='1146/annurev‐ bioeng‐071516‐044442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfpwiN/content/2301.04032v1.pdf'} diff --git a/KNE1T4oBgHgl3EQfsQX7/vector_store/index.faiss b/KNE1T4oBgHgl3EQfsQX7/vector_store/index.faiss new file mode 100644 index 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+Ordinary versus non ordinary weak del Pezzo surfaces . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.2 +The blow-up model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.3 +The anticanonical model Xs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.3.1 +The morphism induced by the anticanonical divisor −KX . . . . . . . . . . . . . . . . . . . +5 +2.3.2 +Cartier versus Weil divisors and class groups +. . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.3.3 +Lattice computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +3 +Codes from surfaces: construction and tools for their study +8 +3.1 +Evaluation codes from surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Blowing up, divisors and (non) complete linear systems +. . . . . . . . . . . . . . . . . . . . . . . . +8 +3.3 +Blowing up and evaluation codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Anticanonical codes from weak del Pezzo surfaces +10 +4.1 +General description of the codes and of the main steps of their studies . . . . . . . . . . . . . . . . +10 +4.2 +Degree 6, singularity of type A1 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +4.3 +Degree 5, singularity of type 2A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.4 +Degree 4, singularity of type A1 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +4.5 +Degree 4, singularity of type 4A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +4.6 +Degree 4, singularity of type A2 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +4.7 +Degree 4, singularity of type D5 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +4.8 +Degree 3, singularity of type A1 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +4.9 +Degree 3, singularity of type 3A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +1 +Introduction +The geometric construction of error correcting codes goes back to Reed-Solomon and Goppa for curves and to +Reed-Muller for affine or projective spaces. In this work, we focus on evaluation codes from algebraic surfaces +whose construction works as follows. Given X an algebraic surface over Fq and D a (Cartier) divisor on X, we +denote by X(Fq) the set of rational points of X and by H0(X, D) the space of global sections of the divisor D. The +associated evaluation code is the code whose codewords are the evaluations of the functions of H0(X, D) at the +points of X(Fq) (see definition 3.1). The length of such a code is thus #X(Fq), the dimension is dim +� +H0(X, D) +� +(at least if the evaluation map is injective) and the third invariant, the minimum distance, is more difficult +to control. It is related to the maximum of rational points that can contain a curve in the linear system |D| +(proposition 3.2). The lower this maximum, the better is the minimum distance. +The well known Reed-Muller code of degree d for the projective plane P2 over Fq is a nice example. Its +codewords are the evaluations of the homogeneous polynomials of degree d at the rational points of P2(Fq). In +the geometric setting above, this is the evaluation code associated to the algebraic surface P2, the divisor dℓ +where ℓ denotes any line, and the whole set of rational points P2(Fq). Its parameters are well known when q > d: +the length is the number of rational points #P2(Fq) = q2 + q + 1, the dimension is the dimension of the space of +global section dim +� +H0(P2, dℓ) +� += +�d+2 +2 +� +, and the minimum distance is q(q − d + 1). This minimum distance can +be written (q2 + q + 1) − (1 + dq) and (1 + dq) is nothing else than the maximal number of rational points of a +curve lying in the linear system |dℓ| (i.e. the set of plane curves of degree d). In fact, this is the number of points +of the union of d lines of P2 meeting at one point. The existence of this kind of extremely reducible curve over Fq +impacts negatively the minimum distance, since they contain too many rational points. Among other things, this +is why evaluation codes associated to more general algebraic surfaces have been considered. +1 + +One can distinguish several strategies in the literature to get rid of reducible curves with many components in +the linear system. The idea of Couvreur [Cou11] is to work with sublinear systems of P2 by adding constraints +that remove the very reducible sections. In fact by choosing carefully the sublinear system, this kind of sections +is no longer defined over the base field, but only over an extension. The number of rational points of irreducible +curves that are not absolutely irreducible may fail drastically. In the preceding example of the d intersecting lines, +if they are not defined over Fq but only conjugate over Fq, then one can easily convince ourselves that their union +only contains one rational point, their meeting point. Some other examples can be found in Edoukou [Edo08] or +Couvreur & Duursma [CD13]. +Following Zarzar [Zar07], another fairly repeated strategy is to concentrate on surfaces whose (arithmetic) +Cartier class group is free of rank 1. Indeed, this is a natural way to overcome the difficulty of the existence of +(very) reducible sections in the linear system |D|. Little and Schenck [LS18] have studied anticanonical codes on +degree 3 and 4 del Pezzo surfaces having rank 1. In our previous work [BCH+20], we could say that we fill a +gap in the study of algebraic geometric codes constructed from del Pezzo surfaces of rank 1. Let us remark that, +even if the elements of the linear system |D| are all irreducible, some of them may be absolutely reducible. As in +the example of conjugate lines, it is expected that these configurations do not contain too many points but this +requires a proof. +In this work, we continue the investigation of codes constructed from del Pezzo surfaces. We do not restrict +ourselves to rank one surfaces but above all we consider more general surfaces, that is non ordinary weak del Pezzo +surfaces. As ordinary del Pezzo surfaces, non ordinary weak del Pezzo surfaces admit a blowing-up description; in +the ordinary case, the points that are blown up are in general position but in the non ordinary case, they are only +in almost general position (three points can be colinear and six points can be conconic). The main consequences +of these weaker hypotheses on the configuration of points are twofold. First, the surface contains −2-curves (and +not only −1-curves). Secondly, the anticanonical divisor is not ample anymore but only big and nef and the +anticanonical model is singular with rational double points. +In a concomitant work [BH22], we have computed explicit models for all the arithmetic types of weak del +Pezzo surfaces of degree at least 3 over a finite field (these types lead to a classification that is coarser than the +isomorphism one but that permits to distinguish the main arithmetic properties of the weak del Pezzo surfaces). +Taking advantage of this knowledge, we select eight types of (non ordinary) weak del Pezzo that are well suited +for coding applications. More precisely, we consider X a (smooth) weak del Pezzo surface of degree d over Fq +and we denote by Xs its (singular) anticanonical model; this is the image of the surface X by the morphism ϕ +associated to −KX the anticanonical divisor of X. Since −KX is not ample, the surface Xs is singular with a +finite number of rational double points. We study the evaluation code associated to the (singular) surface Xs, +the Cartier divisor −KXs = ϕ∗(−KX) and the whole set of rational points of Xs (definition 4.1). Except for +small values of q, this code has length n = #Xs(Fq), dimension k = d + 1. The last invariant, the minimum +distance dmin, is much more subtle to control and requires preparatory calculations. +Before going into details, let us discuss the advantages and disadvantages of considering such weak Del Pezzo +surfaces. In the process of construction of a del Pezzo surface, there are blowing-up and blowing-down. The +blowing-up may add rational points and thus may increase the length. The blowing-down permits to contract +some lines and thus decreases the types of reducible configurations. Since the anticanonical model is no longer +smooth, besides the exceptional curves some other curves, in fact the effective roots, can be contracted. If these +curves are components of the most reducible sections of the anticanonical divisor on the weak del Pezzo, the +parameters of the code could be improved. This is the positive aspect of considering anticanonical model of weak +Del Pezzo surfaces. But we should also mention a negative one: because of the singularity of Xs, the notions of +Cartier and Weil divisors are not equivalent and this makes it difficult to calculate the minimum distance dmin as +we will see below. +The computation of dmin reduces to compute the number: +Nq (−KXs) = max {#C(Fq) | C ∈ |−KXs|} . +Since every curve of the linear system |−KXs| is of arithmetic genus 1 (adjunction formula), all its absolutely +irreducible curves have a number of rational points which is bounded above by the classic: +Nq(1) = max {#C(Fq) | C absolutely irreducible, smooth, genus 1, curves over Fq} . +By the Weil-Serre bound, we know that Nq(1) ≤ q + 1 + ⌊2√q⌋; in fact, except for very special values of q, the +Weil-Serre bound turns to be sharp: +Nq(1) = +� +q + ⌊2√q⌋ +if q = pe, e ≥ 5, e odd and p | ⌊2√q⌋, +q + 1 + ⌊2√q⌋ +otherwise +([Ser20, Chap 2, Th 6.3]). +Anyway, this bound does not permit to control the number of rational points of +reducible or absolutely reducible curves of |−KXs|. Due to the singularities of Xs or more specifically to the +difference between the Cartier or Weil divisors or class groups, the expectation that irreducible, but absolutely +2 + +reducible curves in the linear system do not contain too many points is more difficult to verify. Even if the Cartier +class group CaCl(Xs) is free of rank 1, generated by −KXs, this does not mean that the curves of the linear +system |−KXs| are all irreducible since they can decompose in the Weil class group Cl(Xs), that is into a sum of +Weil irreducible divisors that are not Cartier divisors. To overcome this difficulty, we took fall advantage of the +fact that in the context of weak del Pezzo surfaces, explicit models of all the class groups can be computed. This +permits us to accurately measure the difference between the Cartier and the Weil divisors. This step uses some +basic methods on lattices computations. Then to explicitly compute the maximum Nq (−KXs), we list all the +kinds of decompositions into irreducible components that may appear in the linear system |−KXs|. In general, +this can be a difficult issue but in our context this task is greatly facilitated by the fact that all the considered +surfaces are blowing-up and down of the projective plane: as explained in Hartshorne’s classic [Har77, Chap V, +beginning of §4 & Remark 4.8.1], we are brought back to the study of some sub-linear systems of plane curves. +We choose examples that illustrate the variety of situations that may occur. In the column CaCl(Xs) ֒→ Cl(X) +in the tabular below, we see that the Cartier class group CaCl(Xs) always embeds in the Weil class group Cl(Xs), +and via this embedding CaCl(Xs) may be equal to Cl(Xs), or of finite index into Cl(Xs), or of positive co-rank +into Cl(Xs). Note also that the lattice CaCl(Xs) is always free, whereas Cl(Xs) may have a torsion subgroup. In +the column Nq (−KXs), one can see that this is not always the absolutely irreducible curves of the linear system +that contains the maximum of rational points. Only a case-by-case proof and a carefully study of all the geometric +properties permits to estimate the three invariants [n, k, dmin] that are contained in the last column. +Deg. +Sing. +CaCl(Xs) ֒→ Cl(X) +Nq (−KXs) +[n, k, dmin] +§4.2 +6 +A1 +2Z ֒→ Z +2q + 1 +� +q2 + 1, 7, q2 − 2q +� +§4.3 +5 +2A1 +Z ⊕ 2Z ֒→ Z ⊕ Z +2q + 2 +� +q2 + q + 1, 6, q2 − q − 1 +� +§4.4 +4 +A1 +Z ≃ Z +≤ Nq(1) +� +q2 − q + 1, 5, ≥ q2 − q + 1 − Nq(1) +� +§4.5 +4 +4A1 +Z ֒→ Z ⊕ Z/2Z +≤ Nq(1) +� +q2 + 1, 5, ≥ q2 + 1 − Nq(1) +� +§4.6 +4 +A2 +Z ≃ Z +≤ Nq(1) +� +q2 + 1, 5, ≥ q2 + 1 − Nq(1) +� +§4.7 +4 +D5 +4Z ֒→ Z +2q + 1 +� +q2 + q + 1, 5, q2 − q +� +§4.8 +3 +A1 +Z ≃ Z +≤ Nq(1) +� +q2 + 1, 4, ≥ q2 + 1 − Nq(1) +� +§4.9 +3 +3A2 +Z ֒→ Z ⊕ Z/3Z +≤ Nq(1) +� +q2 + q + 1, 4, ≥ q2 + q + 1 − Nq(1) +� +In the tabular above, the inequality Nq (−KXs) ≤ Nq(1) means that the curves of the linear system |−KXs| that +contain the maximum number of points are the absolutely irreducible ones. They are all of arithmetic genus 1, +but it may happen that none of these curves is maximal (i.e. has a number of rational points equal to Nq(1)). +This is why in these cases, one can only give an upper bound for Nq (−KXs) and thus a lower bound for dmin. +It turns out that we recover two examples of Koshelev [Kos20] (§4.5 and 4.9), where he proved that the linear +systems cannot contain a maximal genus one curve for certain finite fields. This permits to increase the lower +bound of the minimum distance by one over these fields. +All the presented codes can be easily constructed using a mathematics software system. On the second author’s +webpage, we put a magma program that permits to construct all our codes. +2 +Generalities on weak del Pezzo surfaces +Let k be a finite field (most of the results remain true on any field), k its algebraic closure, Γ = Gal(k/k) its +absolute Galois group and let σ be the Frobenius automorphism. +In this section, we recall the classical properties of del Pezzo surfaces. In particular, we focus on the specificities +of non ordinary weak del Pezzo surfaces compared to the ordinary ones. The essential references for the content +of this section are the book of Manin [Man74] or the more recent one of Dolgachev [Dol12, §8]. +2.1 +Ordinary versus non ordinary weak del Pezzo surfaces +There are several definitions of a del Pezzo surfaces, even in the Dolgachev’s classic [Dol12]; let us start with the +definition 8.1.18 of this book. +Definition 2.1. A smooth projective surface X is a weak del Pezzo surface if its anticanonical divisor −KX +is: +(i) big, which means that K·2 +X > 0, +(ii) and nef, which means that (−KX) · D ≥ 0 for any effective divisor D on X. +3 + +The self-intersection K·2 +X is the degree of the del Pezzo surface X. +Thanks to the Nakai-Moishezon criterion [Har77, Chap V, Theorem 1.10], these kinds of surfaces are divided +into two cases: +(i) either the inequalities in (ii) are all strict ((−KX) · D > 0): the anticanonical divisor is thus ample and we +say that the del Pezzo surface is an ordinary one; +(ii) or there exists an effective divisor D such that (−KX) · D = 0: the anticanonical divisor is not ample and +we say that the del Pezzo surface is a non ordinary one. +These properties have consequences on the negative curves on X, those whose self-intersection is negative. +Indeed, let C be an absolutely irreducible curve on X of arithmetic genus γ(C). By adjunction formula, we know +that C·2 = 2γ(C)−2+C·(−KX) and since γ(C) ≥ 0 and C·(−KX) ≥ 0, we deduce that C·2 ≥ −2. Thus, negative +curves on X have self-intersection −2 or −1. Moreover C·2 = −2 if and only if γ(C) = 0 and C · (−KX) = 0; +this means that only non ordinary del Pezzo surfaces can contain (−2)-curves. We also prove the same way that +(−1)-curves on weak del Pezzo surfaces must have arithmetic genus equal to 0. This motivates the following +definition which deals with negative curves but also divisor classes of curves. +Definition 2.2. Let X be a weak del Pezzo surface over a field k, let X = X ⊗ k be its extension to the algebraic +closure k and let Cl(X) denote the divisor class group of X. +(i) A divisor class D ∈ Cl(X) is an exceptional class if D·2 = D·KX = −1; an absolutely irreducible curve C +on X whose class is exceptional is an exceptional curve. +(ii) A divisor class D ∈ Cl(X) is a root if D·2 = −2 and D · KX = 0; a curve C on X whose class is a root is +an effective root and if such a curve is absolutely irreducible then C is called a (−2)-curve. +It is well known that the geometry of weak del Pezzo surfaces depends to a large extent of these negative +curves. For example, if X is a weak ordinary del Pezzo surface then all the exceptional classes are the classes +of a (unique) exceptional curve and no root is effective. On the contrary, if X is weak non ordinary del Pezzo +surface then some exceptional classes may be represented by reducible curves and some roots are effective. These +differences of behaviours appear naturally in the blowup description of the generalized del Pezzo surfaces. +2.2 +The blow-up model +Over k, every del Pezzo surface can be obtained by a sequence of blowing ups starting from the projective plane P2. +This description makes most of the invariants of the surface very explicit. +Recall that if π : Y → X is the blowing up of a smooth surface X at a point p, with exceptional divisor E, +then Cl(Y ) = π∗ Cl(X)⊕ZE, the intersection pairing on Y satisfying E2 = −1, π∗D·E = 0 and π∗D·π∗D′ = D·D′ +for all divisors D and D′ of X (the blowing up is an isometry for the intersections pairings). Moreover KY = +π∗KX + E. +We recall that r ≤ 8 points in P2(k) are said to be +• in general position if and only if no three lie on a line, no six lie on a conic, and there is no cubic through +seven of them having a singular point at the eighth; +• in almost general position if and only if no four lie on a line and no seven lie on a conic. +Del Pezzo surfaces can always be described as follows [Dol12, Th 8.1.15]. +Theorem 2.3. Let X be a generalized del Pezzo surface over k and let X = X ⊗k k its extension to k. If X is of +degree d, with 3 ≤ d ≤ 6, then X is the blowing up of P2 at r = 9 − d points p1, . . . , pr in almost general position; +more precisely X results in r successive blowups π1, . . . , πr: +X +πr +−→ Xr −→ · · · −→ X2 +π1 +−→ X1 := P2 +k +where pi ∈ Xi are in almost general position. +Let E0 be the class of a line in P2 and let E1, . . . , Er be the exceptional curves at each stage. Then the divisor +class group of X with its intersection pairing can be easily described: +Cl(X) = ZE0 ⊕ ZE1 ⊕ · · · ⊕ ZEr +Mat ( · , (Ei)0≤i≤r) = Diag(1, −1, . . ., −1), +where Diag denotes the diagonal matrix. This also gives explicitly the canonical class: +KX = −3E0 + +r +� +i=1 +Ei. +(1) +The negative classes can be expressed in terms of the basis E0, E1, . . . , Er [Dol12, §8.2]. +4 + +Name +Exceptional classes E +Roots R +Conditions +E·2 = −1 and E · (−KX) = 1 +R·2 = −2 and R · KX = 0 +Expression +Ei, +i ∈ {1, . . ., r} +Rij = Ei − Ej, +{i, j} ⊂ {1, . . ., r} +in terms of the Ei +Eij = E0 − Ei − Ej, {i, j} ⊂ {1, . . ., r} +Rijk = E0 − Ei − Ej − Ek, −Rijk +Ei1···i5 = 2E0 − �5 +j=1 Eij +{i, j, k} ⊂ {1, . . ., r} +{i1, i2, i3, i4, i5} ⊂ {1, . . ., r} +Ri1···i6 = 2E0 − �6 +j=1 Eij, −Ri1···i6 +{i1, i2, i3, i4, i5, i6} ⊂ {1, . . ., r} +We note that, in the notation Eij, the indices are unordered (which leads to +�r +2 +� +possibilities), whereas they +are ordered in the notation Rij since Rji = −Rij (which leads to 2 +�r +2 +� +possibilities). +Not all these divisor classes are effective and the effectiveness of certain of these classes differentiate some +types of Del Pezzo surface. +• In the ordinary case, each exceptional class of divisor is represented by a unique irreducible curve. Either it +is one exceptional curve Ei for some 1 ≤ i ≤ r or the strict transform of the line of P2 passing through pi and pj +for the class Eij or the strict transform of the (unique) conic of P2 passing through the pi1, . . . , pi5 for Ei1i2i3i4i5. +Their intersection graph is an important invariant of the ordinary Del Pezzo surfaces; figures of these graphs +for 3 ≤ r ≤ 5 can be found in Manin [Man74, §26.9] or in Dolgachev [Dol12, §8.6.3, Figure 8.5]. As for the root +classes, no one is effective. +• In the non ordinary cases, where the points are no longer in general position but only in almost general +position, the exceptional divisors are still effective but not necessarily represented by irreducible curves anymore. +For example, if p1, p2, p3 are collinear then the root R123 becomes effective since it is the class of the strict +transform of the line passing through p1, p2, p3 and the four exceptional classes E12, E13, E23, E12345 are represented +by reducible curves since +E12 = R123 + E3, +E13 = R123 + E2, +E23 = R123 + E1, +E12345 = R123 + E45. +In this case, all other exceptional divisors are still represented by irreducible curves. +Another simple example: if p2 is chosen to be on E1, p2 ≻ p1, then the root E1 − E2 becomes effective since it +is the strict transform of E1. The exceptional classes E1 and E1j j ̸= 1, 2 are no longer represented by irreducible +curves. +In general, a result of Demazure states that exceptional divisors that are represented by irreducible curves are +characterized by the fact that they intersect non negatively (≥ 0) all the irreducible roots [CT88, Proposition 5.5]. +Another general result states that the set of irreducible roots (the effective classes represented by an irreducible +curve) is necessarily a free family in Cl(X⊗Fq) (see loc. cit.). In particular, there are at most r effective irreducible +roots. The lattice generated by the effective roots plays a crucial role. +Definition 2.4. Let X be a generalized del Pezzo surface over k and let X = X ⊗k k be its extension to k. +We denote by R the sub-lattice of Cl(X) generated by the effective roots and by R sub-lattice of Cl(X) defined +by R = R +Γ. +Following Coray and Tsfasman (see loc. cit.), an important invariant of a weak Del Pezzo surface is the graph +of negative curves, which is an analog of the intersection graph of the exceptional divisors/curves introduced above +in the ordinary case. To take into account the fact that the surface may be non ordinary, the set of vertices is +modified: the vertices corresponding to reducible exceptional divisors are cancelled, while vertices corresponding +to effective and irreducible roots are added. +2.3 +The anticanonical model Xs +2.3.1 +The morphism induced by the anticanonical divisor −KX +Let X be a del Pezzo surface of degree d, with 3 ≤ d ≤ 6 and whose canonical divisor is denoted by KX. In the +ordinary case, the anticanonical divisor −KX is known to be very ample and it induces a projective embedding +of X into Pd = P(H0(X, −KX)) (see §3.1 for a review about the space of global sections of a divisor) In the non +ordinary case, the anticanonical class −KX is no longer ample but its linear system remains base point free and +gives a morphism from X to a projective space: +Definition 2.5. Let X be a weak del Pezzo surface of degree d, with 3 ≤ d ≤ 6. The image ϕ(X), where ϕ : +X → P +� +H0(X, −KX) +� += Pd is the projective morphism associated to the anticanonical divisor −KX is called +the anticanonical model of X and is denoted by Xs. We put KXs = ϕ∗(KX). +This kind of del Pezzo surface corresponds to the definition 8.1.5 in Dolgachev [Dol12]. The fifth talk of +Demazure [Dem80, Expos´e V] on del Pezzo surfaces contains all the main properties of this anticanonical model. +Proposition 2.6. The morphism ϕ satisfies: +5 + +(i) it is not a projective embedding (since −KX is not ample) but the image Xs is a normal surface whose +singularities are rational double points; +(ii) it is the minimal desingularization of Xs, it contracts all the irreducible effective roots on X into the singular +points and nothing else; +(iii) the Weil divisor KXs = ϕ∗(KX) is a Cartier divisor of Xs which satisfies ϕ∗(KXs) = KX; +For each singularity, the exceptional divisor of its minimal resolution is a sum of irreducible effective roots (Ri) +with Ri · Rj ∈ {0, 1} for any i ̸= j. As usual in the ADE classification of rational double points, we describe the +type of a singularity by its dual graph: its vertices correspond to the above roots, and there is an edge between +the two vertices when the corresponding roots meet. In the examples below, the types of rational double points +that appear correspond to the graphs: +• +A1 +• +• +A2 +• +• +D5 +• +• +• +As mentioned in the last item, since the anticanonical model of a non ordinary weak del Pezzo surface is not +smooth but only normal, a Weil divisor may not be Cartier and the class groups of Cartier or Weil divisor may +differ. +2.3.2 +Cartier versus Weil divisors and class groups +Let X be a normal surface; let k(X) be its field of rational functions and OX its structural sheaf. We need to +review some general facts about divisors in such surfaces (see Liu [Liu02, §7.1 & 7.2] for more details). +• A prime Weil divisor X is a prime closed sub-variety of codimension 1 and the group of Weil divi- +sors WDiv(X) is the free abelian group generated by prime Weil divisors. A Weil divisor D can be written � +i niCi +where the Ci’s are irreducible curves on X and where the ni are integers of which only a finite number are non +zero. Such a divisor is said effective if ni ≥ 0 for all i. Since X is normal, it is regular in codimension 1 and +to each rational function f ∈ k(X), one can associate a Weil divisor (f) which is called principal. The set of +principal divisors is a sub-group of WDiv(X). +• A Cartier divisor, or a locally principal divisor D is a global section of the sheaf k(X)×/O× +X; it consists +in a collection (Ui, fi)i∈I where (Ui)i∈I is an open covering of X and where the fi’s are rational functions such +that the quotients fi/fj have neither zeroes nor poles on Ui ∩ Uj, i.e. such that fi/fj ∈ O× +X(Ui ∩ Uj). Two +collections (Ui, fi)i∈I and (Vj, gj)j∈I represent the same Cartier divisor if on Ui ∩ Vj, the functions fi and gj +differ by a multiplicative factor in O× +X(Ui ∩ Vj) for every i, j. The set of Cartier divisors can be turned into an +abelian group which we denote by CDiv(X). A Cartier divisor is called effective if it can be represented by a +collection (Ui, fi) with fi ∈ OX(Ui) for every i. A principal Cartier divisor is represented by a collection (X, f), +where f ∈ k(X)×. The set of principal divisors is also a sub-group of CDiv(X). +• To each Cartier divisor D one can associate a Weil divisor and this correspondence induces a group ho- +morphism CDiv(X) → WDiv(X); since X is supposed to be normal, this morphism is injective [Liu02, Chap 7, +Prop 2.14] and it sends an effective Cartier divisor to an effective Weil one. +• The quotients of the divisor groups CDiv(X) and WDiv(X) by the principal divisors are denoted CaCl(X) +and Cl(X). The previous correspondence induces an injective homorphism CaCl(X) → Cl(X). +These are general facts, but in the context of weak del Pezzo surfaces, we are able to be much more explicit. +In particular, one can relate the two groups CaCl(Xs) and Cl(Xs) to the group Cl(X) = CaCl(X). +Proposition 2.7. Let X be a weak del Pezzo surface over k and let Xs be its anticanonical model. Over k, one +has the two exact sequences: +0 −→ R −→ Cl(X) −→ Cl(Xs) −→ 0 +=⇒ +Cl(Xs) = Cl(X)/R, +(2) +0 −→ CaCl(Xs) −→ Cl(X) −→ Hom +� +R, Z +� +=⇒ +CaCl(Xs) = R +⊥, +(3) +where the arrow Cl(X) → Hom +� +R, Z +� +is given by D �→ [R �→ D · R], and where R +⊥ = {D ∈ Cl(X) | D · R = +0, ∀R ∈ R}. Over k, one has Cl(X) = CaCl(X) = CaCl(X)Γ = Cl(X)Γ for X, and for its anticanonical model: +0 −→ R −→ Cl(X) −→ Cl(Xs) −→ 0 +=⇒ +Cl(Xs) = Cl(X)/R, +(4) +0 −→ CaCl(Xs) −→ Cl(X) −→ Hom +� +R, Z +�Γ +=⇒ +CaCl(Xs) = R⊥ +(5) +Moreover we have an isomorphism Cl(Xs) ≃ Cl(Xs)Γ. +6 + +Proof. Let R be the union of effective roots in X and let U = X \ R be the open complementary. By a result of +Hartshorne [Har77, Chap II, Prop 6.5], we have the exact sequence: +0 −→ R −→ Cl(X) −→ Cl(U) −→ 0. +Let Us be the smooth locus of Xs. This open set is of codimension 2 in Xs and thus Cl(Xs) ≃ Cl(Us) (see loc. +cit.). Since the anticanonical map ϕ−KX induces an isomorphism from U to Us one has Cl(Us) ≃ Cl(U) and the +sequence (4) follows. Sequence (2) also follows by extending scalars to k. +On the other hand, we note that the module R being induced [Man74, Chap IV,§29], we know that H1(Γ, R) = +0; thus taking the Galois invariants of (2) leads to: +0 −→ R −→ Cl(X)Γ −→ Cl(Xs)Γ −→ 0 +Now X is smooth and we have Cl(X) = Cl(X)Γ [Sta18, Tag 0CDS]; we deduce the last isomorphism Cl(Xs) ≃ +Cl(Xs)Γ. +The exact sequence (3) comes from Bright [Bri13, Prop 1]. We deduce the equality CaCl(Xs) = R +⊥, and +from [Sta18, Tag 0CDS], we deduce that CaCl(Xs) = CaCl(Xs)Γ = (R +⊥)Γ. +Finally, taking the Galois invariants in the sequence (3) gives the exact sequence in (5). +Now since the +intersection product is invariant under the Galois action, a divisor in Cl(X) is orthogonal to R if and only if it is +orthogonal to R, and we get the isomorphism in (5). +2.3.3 +Lattice computations +One of the key step of the study of codes from weak del Pezzo surfaces is the explicit computation of the divisor +class groups as in (2) and (3). Such computations take place in the group Cl(X), which is known to be a free Z- +module of finite type endowed with the (non degenerate) intersection bilinear form and involve the root lattices R, +which is given by some explicit generators and which satisfies R ∩ R +⊥ = {0} (the orthogonal is relative to the +intersection pairing). +This is a general issue and let us consider C (for the “class group”) a free Z-module of finite rank with a non +degenerate symmetric bilinear form (x, y) �→ x · y (for the intersection product). Recall that a submodule M of C +is a direct summand (or is complemented) if there exists a submodule N of C such that C = M ⊕ N; in this case, +the submodules M and N are called complementary submodules of C [AW92, §3.8,§6.1]. Let R be a submodule +of C such that R ∩ R⊥ = {0} (for the root lattice). +In the context of modules over a principal ideal domain, contrary to what is happening in vector spaces over +a field, even if R ∩ R⊥ = {0}, the orthogonal submodules R and R⊥ may not be complementary submodules. +There are at least two different kinds of obstructions for this. Either the submodule R is not a direct summand +or both submodules R and R⊥ are direct summands but they are not complementary submodules. +In any case the smallest submodule containing R which is a direct summand is called the hull of R and is +denoted R♯. As for the submodule R⊥, since it is the kernel of a morphism of free modules, it is always a direct +summand. In the same way, even though R♯ and R⊥ are direct summand, they may or may not be complementary +submodules. These phenomenes make the description of the exact sequence: +0 −→ R⊥ −→ C/R −→ C/R ⊕ R⊥ −→ 0 +a little bit tricky (ie the comparison between the groups of Weil classes and Cartier classes). The main tool for +the explicit computation of this sequence is the Invariant factor theorem for submodules that will be used twice +(see [AW92, Theorem 6.23]). +• First, we apply this result to the submodule R ⊂ C: there exists a Z-basis e1, . . . , en of C and α1 | · · · | αr +(r ≤ n) a sequence of positive integers, called invariant factors, such that R = Zα1e1 ⊕ · · · ⊕ Zαrer and R♯ = +Ze1 ⊕ · · · ⊕ Zer. The submodule R is a direct summand of C if and only if R♯ = R, if and only if the invariant +factors α1, . . . , αr are all equal to 1. +Put M = Zer+1 ⊕ · · · ⊕ Zen then R♯ and M are complementary submodules, C = R♯ ⊕ M, and the projec- +tions ιtors and ι onto each factors lead to an isomorphism +C/R +≃ +−→ +R♯/R +⊕ +M +x mod R +�−→ +ιtors(x) mod R ++ +ι(x) +In other words, the projection ιtors gives an isomorphism from the torsion submodule of C/R to the quotient +module R♯/R which is isomorphic to Z/α1Z × · · · × Z/αrZ; the projection ι gives an isomorphism from the +torsion-free submodule of C/R to the submodule M of C. +• Since R ∩ R⊥ = {0}, we know that R⊥ canonically embeds in the quotient C/R; being free of torsion it +is a submodule of the torsion-free submodule of C/R. Via ι it thus embeds in M. Using the Invariant factor +theorem again, one can choose the basis er+1, . . . , en of M, in such a way that there exists βr+1 | · · · | βn such +7 + +that ι(R⊥) = Zβr+1er+1 ⊕ · · · ⊕ Zβnen. The cokernel C/R ⊕ R⊥ is then isomorphic to Z/βr+1Z × · · · × Z/βnZ. +In particular, the canonical embedding of R⊥ inside M induces an isomorphism if and only if βr+1, . . . , βn are +all equal to 1. +In the sequel, we do not give names to the projection morphisms ιtors, ι. +3 +Codes from surfaces: construction and tools for their study +In this section k is a finite field Fq. +3.1 +Evaluation codes from surfaces +Cartier divisors, their spaces of global sections, and the associated complete linear systems are the main ingredients +to define and to characterize the parameters of the evaluation codes from an algebraic surfaces. Let us recall the +definitions and the basic facts concerning these objects. We consider X a (not necessarily smooth, but in fact +at least normal here) irreducible surface over k. We denote by k(X) its function field and by OX its structural +sheaf. Let D = (Ui, fi)i∈I be a Cartier divisor on this surface X. +A global section of D is a function s ∈ k(X) such that for every i ∈ I, the product sfi is regular on Ui, that +is sfi ∈ OX(Ui) for all i ∈ I. We denote by H0(X, D) the set of these sections; this is a vector space which is +known to have finite dimension. +By definition, if s ∈ H0(X, D) is a global section of D then the Cartier divisor (Ui, sfi)i∈I is effective. It can +be shown that two global sections of D lead to the same effective Cartier divisor if and only if they differ by a non +zero constant. This means that there is a one-to-one correspondence between the projective space P(H0(X, D)) +and the set of effective Cartier divisors linearly equivalent to D. This last set is called the complete linear system +associated to the divisor D and is currently denoted by |D|, so we have |D| = P(H0(X, D)). An important +invariant of the divisor (or linear system) for our purpose is the maximum of rational points that can contain a +curve of |D|; we put: +Nq(D) = max{#C(Fq) | C ∈ |D|}. +(6) +Definition 3.1. Let X be a (not necessarily smooth) irreducible surface over k, let D be a Cartier divisor of X +and let P = {p1, . . . , pn} be a set of rational points of X. The evaluation code CX(D, P) is the image of the +evaluation map +H0(X, D) +−→ +kn +s +�−→ +(sfip)(p) +where for each point p, the index ip is chosen in such a way that p ∈ Uip. +In the preceding definition, the choice of ip may be not unique but different choices ip, jp of these indices lead +to homothetic codes since the quotients fip/fjp are non vanishing regular functions on Uip ∩ Ujp. +The usual parameters of the evaluation code are related with some invariants of the surface. +Proposition 3.2. If the evaluation map is injective, a CX(D, X(Fq)) has +(i) length equal to #X(Fq) the number of rational points of X, +(ii) dimension equal to the dimension of the space H0(X, D) of global sections, +(iii) minimum distance bounded below by n − Nq(D), where Nq(D) is defined in (6). +Thanks to this proposition, it is worth noticing that bounding below the minimum distance of an evaluation +code CX(D, X(k)) from a surface X reduces to bounding above Nq(D) the number of points of the curves of the +linear system |D| associated to the divisor D. The fewer the number of rational points of the curves in the linear +system |D|, the higher the minimum distance. +3.2 +Blowing up, divisors and (non) complete linear systems +One of the key tools of the construction of the codes from Del Pezzo surfaces are the blowing-up or the blowing +down depending on the sense of the arrows. +Let π : Y → X be a sequence of blowing ups where all the surfaces involved are supposed to be smooth, +except the last one X which is only supposed to be normal. Such a morphism leads to two natural maps involving +different kinds of divisors and divisor class groups. +• First, starting from a Cartier divisor of X, the pullback π∗D is the Cartier divisor on Y defined locally +by (π−1Ui, fi ◦ π). This lead to a morphism π∗ : CDiv(X) −→ CDiv(Y ). +• Secondly, it can be shown that if C irreducible effective Weil divisor of Y , then π(C) is either a point or an +irreducible effective Weil divisor of X. Then the map π∗ : WDiv(Y ) −→ WDiv(X) defined by π∗(C) = 0 if π(C) +is a point and π∗(C) = π(C) otherwise extends to a group homorphism [Liu02, Chap 9, Lem 2.10]. +8 + +• Moreover, for every Cartier divisor D of X, one has π∗ (π∗D) = D [Liu02, Chap 9, Prop 2.11] where π∗ is +applied to the Weil divisor of Y associated to the Cartier divisor π∗D. +• These two maps induce two homomorphisms π∗ : CaCl(X) → CaCl(Y ) [Liu02, Chap 7, Def 1.34] and π∗ : +Cl(Y ) → Cl(X) [Ful98, Chap 1, Th 1.4]. +• At the level of global sections and linear systems, the map π∗ also induces isomorphisms: +H0(X, D) +≃ +−→ +H0(Y, π∗D) +s +�−→ +s ◦ π +|D|X +≃ +−→ +|π∗D|Y +C +�−→ +π∗C +where D is a Cartier divisor on X ([Dem80, Expos´e V, Cor 2]). Since π∗ (π∗C) = C, the inverse of the right +isomorphism is nothing else than π∗ |π∗D|Y −→ |D|X. +We will need to describe the one-to-one correspondence |D|X −→ |π∗D|Y , when the right divisor is replaced +by a divisor of the form π∗D − E. Some natural sublinear systems appear. Let us go step by step. +• Let π : Y → X be the blowing-up of a smooth surface X at a point p ∈ X and let E be its exceptional divisor +on Y . Since the surfaces are supposed to be smooth, we do not have to distinguish Cartier and Weil divisors. +We mainly focus on effective divisors and we call them curves. Given C a curve on X, then the pullback π∗C is +called the total transform of C, the closure in Y of π−1(C \ {p}), denoted �C, is called the strict transform of C. +These two curves on Y are related by the relation: +π∗C = �C + mp(C)E, +where mp(C) denote the multiplicity of C at the point p. In particular, for any n ≥ 0, the divisor π∗C − nE is +effective if and only if mp(C) ≥ n. +This permits to relate the complete linear system |π∗D−nE| on Y to an uncomplete one on X, that is |D−np| +the space of curves of |D| which pass through p with multiplicity at least n. In fact this shows that the map C �→ +π∗C − nE leads to a one-to-one correspondence from |D − np| to |π∗D − nE| (the other way around, it says that +the blowing-up permits to turn uncomplete linear systems into complete ones). +• The same is true if we blow up several points. +Let π : Y → X be the blowing-up of a smooth sur- +face X at some points p1, . . . , pr, and let E1, . . . , Er be the exceptional divisors. +For D a divisor on X. +Let |D − n1p1 − · · · − nrpr| denotes the sub-linear system of the complete linear system |D| consisting of curves +of |D| which pass through p1, . . . , pr with multiplicities at least n1, . . . , nr. The blowing-up permits to turn this +incomplete linear system into a complete one: there is a one-to-one correspondence between ([Har77, loc. cit.], +[CA00]), +|D − n1p1 − · · · − nrpr| +−→ +|π∗D − n1E1 − · · · − nrEr| +C +�−→ +C♯ +where +C♯ def. += π∗C − n1E1 − · · · − nrEr. +(7) +This curve C♯ is sometime called the virtual transform of C. The total, strict and virtual transforms are thus +related by: +π∗C = �C + +r +� +i=1 +mpi(C)Ei = C♯ + +r +� +i=1 +niEi +=⇒ +C♯ = �C + +r +� +i=1 +(mpi(C) − ni) Ei. +In particular the virtual and the strict transforms coincide when mpi(C) = ni for all i. +• This one-to-one correspondence is still true if some points in the sequence of blowing ups are infinitely near +points, that is when some pj lies on the exceptional divisor of the blow up of another point pi. In order to describe +this, we need to carefully define the sub-linear system associated to a family of infinitely near points. Let us start +with only two points: if p1 ≺ p2, that is if p2 lies on the exceptional curve E1 above p1, then for n1, n2 > 0, the +sub-linear system of curves passing through p1 and p2 with multiplicities at least n1 and n2 is defined by: +|D − n1p1 − n2p2| +def. += {C ∈ |D − n1p1| , mp2(π∗(C) − n1E1) ≥ n2} = +� +C ∈ |D − n1p1| , mp2(C♯) ≥ n2 +� +. +In particular the sub-system |D − p1 − p2| contains all the curves of |D| that pass through p1 with tangent line at p1 +equal to p2 union all the curves of |D| singular at p1; indeed, in the last case C♯ = π∗(C)−E1 = �C+(mp1(C) − 1) E1 +has E1 as a component and thus passes through p2 (one can check that the conditions p1 ∈ C and p2 ∈ �C are not +linear, which is why we choose p2 ∈ C♯ instead). +In the same way, if p1 ≺ p2 ≺ · · · ≺ pr, one can define recursively, the sub-linear system |D − n1p1 − · · · − nrpr|. +With this definition, the one-to-one correspondence (7) is still true. +Let us end by an example: the case X = P2. If ℓ denotes the class a line, and if E0 is the pullback of ℓ in Y , then +the curves of the complete linear |dE0 −n1E1−· · ·−nrEr|Y on Y corresponds bijectively to |dℓ−n1p1−· · ·−nrpr| +the (projective) vector space consisting of plane curves of degree d passing through p1, . . . , pr with multiplicities +at least n1, . . . , nr. For small degrees, it turns out that the irreducible decompositions of such curves can be easily +described. +9 + +3.3 +Blowing up and evaluation codes +Let us return to codes and compare the evaluation codes CX(D, X(k)) and CY (π∗D, Y (k)). +Proposition 3.3. Let X be a normal surface, let p be a point of X and let π : Y → X be the blowing-up of X +at p. We denote by E the divisor sum of the exceptional curves. +(i) If p is of degree > 1, then the codes CX(D, X(k)) and CY (π∗D, Y (k)) are equivalent; moreover the code CY (π∗D− +nE, Y (k)) can be identified with the sub-code of CX(D, X(k)) where only the global section having multiplicity +at least n at p are evaluated. +(ii) If p is rational, then the code CY (π∗D − nE, Y (k)) can be identified with the sub-code of CX(D, X(k) \ {p}) +where only the global sections having multiplicity at least n at p are evaluated and to which we add the +following (q + 1) coordinates: the evaluations at rational points of P1 of the homogeneous component of +degree n of the local equation at p of the section. +Proof. (i) The map s �→ s◦π is a one-to-one correspondence from the spaces of functions H0(X, D) to H0(Y, π∗D). +Since the blown points are not rational, the map π induces a one-to-one correspondence from Y (k) to X(k). Thus +the codes CX(D, X(k)) and CY (π∗D, Y (k)) must be equivalent. By the previous correspondence the global sections +of H0(Y, π∗D − nE) are in bijection with the global sections of H0(X, D) that pass through p with multiplicity +at least n and the last statement follows. +(ii) The set Y (k) is in one-to-one correspondence with (X(k) \ {p}) ∪ E(k) and we only have to compute +the evaluations at the points of E(k). We choose an open neighbourhood U ⊂ A2 +(x,y) of p in which p = (0, 0) +and D has local equation f(x, y) = 0. Then π−1(U) ⊂ U × P1 +(u:v) with equation xv = yu; there are two affine +charts, π−1(U) = V1 ∪ V2, with V1 ⊂ A2 +(y,u) (resp. V2 ⊂ A2 +(x,v)) with π(y, u) = (yu, y) (resp. π(x, v) = (x, xv)). +On V1, the divisor π∗D − nE has local equation f◦π +yn = f(yu,y) +yn +. Let s◦ π ∈ H0(Y, π∗D − nE) then sf ∈ OX(U) has +multiplicity at least n at p, that is sf(x, y) = pn(x, y) + pn+1(x, y) + · · · , where pn is homogeneous of degree n. +Thus sf◦π +yn += pn(yu,y)+pn+1(yu,y)+··· +yn += pn(u, 1) + yq(u, y). Evaluating at the point (0, u) ∈ E ∩ V1, the section- has +value pn(u, 1). The same is true on V2. +The examples below provide many examples of this blowing-up operation, especially the one in section 4.7. +4 +Anticanonical codes from weak del Pezzo surfaces +In this section we describe some evaluation codes from weak del Pezzo surfaces, we compute their parameters +and for some of them a generator matrix. The base field is a finite field Fq without any other hypothesis excepts +sporadically not being too small (F2 or F3). +In the first subsection, the general construction is given. We also fix many notations that will be used until +the end of the paper. +4.1 +General description of the codes and of the main steps of their studies +The evaluation codes (definition 3.1) studied in the sequel are the ones corresponding to the following choices. +Definition 4.1. Let X be a weak del Pezzo surface over Fq. We call anticanonical code associated to X +the evaluation code CXs (−KXs, Xs(Fq)), where Xs is the anticanonical model of X, −KXs is the anticanonical +(Cartier) divisor on Xs, and where Xs(Fq) denotes the set of rational points of Xs. +Note that we could have considered the evaluation codes CX(−KX, X(Fq)) with the same del Pezzo surfaces, +but this leads to worth codes. +In a concomitant work [BH22], we have computed explicit models for all the arithmetic types of del Pezzo +surfaces over a finite field (these types lead to a classification that is coarser than the isomorphism one but that +permit to distinguish the main arithmetic properties of the weak del Pezzo surfaces). Taking advantage of this +knowledge, we select eight types of weak del Pezzo that are well suited for coding applications. For each example, +our starting point is a blowing-up model of the weak del Pezzo surface, then we study the parameters length, +dimension, minimum distance ([n, k, dmin]q) of the associated anticanonical code and last we give a generator +matrix (or a program to compute it). +Configuration to blow-up. — +The explicit description of the surfaces X and Xs always starts from the +projective plane P2: we first blow up a family of (possibly infinitely near) points p1, . . . , pr to obtain a smooth +surface Y ; then we may blow down a family of (non intersecting) exceptional curves on Y to obtain the smooth +10 + +surface X. Last X is mapped to a projective space corresponding to the anticanoncial divisor to lead to the +singular surface Xs. To sum up, we have the following diagram: +Y +X +P2 +Xs ⊂ Pdeg(X) +π +χ +ϕ +ε +π +is a sequence of blowing ups at points p1, . . . , pr, +χ +is a sequence of contractions of +(−1)-curves F1, . . . , Fs, +ϕ +is the morphism ϕ−KX associated to +the anticanonical divisor −KX of X, +deg(X) +is the degree of the del Pezzo surface X, i.e. K·2 +X. +(8) +All the surfaces and maps are defined over the base field Fq. The solid arrows π, χ, ϕ denote maps that are +morphisms whereas the dashed arrow ε denotes a map which is a rational one. +The need to introduce the +auxiliary surface Y is due to the fact that some times, the surface X we want to work with cannot be constructed +directly by blowing up the plane at some points. Some contractions may be necessary in order to work with +applications that are defined over Fq (and not only over Fq); however this detour is not always useful and in some +examples, one has X = Y and the map χ is only the identity. +Two of the parameters [n, k, dmin]q of the associated anticanonical code are easy to compute. +• The length is nothing else than #Xs(Fq). Following the process of blowing ups and down above, it is not +difficult to compute this number since blowing up a point adds q rational points or does not change the +number of rational points depending on whether the point is rational or not. +• The dimension is nothing else than d + 1, where d is the degree of the del Pezzo surface X, unless the +evaluation map is not injective. This can only occur if #Xs(Fq) ≤ Nq (−KXs) and we compute last number +to estimate the minimum distance. It turns out that the evaluation map is always injective except if the +base field is F2 or F3 in some cases that are excluded. +As usual, the last parameter, the minimum distance, requires much more preparatory works. +Computation of the divisor class groups. — +For these computations, the general ambient space is the +geometric divisor class group of Y , which is known to be equal to Cl(Y ) = ZE0 ⊕ ZE1 ⊕ · · · ⊕ ZEr, where, as +usual, Ei denotes the exceptional curve above pi in the sequence of blowing ups π. In this lattice, one can easily +identify the effective roots in Y , but also in X and we are able to give a basis of the sub-lattice R generated by +the effective roots of X over Fq. The other (geometric) Cartier and Weil divisor class groups are then given by: +Cl(X) = (ZF1 ⊕ · · · ⊕ ZFr)⊥ , +CaCl(Xs) = R +⊥, +Cl(Xs) = Cl(X)/R. +(the left orthogonal is computed in the whole Cl(Y ), the middle one in the sub-lattice Cl(X)). Using tools of +section 2.3.3, explicit bases and canonical embeddings of these geometric divisor class groups can be computed. +Taking into account the Galois action, one can also give bases and explicit canonical embedding bases of all the +arithmetic divisor class groups. Depending on the examples, the computations are carried out in the geometric +groups Cl(X) and the Galois invariants are taken in the last step to return in Cl(X) or we start to compute the +Galois invariants and then perform all the computations in Cl(X). Thanks to Proposition 2.7, these two ways +lead to the same results. +Types of decomposition into irreducible components in |−KXs|. — +The minimum distance is related +to the maximum number of rational points that can contain a (effective) curve in the linear system |−KXs|. To +bound above this number of rational points, one way is to study how the curves in this linear system decompose +into irreducible components and use the exact number of points if known or the Weil bound if not on each +components. Thanks to section 3.2, and since ϕ∗KXs = KX, we have the following one-to-one correspondences: +|−χ∗KX|Y +≃ +−→ +|−KX|X +≃ +−→ +|−KXs|Xs +C +�−→ +χ∗(C) +�−→ +ϕ∗ (χ∗(C)) +The first arrow consists in contracting the family of non-meeting exceptional curves Fi, 1 ≤ i ≤ s, the second in +contracting the effective roots of X. Thus we are reduced to study the types of decompositions into irreducible +components on the smooth surface Y , which is easier. Indeed, we know that −χ∗KX = dE0 − �r +i=1 niEi for +some explicit d and ni’s; in fact, in all examples, d ∈ {3, 4} and ni ∈ {1, 2}. Since Y is the blowing up of P2 at a +family of points, thanks to section 3.2, curves of |−KY | are in one-to-one correspondence to the plane curves of +a well specified (non complete) linear system of P2: +|dℓ − n1p1 − · · · − nrpr| +≃ +−→ +|−χ∗KX| +C +�−→ +C♯ +11 + +We are thus reduced to list all the types of decompositions into irreducible components of the plane curves of +degree d passing through pi with multiplicity ni. Since d ≤ 4, these absolutely irreducible components must be +plane lines, conics, cubics or quartics and an enumeration case by case can be done. More specifically, we follow +the steps: +• degree by degree, we list all the possible absolutely irreducible curves that pass through some of the points pi; +• we compute their Galois-orbits since if an absolutely irreducible component not defined over Fq appears in +the decomposition with multiplicity m, then the same holds for all its conjugates (this permits to get rid of +many curves because of their too high degree); +• we combine all these irreducible curves to obtain plane curves in the expected sub-linear system. +In order to make easier this step, we adopt the following notations and conventions. The letters ℓ, q, c, t respectively +denote plane lines, quadrics (or conics), cubics and quartics. The indices below these letters are the numbers +of the points through which the curve passes. For example, ℓ1 denotes a line that passes through p1 (but not +through any other point), ℓ123 a line that passes through p1, p2, p3 (if it exists), q123456 a conic passing through +the six points p1, . . . , p6 and ℓ or q a line or quadric that do not pass through any pi. The goal is then to combine +all these irreducible plane curves to obtain a curve in the expected linear system. +At the end of this step, we are able to compute the maximum Nq (−KXs) to which the minimum distance is +related (proposition 3.2). Comparing with the number #Xs(Fq), this also permits us to exclude some too small +values of q for which the evaluation map may fail to be injective. +Computation of the global sections from P2. — +Last, if we want to explicitly compute a generator matrix +of the code, we need to exhibit a basis of the sub linear system |dℓ − n1p1 − · · · − nrpr|. Then, by construction +we know to which points of P2 these functions have to be evaluated; in some cases we also need to add some +extra evaluation points corresponding to points on some exceptional curves. In any cases, one can compute a +generator matrix. This last (concrete) description turns the code into a code close to a Reed-Muller one: the +space of polynomials to be evaluated has been restricted, some of the evaluation points have been deleted, some +others have been added. +If some readers want to use our code, we put on the second author’s webpage, a magma program that permits +to construct all the codes presented below. +4.2 +Degree 6, singularity of type A1 +This example corresponds to the type number 3 in degree 6 [BH22]. +Configuration to blow-up. — +We blow up P2 at three collinear points that are conjugate over Fq. +ℓ123 +•p1 +•p2 +•p3 +p2 = pσ +1, +p3 = pσ2 +1 +The resulting surface is a weak del Pezzo surface X whose anticanonical model is denoted Xs. It has a unique +singular point of type A1 which is necessarily rational. +Computation of the divisor class groups. — +Over Fq, one has +Cl(X) = ZE0 ⊕ ZE1 ⊕ ZE2 ⊕ ZE3 +and +− KX = 3E0 − E1 − E2 − E3. +There is a unique effective root, the strict transform of the line ℓ123 passing through the three points p1, p2, p3, +and its class is E0 − E1 − E2 − E3. Then +R = Z(E0 − E1 − E2 − E3) +R +⊥ = {a0E0 + a1E1 + a2E2 + a3E3 | a0 + a1 + a2 + a3 = 0} += Z(E0 − E1) ⊕ Z(E0 − E2) ⊕ Z(E0 − E3) +Both R and R +⊥ are direct summand but R and R +⊥ are not complementary submodules since R ⊕ R +⊥ is of +index 2 in Cl(X). For a submodule complement to R, one can choose: +Cl(X) += +R +⊕ +(ZE1 ⊕ ZE2 ⊕ ZE3) +a0E0 + a1E1 + a2E2 + a3E3 += +a0(E0 − E1 − E2 − E3) ++ +(a1 + a0)E1 + (a2 + a0)E2 + (a3 + a0)E3 +This leads to the following isomorphism: +Cl(X)/R +≃ +−→ +ZE1 ⊕ ZE2 ⊕ ZE3 +a0E0 + a1E1 + a2E2 + a3E3 mod R +�−→ +(a0 + a1)E1 + (a0 + a2)E2 + (a0 + a3)E3. +12 + +Via this isomorphism, the submodule CaCl(Xs) = R +⊥ identifies with Z(E1 + E2) ⊕ Z(E2 + E3) ⊕ Z(E1 + E3) of +invariant factors 1, 1, 2 in ZE1 ⊕ ZE2 ⊕ ZE3. +Over Fq, to recover the class groups Cl(Xs) and CaCl(Xs), we only need to take the invariants under the +Galois action (E0)(E1E2E3), what is easy here. One has +CaCl(Xs) = CaCl(Xs)Γ = +� +R +⊥�Γ +≃ Z(3E0 − E1 − E2 − E3) = Z(−KX), +Cl(Xs) = Cl(Xs)Γ = +� +Cl(X)/R +�Γ ≃ Z(E1 + E2 + E3). +With these identifications, the canonical embedding of CaCl(Xs) into Cl(Xs) becomes: +0 +−→ +CaCl(Xs) +−→ +Cl(Xs) +−KX +�−→ +2(E1 + E2 + E3) +Thus both CaCl(Xs) and Cl(Xs) are free of rank 1, but CaCl(Xs) is of index 2 into Cl(Xs). This index has the +following consequence: even if CaCl(Xs) is free of rank one generated by −KXs, a Cartier divisor may decompose +into a sum of equivalent Weil irreducible divisors. This explains why we need to investigate how elements of |−KX| +can decompose into irreducible components and how the non ordinary weak del Pezzo surfaces we consider here +differ from ordinary ones (compare with [BCH+20]). +Types of decomposition into irreducible components in |−KXs|. — +In this example, there is no need +to introduce an auxiliary surface Y (one has Y = X and χ is the identity with the notation of the beginning of +this section). Since −KX = 3E0 − E1 − E2 − E3, the virtual transform composed with the push forward lead to +a one-to-one correspondence: +|3ℓ − p1 − p2 − p3| +−→ +|3E0 − E1 − E2 − E3| +−→ +|−KXs| +C +�−→ +C♯ +�−→ +ϕ∗(C♯) +(the left linear system is on P2, the middle one on X and the right one on Xs). Then we are reduced to list all the +types of decompositions into irreducible components of the curves of |3ℓ − p1 − p2 − p3|, the sub linear system of +cubics passing through the points p1, p2, p3. The orbits of lines, conics, cubics which have degree at most 3 and +pass through some of the points pi’s are +ℓ1 ∪ ℓ2 ∪ ℓ3, +ℓ123, +c123, +(of course, implicitly ℓ2 = ℓσ +1, ℓ3 = ℓσ2 +1 +where σ is a generator of Gal(Fq/Fq)). Indeed an absolutely irreducible +conic qi or qij cannot be defined over Fq and they have at least three conjugates; combining these curves with their +conjugates lead to plane curves of degree greater than 6 and thus they cannot appear in our case. A conic q123 +cannot be absolutely irreducible otherwise it would have three intersection points with the line ℓ123. +Let us +combine these rational irreducible decompositions in order to construct plane curves in the expected sub-linear +system. +First suppose that the decomposition contains a line. +• If this line is ℓ1, then its conjugates ℓ2, ℓ3 must also be geometric components; the only possibility is ℓ1∪ℓ2∪ℓ3 +(line 1 in the tabular below) which is an element of |3ℓ − p1 − p2 − p3|. +• A component ℓ cannot be completed by a conic passing through the three points and thus if there is a line +in the geometric decomposition, ℓ123 must be one of them. Since ℓ123 already passes through p1, p2, p3 it +can be completed by any conic (irreducible or not); this leads to the decompositions of the lines 3 to 7 in +the tabular below. +Last, if the decomposition does not contain any line, it must be an irreducible cubic which passes through the +13 + +three points; this cubic can be smooth or not and we recover the two last lines of the tabular. +|3ℓ − p1 − p2 − p3| +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +ℓ1 ∪ ℓ2 ∪ ℓ3 +�ℓ1 ∪ �ℓ2 ∪ �ℓ3 +�3 +i=1 ϕ∗(�ℓi) +1 +2 +ℓ123 ∪ q, ℓ123 ∩ q ̸⊂ P2(Fq) +�ℓ123 ∪ �q +ϕ∗(�q) +q + 2 +3 +ℓ123 ∪ q, ℓ123 ∩ q ⊂ P2(Fq) +�ℓ123 ∪ �q +ϕ∗(�q) +q +4 +ℓ123 ∪ ℓ ∪ ℓ′, ℓ123 ∩ ℓ ∩ ℓ′ ̸= ∅ +�ℓ123 ∪ �ℓ ∪ �ℓ′ +ϕ∗(�ℓ) ∪ ϕ∗(�ℓ′) +2q + 1 +5 +ℓ123 ∪ ℓ ∪ ℓ′, ℓ123 ∩ ℓ ∩ ℓ′ = ∅ +�ℓ123 ∪ �ℓ ∪ �ℓ′ +ϕ∗(�ℓ) ∪ ϕ∗(�ℓ′) +2q +6 +2ℓ123 ∪ ℓ +2�ℓ123 ∪ �ℓ ∪ �3 +i=1 Ei +ϕ∗(�ℓ) ∪ �3 +i=1 ϕ∗(Ei) +q + 1 +7 +3ℓ123 +3�ℓ123 ∪ �3 +i=1 2Ei +�3 +i=1 2ϕ∗(Ei) +1 +8 +c123, singular +�c123 +ϕ∗(�c123) +q + 2 +9 +c123, smooth +�c123 +ϕ∗(�c123) +Nq (1) +Some comments about the three first columns of the previous tabular. The unique irreducible effective root of X +is nothing else than the strict transform ℓ123 and this explains why this curve disappears in the third column: +this line on X is mapped by ϕ∗ to the unique singular point s ∈ Xs. Note also that except in the cases 6 and 7, +all the curves have exactly multiplicities 1 at the pi and thus their strict or virtual transforms are equal. On +the contrary, in the remaining cases, the curves on X are the virtual transforms of the ones on P2. Last, in +the decomposition ϕ∗(�ℓ) ∪ ϕ∗(�ℓ′), it is worth noticing that irreducible components involves divisors that are not +Cartier divisors but only Weil ones on Xs. Indeed the class of �ℓ in Cl(X) is E0, which is mapped to E1 + E2 + E3 +in Cl(Xs), which is not an element of CaCl(Xs) (equivalently E0 ̸∈ R⊥). +Now we make some comments on the numbers of rational points. +Case 1. Since the lines ℓ1, ℓ2, ℓ3 are conjugate a rational point on their union must be at their intersection +which contains at most one point. +On X the strict transforms �ℓ1, �ℓ2, �ℓ3 do not meet the root ℓ123 and the +contraction does not add any point. +Cases 2, 3, 4 & 5. If the two points of q ∩ ℓ123 are not rational then they are still unrational on �q ∩ �ℓ123 and +they are contracted to the singular point s in Xs and thus the image ϕ∗(�q) has one more rational point; otherwise, +if the two points of q ∩ ℓ123 are rational then they are contracted in Xs and thus the image ϕ∗(�q) looses a rational +point. The same is true on lines 4 and 5. +Cases 6 & 7. The line �ℓ123 is contracted by ϕ∗ and there are no rational points on the lines Ei. +Cases 8 & 9. The starting cubic c123 has (q + 1) or less than Nq (1) rational points depending on whether it +is singular or smooth. On Xs the number of rational points of ϕ∗(�c123) is increased by 1 since the line ℓ123 meets +the cubic at three conjugate points. The multiplicity of intersection of c123 and ℓ123 at each point pi is one (since +otherwise, these two curves would have too many intersection points counting with multiplicities). Therefore, the +blowing ups at p1, p2, p3 separate the strict transforms �ℓ123 and �c123. Thus �c123 and ϕ∗(�c123) are isomorphic and +have the same number of rational points. Finally, we remark that for every q, one has q + 1 + ⌊2√q⌋ ≤ 2q + 1 +(with equality if and only if q ∈ {2, 3, 4}) and thus: +Nq (−KXs) = 2q + 1. +Last we note that, except for the cases 1 and 9, all the maximum numbers of points are in fact exact numbers of +points. Thus we are not far from having the distribution of weights if the code. +Since p1, p2, p3 are not rational, the three blowing ups do not add any rational point and #X(Fq) = q2 +q +1. +Then, the root �ℓ123 is contracted via the anticanonical morphism and thus #Xs(Fq) = q2 + 1. Except if q = 2, +one has #Xs(Fq) > Nq(−KXs) and the evaluation map is injective. With this choice of weak del Pezzo surface, +the code of definition 4.1 satisfies the following proposition. +Proposition 4.2. Let p1, p2, p3 be conjugate collinear point in P2 +Fq, with q ̸= 2. The anticanonical code associated +to the weak del Pezzo surface obtained by blowing up these points has parameters [q2 + 1, 7, q2 − 2q]. +Computation of the global sections from P2. — +To construct this kind of codes, one can choose ℓ123 to +be the line of equation Y = 0 in P2. For any ζ ∈ Fq3 \ Fq, the point p1 = (ζ : 0 : 1) ∈ P2 is a degree 3 point +whose conjugates p2 = (ζσ : 0 : 1) and p3 = (ζσ2 : 0 : 1) are also in ℓ123. Let X3 + a2X2 + a1X + a0 ∈ Fq[X] be +the minimal polynomial of ζ over Fq. Then, we easily verify that +|3ℓ − p1 − p2 − p3| = +� +Y 3, Y 2X, Y 2Z, Y X2, Y Z2, Y XZ, X3 + a2X2Z + a1XZ2 + a0Z3� +Fq . +14 + +Last, the evaluation points are nothing else than the points of P2(Fq) \ ℓ123(Fq), plus one point of ℓ123(Fq) since +the strict transform of ℓ123 is contracted via the anticanonical morphism. Let us denote (xi : 1 : zi), 1 ≤ i ≤ q2, +the first q2 points, and let us choose (0 : 0 : 1) ∈ ℓ123(Fq), then the corresponding generator matrix of the code is: + + + + + + + + + + +1 +· · · +1 +0 +x1 +· · · +xq2 +0 +z1 +· · · +zq2 +0 +x2 +1 +· · · +x2 +q2 +0 +z2 +1 +· · · +z2 +q2 +0 +x1z1 +· · · +xq2zq2 +0 +P(x1, 1, z1) +· · · +P(xq2, 1, zq2) +a0 + + + + + + + + + + +, +P(X, Y, Z) = X3 + a2X2Z + a1XZ2 + a0Z3. +We recover the classical Reed-Muller code on A2 of degree 2, augmented by one point. +4.3 +Degree 5, singularity of type 2A1 +This example corresponds to the type number 5 in degree 5 [BH22]. +Configuration to blow-up. — +We blow up p1 ≺ p2 and p3 ≺ p4, where p1, p3 and p2, p4 are conjugate points +of degree 2. +ℓ12 +p2 +ℓ34 +p4 +• +p1 +• +p3 +ℓ13 +p3 = pσ +1 +p4 = pσ +2. +Since the points p2, p4 are infinitely near the points p1, p3, they are represented by tangent lines or directions on +the picture above. The anticanonical model Xs has two singular points of type A1 that are conjugate points. +Computation of the divisor class groups. — +Over Fq, one has: +Cl(X) = ZE0 ⊕ ZE1 ⊕ ZE3 ⊕ ZE2 ⊕ ZE4 +and +− KX = 3E0 − E1 − E2 − E3 − E4. +There are two conjugate effective roots, the strict transforms of E1 and E3 in the sequence of blowing ups; their +classes are E1 − E2 and E3 − E4 in such a way that: +R = Z(E1 − E2) ⊕ Z(E3 − E4), +R +⊥ = {a0E0 + a1E1 + a2E2 + a3E3 + a4E4 | a1 = a2, a3 = a4} += ZE0 ⊕ Z(E1 + E2) ⊕ Z(E3 + E4). +The sub-module R is a direct summand, and as a complementary sub-module one can choose: +Cl(X) += +R +⊕ +ZE0 ⊕ ZE2 ⊕ ZE4 +�4 +i=0 aiEi += +a1(E1 − E2) + a3(E3 − E4) ++ +a0E0 + (a1 + a2)E2 + (a3 + a4)E4. +We deduce the isomorphism: +Cl(Xs) ≃ Cl(X)/R +≃ +−→ +ZE0 ⊕ ZE2 ⊕ ZE4 +�4 +i=0 aiEi mod R +�−→ +a0E0 + (a1 + a2)E2 + (a3 + a4)E4 +. +Since CaCl(Xs) ≃ R +⊥, this class group is a rank 3 free sub-group of Cl(X). Via the previous isomorphism it is +mapped to the sub-group ZE0 ⊕ Z2E2 ⊕ Z2E4, of invariant factors 1, 2, 2. +The arithmetic groups CaCl(Xs) and Cl(Xs) can be computed by taking the invariants under the Galois action +which is (E0)(E1E3)(E2E4). Via the previous isomorphism, if we set E := E2 + E4, the canonical embedding 0 → +CaCl(Xs) → Cl(Xs) is only: +ZE0 ⊕ Z2E +� +�� +� +≃CaCl(Xs) +⊂ ZE0 ⊕ ZE +� +�� +� +≃Cl(Xs) +. +In other terms, CaCl(Xs) and Cl(Xs) are both free of rank 2 and via the canonical embedding, the first one has +invariant factors 1, 2 inside the second one. +15 + +Types of decomposition into irreducible components in |−KXs|. — +Since the two class groups are +not rank one, one expects to find a wide variety of possible decompositions into irreducible components for the +curves in the linear system |−KXs|. In order to list all these types, we start form P2 and use the one-to-one +correspondences: +|3ℓ − p1 − p3 − p2 − p4| +−→ +|3E0 − E1 − E3 − E2 − E4| +−→ +|−KXs| +C +�−→ +C♯ +�−→ +ϕ∗ +� +C♯� +. +The curves of the left linear system are nothing else than the plane cubics over Fq passing through p1, p3 that are +either smooth at p1, p3 with tangent lines p2, p4 respectively or singular at these points. +Our notations are the same: ℓ13 is the line (p1p3) which is rational, ℓ12 and ℓ34 are the lines (p1p2) respec- +tively (p3p4) (that is the lines of P2 passing through p1, respectively p3, whose strict transform pass through p2, +respectively p4); these last two lines are conjugate. The orbits of lines, conics, cubics having degree less than 3 +and passing through some of the points pi’s are +ℓ1 ∪ ℓ3, +ℓ13, +ℓ12 ∪ ℓ34, +q13, +q1234, +c13, +c1234 +(of course, implicitly ℓ3 = ℓσ +1, ℓ34 = ℓσ +12). We have just to combine these rational irreducible decompositions in +order to construct plane curves in the expected sub-linear system. +Suppose that there is at least one line in the absolute irreducible decomposition. +• If this line is ℓ12, then by rationality, ℓ34 is also an absolute irreducible component. Since ℓ12 ∪ ℓ34 already +passes through p1, p2, p3, p4, one can complete by any rational line ℓ or by the line ℓ13 (see cases 1 and 2 in +the tabular below). +• If this line is ℓ13, then the two incidence conditions at p1 and p3 are satisfied. The complement component +must be a (maybe reducible) conic whose strict transform passes through p2 and p4; this conic must neces- +sarily pass through p1, p3. This conditions suffice since the union of ℓ13 with any conic passing through p1, p3 +is singular. The complement can be the union ℓ12 ∪ ℓ34 (same as case 2), or ℓ1 ∪ ℓ3 = ℓσ +1, or ℓ13 itself union +any other line, or twice ℓ13, or q13, or q1234. +• If this line is ℓ a line that does not pass through the pi’s, then the complement conic must be either ℓ12 ∪ℓ34 +as in first case, or a conic passing through the four points. +Last, if there is not any line in the absolute irreducible decomposition, then the cubic must be absolutely irreducible +and it has to pass through the four points. +That being, the possible cubics are listed below. The irreducible effective roots of X are the (conjugate) strict +transforms �E1 and �E3; since they do not meet, their contraction lead to two (conjugate) singular points s and sσ +on Xs. +���3ℓ − �4 +i1 pi +��� +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +ℓ12 ∪ ℓ34 ∪ ℓ +� +ℓ12 ∪ � +ℓ34 ∪ �ℓ +ϕ∗(� +ℓ12) ∪ ϕ∗(�ℓ34) ∪ ϕ∗(�ℓ) +q + 2 +2 +ℓ12 ∪ ℓ34 ∪ ℓ13 +� +ℓ12 ∪ � +ℓ34 ∪ �ℓ13 ∪ �E1 ∪ �E3 ∪ E2 ∪ E4 +ϕ∗(� +ℓ12) ∪ ϕ∗(�ℓ34) ∪ ϕ∗(�ℓ13) ∪ ϕ∗(E2) ∪ ϕ∗(E4) +q + 2 +3 +ℓ13 ∪ ℓ1 ∪ ℓ3 +�ℓ13 ∪ �ℓ1 ∪ �ℓ3 ∪ �E1 ∪ �E3 +ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ1) ∪ ϕ∗(�ℓ3) +q + 2 +4 +2ℓ13 ∪ ℓ +2�ℓ13 ∪ �ℓ ∪ �E1 ∪ �E3 +2ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ) +2q + 1 +5 +3ℓ13 +3�ℓ13 ∪ 2 �E1 ∪ 2 �E3 ∪ E2 ∪ E4 +3ϕ∗(�ℓ13) ∪ ϕ∗(E2) ∪ ϕ∗( �E2) +q + 1 +6 +ℓ13 ∪ q13 +�ℓ13 ∪ �q13 ∪ �E1 ∪ �E3 +ϕ∗(�ℓ13) ∪ ϕ∗(�q13) +2q + 2 +7 +ℓ13 ∪ q1234 +�ℓ13 ∪ �q1234 ∪ �E1 ∪ �E3 ∪ E2 ∪ E4 +ϕ∗(�ℓ13) ∪ ϕ∗(�q1234) ∪ ϕ∗(E2) ∪ ϕ∗(E4) +2q + 2 +8 +ℓ ∪ q1234 +�ℓ ∪ �q1234 +ϕ∗(�ℓ) ∪ ϕ∗(�q1234) +2q + 2 +9 +c1234 +�c1234 +ϕ∗(�c1234) +Nq (1) +We draw all the preceding decompositions in order to illustrate what is going on. The blowing up π : X → P2 +is decomposed into two blowing ups π = π2 ◦ π1, where π1 : X1 → P2 is the blowing up at p1 and p3, and +where π2 : X → X1 is the blowing up at p2 and p4. The left column is the drawing of the starting configuration +in P2, the middle one the configuration after having blowing up p1 and p3, the right one the configuration in X. +The operation from a column to the next one is the virtual transform. Curves drawn in gray are not part of +virtual transform, curves drawn in red are the effective roots; these curves are contracted in Xs (but we do not +16 + +draw this step). In brackets, to the right of the name of a curve, we put its self-intersection. We draw all the +cases of the preceding tabular, except the cubic case. +In any case, one can verify that the union of the black curves passes through p1, p3, p2, p4 and that the divisor +class is equal to −KX = 3E0 − E1 − E3 − E2 − E4. +ℓ(1) +p1 +• +p3 • +ℓ12(1) +p2 +ℓ34(1) +p4 +�ℓ(1) +p2 +• +p4 • +�ℓ12(0) +�ℓ34(0) +E1(−1) +E3(−1) +�ℓ(1) +E2(−1) +E4(−1) +�E1(−2) +�E3(−2) +�ℓ12(−1) +�ℓ34(−1) +ℓ(1) +p1 +• +p3 • +ℓ12(1) +p2 +ℓ34(1) +p4 +�ℓ(−1) +�ℓ12(0) +�δ34(0) +p2 +• +p4 • +E1(−1) +E3(−1) +�ℓ(−1) +E2(−1) +E4(−1) +�E1(−2) +�E3(−2) +�ℓ12(−1) +�ℓ34(−1) +ℓ13(1) +p1 +• +p3 • +ℓ1(1) +p2 +ℓσ +1(1) +p4 +�ℓ13(−1) +�ℓ1(0) +�ℓσ +1(0) +p2 +• +p4 • +E1(−1) +E3(−1) +�ℓ13(−1) +�ℓ1(0) +�ℓσ +1(0) +E2(−1) +E4(−1) +E1(−2) +E3(−2) +2ℓ13(1) +p1 +• +p3 • +ℓ(1) +p2 +p4 +2�ℓ13(−1) +E1(−1) +E3(−1) +p2 +• +p4 • +�ℓ(1) +2�ℓ13(−1) +�E1(−2) +�E3(−2) +E2(−1) +E4(−1) +�ℓ(1) +3ℓ13(1) +p1 +• +p3 • +p2 +p4 +3�ℓ13(−1) +2E1(−1) +2E3(−1) +p2 +• +p4 • +3�ℓ13(−1) +2 �E1(−2) +2 �E3(−2) +E2(−1) +E4(−1) +p2 +p4 +q13(4) +• +p1 +• +p3 +ℓ13(1) +�q13(2) +E1(−1) +E3(−1) +• p2 +• +p4 +�ℓ13(−1) +�q13(2) +�E1(−2) +�E3(−2) +E2(−1) +E4(−1) +�ℓ13(−1) +17 + +q1234(4) +p2 +p4 +• p1 +• +p3 +ℓ13(1) +�q1234(2) +E1(−1) +E3(−1) +• p2 +• +p4 +�ℓ13(−1) +�q1234(0) +�E1(−2) +�E3(−2) +�ℓ13(−1) +E2(−1) +E4(−1) +q1234(4) +p2 +p4 +• p1 +• +p3 +ℓ(1) +�q1234(2) +E1(−1) +E3(−1) +• p2 +• +p4 +ℓ(1) +�q1234(0) +�E1(−2) +�E3(−2) +E2(−1) +E4(−1) +ℓ(1) +Some comments about the number of points. +Cases 1, 2, & 3. The unions of lines ℓ12 ∪ℓ34, or ℓ1 ∪ℓ3 (recall that ℓ3 = ℓσ +1), contain a unique rational point, +the intersection point of the two lines. Except ℓ (cases 1 and 2) or ℓ13 (case 3), all the lines in the decomposition +are not defined over Fq and di not contain any rational points. Thus to the previous single point we have to add +the (q + 1) rational points of the line ℓ or ℓ13. +Case 4. The two (black) components have (q +1) rational points but they meet at a rational point, thus their +union contains 2q + 1 rational points. +Case 5. The only component that contains rational points is ϕ∗(�ℓ13). +Cases 6, 7, & 8. In these cases, they are two disjoint components that contain (q + 1) rational points. +Finally: +Nq (−KXs) = 2q + 2. +Since none of the points pi is rational, the surfaces X and Xs still have q2 + q + 1 points. For every q, one +has #Xs(Fq) > Nq (−KXs) and the evaluation map is always injective. The parameters of the code are thus +given by: +Proposition 4.3. Let p1 ≺ p2 and p3 ≺ p4 be such that p1, p3 and p2, p4 are conjugate points of degree 2. The +anticanonical code of the weak del Pezzo surface obtained by blowing up these points has parameters [q2 + q + +1, 6, q2 − q − 1]. +Computation of the global sections from P2. — +Let d ∈ Fq be a non-square and put ζ = +√ +d ∈ Fq2. We +choose p1 = (ζ : 0 : 1) and p2 = (ζ : 1). Then p3 = (−ζ : 0 : 1), the line ℓ13 = (p1p3) has equation Y = 0, +the line ℓ12 = (p1p2) corresponds to the zeros of the linear form L = X − ζ(Y + Z) and the line ℓ34 = (p3p4) +corresponds to the zeros of the linear form Lσ = X + ζ(Y + Z) = 0. The linear forms Y, L, Lσ generate the global +sections of ℓ and we easily prove that +|3ℓ − p1 − p3 − p2 − p4| = +� +Y 3, Y 2L, Y 2Lσ, Y LLσ, L2Lσ, L(Lσ)2� +Fq += +� +Y 3, Y 2(L + Lσ), Y 2ζ(L − Lσ), Y LLσ, LLσ(L + Lσ), LLσζ(L − Lσ) +� +Fq += +� +Y 3, Y 2X, Y 2(Y + Z), Y Π, XΠ, (Y + Z)Π +� +Fq , +where Π = LLσ = X2 − d(Y + Z)2. The evaluation points are nothing else than all the points of P2(Fq). +4.4 +Degree 4, singularity of type A1 +This example corresponds to the type number 8 in degree 4 [BH22]. +Configuration to blow-up and down. — +We blow up six points on a conic, the first two p1 and p2 being +conjugate (or rational), the last four p3, p4, p5, p6 being conjugate. This leads to a degree 3 weak Del Pezzo +surface Y +π +−→ P2 as in (8). In this surface, the strict transform of the line ℓ12 passing through p1 and p2 is a +18 + +rational (−1)-curve that can be contracted: the codomain of the contraction Y +χ +−→ X is the degree 4 weak Del +Pezzo surface we want to work with in this section. +p1 +p2 +p3 +p4 +p5 +p6 +ℓ12 +q123456 +p2 = pσ +1 +p4 = pσ +3 +p5 = pσ2 +3 +p6 = pσ3 +3 +The anticanonical model of this weak del Pezzo surface X has a unique singular point. +Computation of the divisor class groups. — +On Y , one has Cl(Y ) = �6 +i=0 ZEi and 27 exceptional classes +of divisor, namely the 6 exceptional lines Ei, 1 ≤ i ≤ 6, the 15 strict transforms of the lines passing through two +of the six points, Eij = E0 − Ei − Ej, 1 ≤ i < j ≤ 6, and the 6 strict transforms of the quadrics passing through +five of the six points, Qi = 2E0 − � +j̸=i Ej, 1 ≤ 6. Due to weaknesses, among these classes, the quadric ones +are not represented by irreducible curves. Indeed 2E0 − � +j̸=i Ej = Ei + +� +2E0 − �6 +j=1 Ej +� +and the last class is +nothing else than the class of the unique effective root, the strict transform of the quadric q123456 passing through +all the six points. +The group Cl(X) can be identified with Z(E0 − E1 − E2)⊥ via the orthogonal projection onto this space. This +projection is given by: +Cl(Y ) += +Z(E0 − E1 − E2) +⊕ +(Z(E0 − E1) ⊕ Z(E0 − E2) ⊕ ZE3 ⊕ · · · ⊕ ZE6) +�6 +i=0 aiEi += +(−a0 − a1 − a2)(E0 − E1 − E2) ++ +� +(a0 + a2)(E0 − E1) + (a0 + a1)(E0 − E2) + �6 +i=3 aiEi +� , +and thus +Cl(X) = ZL1 ⊕ ZL2 ⊕ ZE3 ⊕ · · · ⊕ ZE6, +where +Li = E0 − Ei, +i = 1, 2. +In particular, the anticanonical divisors are related by +−KY = 3E0 − +6 +� +i=1 +Ei = −(E0 − E1 − E2) + 2L1 + 2L2 − +6 +� +i=3 +Ei +� +�� +� +−KX +Only the exceptional classes of Y that do not meet E0 − E1 − E2 are mapped to exceptional classes on X; +for 3 ≤ i ≤ 6, this leaves the classes: +Ei �−→ Ei, +E1i �−→ L1 − Ej, +E2i �−→ L2 − Ej, +Qi �−→ L1 + L2 − +� +j∈{3,...,6}\{i} +Ej. +As for the unique effective root of Y , it is mapped to the root L1 + L2 − E3 − E4 − E5 − E6 and the last four +exceptional classes are not represented by irreducible curves. Thus one has +R = Z(L1 + L2 − E3 − · · · − E6), +R +⊥ = +� +a1L1 + a2L2 + +6 +� +i=3 +Ei | a1 + a2 + +6 +� +i=3 +ai = 0 +� +. +In order to take into account the Galois action, which acts via (L1L2)(E3E4E5E6), we put L = L1 + L2 and E = +�6 +i=3 Ei. We easily verify that +Cl(X) = CaCl(X) = ZL ⊕ ZE = Z(L − E) ⊕ ZE, +R = Z(L − E), +R⊥ = Z(2L − E) = ZKX. +This leads to the following isomorphism: +Cl(Xs) ≃ Cl(X)/R +≃ +−→ +ZE +aL + bE mod R +�−→ +(a + b)E +Via this isomorphism, the sub-module CaCl(Xs) = R⊥ = Z(2L−E) = ZKX is mapped to ZE itself. In conclusion +both CaCl(Xs) and Cl(Xs) are free Z-module of rank 1 and the canonical embedding turns to be an isomorphism. +19 + +Types of decomposition into irreducible components in |−KXs|. — +Recall that +−KX = 2L1 + 2L2 − +6 +� +i=3 +Ei = 4E0 − 2E1 − 2E2 − +6 +� +i=3 +Ei. +Global sections of −KX are thus related to quartics of P2. More precisely, one has the following one-to-one +correspondences: +���4ℓ − 2p1 − 2p2 − �6 +i=3 pi +��� +Y +−→ +���4E0 − 2E1 − 2E2 − �6 +i=3 Ei +��� +X +−→ +|−KXs|Xs +C +�−→ +χ∗ +� +C♯� +�−→ +ϕ∗ +� +χ∗ +� +C♯�� +, +and we need to list all the quadrics of P2 having multiplicity at least 2 at p1 and p2 and passing through the pi +for 3 ≤ i ≤ 6. +Note that in the correspondences above, we skip the surface Y . +Recall that, as in (8), we +have P2 +π +←− Y +χ +−→ X and the morphism χ here is the contraction of the strict transform of the line passing +through p1 and p2. +The orbits of lines, respectively conics, having degree less than 4 and passing through some of the points pi’s +are +ℓ1 ∪ ℓ2, +ℓ3 ∪ ℓ4 ∪ ℓ5 ∪ ℓ6, +ℓ12, +ℓ35 ∪ ℓ46, +ℓ13 ∪ ℓ24 ∪ ℓ15 ∪ ℓ26, +ℓ14 ∪ ℓ25 ∪ ℓ16 ∪ ℓ23, +respectively: +q1 ∪ q2, +q12, +q35 ∪ q46, +q3456, +q1235 ∪ q1246, +q123456. +The only orbits of cubics or quartics having degree less than 4 that pass through the points pi are c123456 +and t123456. We now combine the rational irreducible decompositions in order to construct plane curves in the +expected sub-linear system. +First, suppose that the decomposition into absolute irreducible components contains a line +• If this line joins one of the first two points to one of the last four points, i.e. a line ℓij with i ∈ {1, 2} +and j ∈ {3, 4, 5, 6}, then this line has degree 4 and it turns out that its orbit under the Galois action lies in +the linear system; (cases 1 and 2 in the tabular below). +• If this line is ℓ12, which is rational, then this line appears with multiplicity at most 2. If 2ℓ12 is a part of the +decomposition then the complementary conic must be rational and pass through the last four points: the +conic must be ℓ35 ∪ ℓ46 or q3456 or q123456 (cases 3, 4, 5). If ℓ12 has multiplicity 1, then the complementary +cubic passes through the six points. Since, except ℓ12, all the lines passing through some pi have even degree, +this cubic cannot be a union of three lines. The remaining cases are thus q123456∪ℓ or c123456 (cases 6 and 7). +• If the line is ℓi for i ≥ 3, then it has degree (at least) 4 and its orbit under Galois has degree 4 (or greater) +without passing through p1 and p2. It does not work. +• If the line is ℓ1 then its conjugate ℓσ +1 passes through p2; the complement is a conic passing through the six +points, and it must be q123456 (case 8). +• Last if this line is ℓ a line passing through none of the six points then the complementary cubic passes +through the six points with multiplicity 2 at p1 and p2. Since an irreducible plane cubic has at most one +singular point, this cubic must be reducible and it is the union of a line and a conic, whose meeting points +are the singular points, that is p1 and p2. Thus the line must be ℓ12, and the conic is q123456 and we recover +case 6. +Secondly, suppose that there are only two absolutely irreducible conics in the decomposition. For the union +of these two conics to be singular at p1 and p2, they must pass through p1 and p2. Taking into account the +rationality, there are only three possibilities, cases 9, 10, 11. +Last if the quartic is absolutely irreducible, then it must pass through the six points with multiplicity 2 at p1 +and p2. +In the tabular below, we summarize all the possibilities. As noted below, the strict transform �ℓ12 in Y is +contracted in X via the morphism Y +χ +−→ X; this explains why the curve disappears in the middle column. +Then from X to Xs, it is the irreducible effective root �q123456 that is contracted by the morphism X +ϕ +−→ Xs. +Thus on Xs, there are two specific rational points, p the image of the contraction of �ℓ12 and s the image of the +20 + +contraction of �q123456. +���4ℓ − 2p1 − 2p2 − �6 +i=3 pi +��� +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +ℓ13 ∪ ℓ24 ∪ ℓ15 ∪ ℓ26 +�ℓ13 ∪ �ℓ24 ∪ �ℓ15 ∪ �ℓ26 +ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ24) ∪ ϕ∗(�ℓ15) ∪ ϕ∗(�ℓ26) +0 +2 +ℓ14 ∪ ℓ25 ∪ ℓ16 ∪ ℓ23 +�ℓ14 ∪ �ℓ25 ∪ �ℓ16 ∪ �ℓ23 +ϕ∗(�ℓ14) ∪ ϕ∗(�ℓ25) ∪ ϕ∗(�ℓ16) ∪ ϕ∗(�ℓ23) +0 +3 +2ℓ12 ∪ ℓ35 ∪ ℓ46 +�ℓ35 ∪ �ℓ46 +ϕ∗(�ℓ35) ∪ ϕ∗(�ℓ46) +2 +4 +2ℓ12 ∪ q3456 +�q3456 +ϕ∗(�q3456) +q + 2 +5 +2ℓ12 ∪ q123456 +�q123456 ∪ E1 ∪ E2 +{p, s} ∪ ϕ∗(E1) ∪ ϕ∗(E2) +2 +6 +ℓ12 ∪ q123456 ∪ ℓ +�q123456 ∪ �ℓ +ϕ∗(�ℓ) +q + 2 +7 +ℓ12 ∪ c123456 +�c123456 +ϕ∗(�c123456) +Nq (1) +8 +ℓ1 ∪ ℓσ +1 ∪ q123456 +�ℓ1 ∪ �ℓσ +1 ∪ �q123456 +ϕ∗(�ℓ1) ∪ ϕ∗(�ℓσ +1) +2 +9 +q12 ∪ q123456 +�q12 ∪ �q123456 +ϕ∗(�q12) +q + 2 +10 +q1235 ∪ q1246 +�q1235 ∪ �q1246 +ϕ∗(�q1235) ∪ ϕ∗(�q1246) +2 +11 +2q123456 +�q123456 ∪ �6 +i=3 Ei +{s} ∪ �6 +i=3 ϕ∗(Ei) +1 +12 +t123456 singular at p1, p2 +�t123456 +ϕ∗(�t123456) +Nq (1) +Some comments about the numbers of points are in order. +Cases 1 & 2. The four lines ℓ13, ℓ24, ℓ15, ℓ26 are conjugate, they do not meet and thus their union in P2 +does not contain any rational point. +In the blowing-up of P2 at the six points, the strict transforms of the +lines ℓ13, ℓ24, ℓ15, ℓ26 no longer meet the strict transform of ℓ12. Thus the contraction of this line does not add any +rational point. Since none of the lines ℓ13, ℓ24, ℓ15, ℓ26 can be a tangent line to q123456 at some pi (otherwise the +line and the quadric would have too many intersection points by Bezout), blowing-up the pi, 1 ≤ i ≤ 6, separates +the strict transforms of the lines and the strict transform of the quadric. The contraction of this quadric neither +add rational points. This proves that these configurations do not contain any rational point. +Case 3. The lines ℓ35 and ℓ46 are conjugate to each other. Their intersection point is the unique rational +point of their union in P2. This point is still on ϕ∗(�ℓ35) ∪ ϕ∗(�ℓ46). The added point is p which comes from the +contraction of �ℓ12. Note that the contraction of �q123456 does not add any point since the lines ℓ35 and ℓ46 are +separated from �q123456 in the blow-ups. This is because none of these lines can be a tangent to q123456 at one of +the six points (otherwise the line and the quadric would have too many intersection points by Bezout). +Case 4. The conics q123456 and q3456 are separated by the blowing up of the last four points and thus the +point s does not belong to the final section. The strict transforms of ℓ12 and of q3456 meet at two points that are +mapped to p by the contraction of �ℓ12. If these two points are not rational, p is an added rational point of the +final section. +Case 5 & 11. The resulting sections contain some exceptional curves Ei in their supports since the multi- +plicities at some points pi are strictly greater than the ones expected. Since the points pi are not rational, none +of the Ei contain rational points and the only rational points are {p, s} or {s}. +Case 6. The point p lies on ϕ∗(�ℓ) but it comes from the intersection point between ℓ and ℓ12 (which cannot +be one of the pi since it is a rational point) so no points are added in the contraction of ℓ12. As for the point s, +it lies also on ϕ∗(�ℓ) and it could add one more point if ℓ meet q123456 at two conjugate points. +Case 7. The cubic must be smooth at p1 and p2; indeed if it would be singular at one of these points, by +Galois conjugation, it would be singular at both of them and it would have too many singular points. Thus, to +make the multiplicity greater than 2 at p1, p2, the complementary line must be ℓ12. This line meets the cubic +at p1 and p2 and a third point which must be rational and not on q123456. Therefore, the contraction of �ℓ12 pass +through p but does not add any rational points to ϕ∗(�ℓ12). As for the contraction of the strict transform of q123456, +it does not add points either since blowing up the six points separates the cubic and q123456. +Case 8. The meeting point of the curves ℓ1 and ℓσ +1 is necessarily rational and it is the unique rational point +of their union in P2. The strict transforms �ℓ1 and �ℓσ +1 do not meet �ℓ12 and thus the point p does belong to the +final section. The contraction of the root �q123456 add the point s except if the meeting of the curves ℓ1 and ℓσ +1 +already belongs to q123456. +Case 9. The conics q12 and q123456 are separated by the blowing ups. If the conic q12 is chosen in such a way +that the tangent line at p1 equals ℓ12, then �q12 and �ℓ12 meet at two unrational points in Y and the contraction +of �ℓ12 adds the rational point p to �q12 in X. This explains why the final number of points is (q + 1) + 1. +Case 10. The two conics are conjugate. Besides p1 and p2, they meet at two other points (Bezout) that can +21 + +be rational. If so these points are the only points of P2(Fq) that belong to the union of the two conics. Blowing +up the six points disconnect the two conics from the strict transforms of ℓ12 and q123456. So no points are added. +Case 12. Since the quartic has at least two singular points, it geometric genus must be at most 1. +As predicted by the class group computations, all the curves on Xs in the linear system are irreducible; they +are not necessarily absolutely irreducible but it turns out that curves that are not absolutely irreducible never +contain too many rational points. +In any case, one has +Nq (−KXs) ≤ Nq(1). +Since none of the points pi is rational, the surface Y has q2 + q + 1 rational points. Since X is obtained by +contracting �ℓ12, it contains q2 + 1 rational points. In the same way, after contracting �q123456, the surface Xs +has q2 − q + 1 rational points. Since q2 − q + 1 ≤ Nq(1) for q ∈ {2, 3}, the evaluation map may be non injective +and we do not consider the codes with these two values. +Proposition 4.4. Suppose that q ̸= 2, 3. Let p1, . . . , p6 ∈ P2 +Fq be six conconic points, such that p1, p2 and p3, p4, p5, p6 +are conjugate. The anticanonical code of the weak del Pezzo surface obtained by blowing up these six points and +then blowing down the strict transform of the line (p1p2) has parameters [q2 − q + 1, 5, ≥ q2 − q + 1 − Nq (1)]. +Computation of the global sections from P2. — +Let Q be a quadratic polynomial that defines q123456 +and Lij a linear form that defines the line ℓij. Then +�����4ℓ − 2p1 − 2p2 − +6 +� +i=3 +pi +����� = ⟨QL12X, QL12Y, QL12Z, L2 +12L35L46, L13L24L15L26⟩Fq +The three first sections are clearly linearly independent. The fourth one cannot be a linear combination of the +three first ones since otherwise Q would be reducible. Last, the fifth one cannot be a linear combination of the +four first ones since otherwise L12 would divide L13L24L15L26. +4.5 +Degree 4, singularity of type 4A1 +This example corresponds to the type number 48 in degree 4 [BH22]. We recover an example already studied by +Koshelev [Kos20, §1.2]. Our point of view slightly differs from Koshelev’s one, so even if this example appears in +the literature, we choose to go into details. +Configuration to blow-up and down. — +The context is still the one described in (8) with a non trivial +map Y +χ +−→ X. +Let p1, p2 = pσ +1, p3 = pσ2 +1 , p4 = pσ3 +1 +∈ P2 be four conjugate points in general position (no three of them +are collinear) and, as usual, let ℓ12, ℓ23, ℓ34, ℓ14 denote the lines (p1p2), (p2p3), (p3p4), (p1p4); they are conjugate. +Let p5 be the intersection point of ℓ12 and ℓ34 and p6 be the intersection point of ℓ23 and ℓ14. They are also +conjugate and we denote by ℓ56 the rational line passing through p5, p6. +p1 +p2 +p3 +p4 +• +p5 +•p6 +p2 = pσ +1 +p3 = pσ2 +1 +p4 = pσ3 +1 +p6 = pσ +5 +We blow up these six points to obtain a degree 3 weak del Pezzo surface Y . The strict transform of the line ℓ56, +of class E0 − E5 − E6, is an exceptional curve that can be contracted to obtain the degree four weak del Pezzo +surface X we consider here. The anticanonical model of this surface has four singular points of type A1 (since +the four irreducible effective roots do not intersect, see below). +Computation of the divisor class groups. — +Over Fq, we know that Cl(Y ) = �6 +i=0 ZEi and that −KY = +3E0−�6 +i=1 Ei. Moreover the surface Y has four irreducible effective roots, the strict transforms of the lines ℓ125, ℓ236, ℓ345, ℓ146 +whose conjugate classes in Cl(Y ) are: +E0 − E1 − E2 − E5, +E0 − E2 − E3 − E6, +E0 − E3 − E4 − E5, +E0 − E1 − E4 − E6. +22 + +The group Cl(X) identifies with Z(E0 − E5 − E6)⊥ inside Cl(Y ). Since +Cl(Y ) += +Z(E0 − E5 − E6) +⊥⊕ +� +Z(E0 − E5) ⊕ Z(E0 − E6) ⊕ �4 +i=1 ZEi +� +�6 +i=0 aiEi += +(−a0 − a5 − a6)(E0 − E5 − E6) ++ +(a0 + a6)(E0 − E5) + (a0 + a5)(E0 − E6) + �4 +i=1 aiEi +(9) +one has +Cl(X) = Z(E0 − E5) ⊕ Z(E0 − E6) ⊕ ZE1 ⊕ ZE2 ⊕ ZE3 ⊕ ZE4. +In particular, +−KX = 2(E0 − E5) + 2(E0 − E6) − E1 − E2 − E3 − E4 = 4E0 − 2E5 − 2E6 − E1 − E2 − E3 − E4. +On X, there are still four effective roots, the image by the contraction of the effective roots on Y : +R = Z(E0 − E1 − E2 − E5) ⊕ Z(E0 − E2 − E3 − E6) ⊕ Z(E0 − E3 − E4 − E5) ⊕ Z(E0 − E1 − E4 − E6). +On X there are 16 exceptional classes, +(E0 − Ei) − Ej, i ∈ {5, 6}, j ∈ {1, 2, 3, 4}, +(E0 − E5) + (E0 − E6) − Ei1 − Ei2 − Ei3, {i1, i2, i3} ⊂ {1, 2, 3, 4}, +and E1, E2, E3, E4. Only these last four classes are represented by irreducible exceptional curves. We recover the +graph number 9 of the Proposition 6.1 of Coray & Tsfasman [CT88]. +The Galois group acts as as a 4-cycle on the roots and as (E1E2E3E4) on the (−1)-curves. Let us put F = +(E0 − E5) + (E0 − E6) and E = E1 + E2 + E3 + E4, in such a way that F·2 = 2, E·2 = −4 and F · E = 0. Then +one has: +� +R = Z2(F − E) +Cl(X) = Z(F − E) ⊕ ZE +=⇒ +Cl(Xs) = Cl(X)/R +≃ +−→ +Z/2Z(F − E) +⊕ +ZE +aF + bE +�−→ +a(F − E) mod R ++ +(a + b)E +As for the Cartier class group, we find CaCl(Xs) = R⊥ = Z(2F − E) = Z(−KX) which embeds in Cl(Xs) +via −KX �→ E. Thus, via the canonical embedding, CaCl(Xs) and the free part of Cl(Xs) are isomorphic. +Types of decomposition into irreducible components in |−KXs|. — +The situation looks like the preceding +one except that the multiplicity is at points p5, p6 here. One has: +���4ℓ − �4 +i=1 pi − 2p5 − 2p6 +��� +Y +−→ +���4E0 − 2E5 − 2E6 − �4 +i=1 Ei +��� +X +−→ +|−KXs|Xs +C +�−→ +χ∗ +� +C♯� +�−→ +ϕ∗ +� +χ∗ +� +C♯�� +. +Thus we are reduced to list all the types of irreducible decompositions of quadrics passing through the six points, +the last two with multiplicities at least 2. +The orbits of lines of degree less than 4 that involve the six points are +ℓ5 ∪ ℓ6, +ℓ1 ∪ ℓ2 ∪ ℓ3 ∪ ℓ4, +ℓ56, +ℓ13 ∪ ℓ24, +and +ℓ125 ∪ ℓ236 ∪ ℓ345 ∪ ℓ146. +There are only two ways (cases 1 and 2 below) to combine these configurations in order to obtain a curve in the +expected linear system. +The orbits of conics of degree less than 4 that involve the six points are q1234, q56 and q1356 ∪ q2456. There are +only two ways (cases 3 and 4 below) to combine the configurations of lines and conics in order to obtain a curve +in the expected linear system. +If the decomposition contains a cubic, it must be smooth at p5 and p6 and the complement must be ℓ56; this +is case 5. This leads to the list below. Let us note that on X the curve �ℓ56 in Y is contracted by χ. On Xs, this +contraction is mapped to a smooth rational point p. This surface contains also four singular points si, 1 ≤ i ≤ 4, +coming from the contraction of the four roots; they are conjugate and of degree 4. +���4ℓ − �4 +i=1 pi − 2p5 − 2p6 +��� +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +2ℓ56 ∪ ℓ13 ∪ ℓ24 +�ℓ13 ∪ �ℓ24 +ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ24) +2 +2 +ℓ125 ∪ ℓ236 ∪ ℓ345 ∪ ℓ146 +�ℓ125 ∪ �ℓ236 ∪ �ℓ345 ∪ �ℓ146 ∪ �4 +i=1 Ei +{si} ∪ �4 +i=1 ϕ∗(Ei) +0 +3 +2ℓ56 ∪ q1234 +�q1234 +ϕ∗(�q1234) +q + 2 +4 +q1356 ∪ q2456 +�q1356 ∪ �q2456 +ϕ∗(�q1356) ∪ ϕ∗(�q2456) +2 +5 +c123456 ∪ ℓ56 +�c123456 +ϕ∗(�c123456) +Nq (1) +6 +t123456 singular at p5, p6 +�t123456 +ϕ∗(�t123456) +Nq (1) +23 + +Some comments on the number of rational points. +Case 1. The lines ℓ13 and ℓ24 are conjugate, their meeting point is the unique rational point of their union. +After the contraction of �ℓ56, these two lines have one more rational point in common, the point p. +Case 3. If q1234 meets ℓ56 at two conjugate points, the contraction of �ℓ56 adds the point p to the other rational +points. +Case 4. Blowing up p5 and p6 separates the strict transforms �q1356 and �q2456 from ℓ56. So the contraction of +this curve do not add any point. On P2 the union of conics �q1356 and �q2456 has at most two rational points: their +meeting points that differ from p5, p6 if they are rational. +Case 5. Necessarily the cubic is smooth at p5, p6 and the tangent lines at these points cannot be equal to ℓ56. +Therefore blowing up p5, p6 separates the curves �ℓ56 and �c123456 above p5 and p6. Besides p5, p6 the curves ℓ56 +and c123456 meet at a third point (Bezout) which is necessarily rational. Via the contraction of �ℓ56, this line +concentrates at this third point and no points are added on �c123456. +Case 6. Necessarily blowing up p5 and p6 disconnects the strict transforms �t123456 and �ℓ56. Moreover, it +desingularizes the quartic t since the singularities at p5 and p6 must be ordinary. Blowing up p1, . . . , p4 also +disconnects �t from all the effective roots. Finally ϕ∗(t123456) turns to be an elliptic curve. +Finally Nq(−KXs) ≤ Nq(1). +Since none of the points pi is rational, #X(Fq) = #P2(Fq) = q2 +q+1. Then contracting the rational line �ℓ56 +decreases the number by q and #Xs(Fq) = q2 + 1. Except for q = 2, this number is stricly greater that Nq(1) +and the evaluation map is injective. +Proposition 4.5. Suppose q ̸= 2. Let p1, p2 = pσ +1, p3 = pσ2 +1 , p4 = pσ3 +1 +∈ P2 +Fq be four conjugate points in general +position (no three of them are collinear) and let p5 (resp. p6) be the point of intersection of the lines (p1p2) +and (p3p4) (resp. (p2p3) and (p1p4)). The anticanonical code of the weak del Pezzo surface obtained by blowing +up these six points and then blowing down the strict transform of the line (p5p6) has parameters [q2 + 1, 5, ≥ +q2 + 1 − Nq (1)]. +As proved by Koshelev [Kos20, §1.2], for some values of q, the minimum distance can be improved by one. +Since the argument is very nice, we choose to briefly sketch it below. The idea is to prove that all the elliptic +curves in our linear system must have a rational 2-torsion point and thus an even number of points. Since for +some q, the maximum Nq(1) is odd, this means that Nq (−KXs) < Nq(1) and our bound for the minimum distance +can be improved by 1. Let c be a cubic passing through the six points. Then, for any choice of the origin, the +alignments of points permit to show that: +p1 + p2 + p5 += +p1 + p4 + p6 += +p2 + p3 + p6 += +p3 + p4 + p5 +=⇒ +p1 − p3 += +p2 − p4 += +p4 − p2 += +p6 − p5 +=⇒ +2(p2 − p4) = 0, +and these points must be rational since p2 − p4 = p6 − p5 with p5, p6 conjugate. The case of the quartics in the +linear system works in the same way but it is a little bit more technical since we need to know the group law on +this kind of curve. +Computation of the global sections from P2. — +In this example, we do not find a nice explicit basis for +the global sections. Instead, we choose to present a magma code that permits to construct the generator matrix. +1 +c l e a r +; +q +:= +7 +; +Fq := +F i n i t e F i e l d ( q ) +; +P2 := +P r o j e c t i v e S p a c e (Fq , +2) +; +XYZ := +C oordi nateRi ng ( P2) +; +6 +phi +:= +map < P2 −> P2 +| +[Xˆq ,Yˆq , Zˆq ] +> +; +// +Some +b a s i c +r o u t i n e s +such +as +v e r i f y i n g +i f +some +p o i n t s +are +i n +g e n e r a l +p o s i t i o n +l oad +” U t i l i t i e s . magma” +; +11 +// +Choose +randomly +a +degree +4 +p o i n t +i n +g e n e r a l +p o s i t i o n . +r e p e a t +p1 +:= +Random( P2 ( ext< Fq +| +4>)) +; +p2 +:= +phi ( p1 ) +; +p3 +:= +phi ( p2 ) +; +p4 +:= +phi ( p3 ) +; +Cl1234 +:= +C l u s t e r ( p1 ) +; +16 +u n t i l +EnPos i ti onGene r al e ( [ p1 , p2 , p3 , p4 ] ) +; +//// +To +compute +the +p o i n t s +p5 , +p6 +we +need +the +extend +the +s c a l a r s +to +Fq4 +P2 Fq4 +:= +BaseChange ( P2 , +ex t < Fq +| +4 +>) +; +21 +p1 Fq4 +:= +P2 Fq4 ! ElementToSequence ( p1 ) +; +p2 Fq4 +:= +P2 Fq4 ! ElementToSequence ( p2 ) +; +p3 Fq4 +:= +P2 Fq4 ! ElementToSequence ( p3 ) +; +p4 Fq4 +:= +P2 Fq4 ! ElementToSequence ( p4 ) +; +L12 +:= +Scheme ( P2 Fq4 , +S e c t i o n s ( LinearSystem ( LinearSystem ( P2 Fq4 , +1 ) , +[ p1 Fq4 , p2 Fq4 ] ) ) [ 1 ] ) +; +L34 +:= +Scheme ( P2 Fq4 , +S e c t i o n s ( LinearSystem ( LinearSystem ( P2 Fq4 , +1 ) , +[ p3 Fq4 , p4 Fq4 ] ) ) [ 1 ] ) +; +26 +p5 Fq4 +:= +Poi nts ( L12 +meet +L34 ) [ 1 ] +; +p5 +:= +P2( ex t < Fq +| +2 +>)! ElementToSequence ( p5 Fq4 ) +; +//// +End +o f +the +computation +ov er +Fq4 +31 +p6 +:= +phi ( p5 ) +; +Cl56 +:= +C l u s t e r ( p5 ) +; +C l 5 6 s q u a r e +:= +C l u s t e r ( P2 , +I d e a l ( Cl56 ) ˆ 2 ) +; +24 + +L +:= +LinearSystem ( LinearSystem ( P2 , +4 ) , +Cl1234 ) +; +36 +L +:= +LinearSystem (L , +C l 5 6 s q u a r e ) +; +T heSecti ons +:= +S e c t i o n s (L) +; +L56 +:= +Scheme ( P2 , +S e c t i o n s ( LinearSystem ( LinearSystem ( P2 , +1 ) , +Cl56 ) ) [ 1 ] ) +; +S +:= +( Poi nts ( P2 ) +d i f f +Poi nts ( L56 ) ) +j o i n +{@ +Poi nts ( L56 ) [ 1 ]@} +; +41 +G := +Matrix ( Fq,5 ,1+ q ˆ2 , +&cat +[ [ Ev al uate ( f , ElementToSequence ( p ) ) +: +p +i n +S ] +: +f +i n +T heSecti ons ] ) +; +TheCode +:= +LinearCode (G) +; +p r i n t f +” Generator +matrix +G =\n%o\n ” , +G +; +46 +l g +:= +Length ( TheCode ) +; +dim +:= +Dimension( TheCode ) +; +d min +:= +MinimumWeight( TheCode ) +; +p r i n t f +”q = %o , \ n [ n , k , d ] += [%o , +%o , +%o ] \ n ” , +q , +l g , +dim , +d min +; +p r i n t f +”What +was +prov i ded +( not +t a k i n g +i n t o +account +Kas hel ev +remark ) += [%o , +5 , +%o ] ” , +qˆ2+1, +qˆ2 − q − +Fl oor (2∗ Sq rt ( q ) ) +; +4.6 +Degree 4, singularity of type A2 +This example corresponds to the type number 30 in degree 4 [BH22]. This type works almost as the one described +in section 4.5. +Configuration to blow-up and down. — +Let p1, p2 = pσ +1, p3 = pσ2 +1 , p4 = pσ3 +1 +∈ P2 be four conjugate points +in general position (no three of them are collinear). The two lines (p1p3) and (p2p4) are conjugate. We choose a +degree 2 point p5 on (p1p3) and we let p6 = pσ +5 which lies on (p2p4). +• +p1 +•p2 +•p3 +• +p4 +•p5 +•p6 +ℓ135 +ℓ246 +ℓ56 +p2 = pσ +1 +p3 = pσ2 +1 +p4 = pσ3 +1 +p6 = pσ +5 +The surfaces of the diagram (8) are the following: we blow up the six points to obtain the degree 3 weak del Pezzo +surface Y . On this surface, the strict transform of the line ℓ56, of class E0 − E5 − E6, is an exceptional curve that +can be contracted to obtain the weak degree four weak del Pezzo surface X defined over Fq. The anticanonical +model has a unique singular point of type A2 (since there are only two irreducible effective root that meet, see +below). +Computation of the divisor class groups. — +Over Fq, we know that Cl(Y ) = �6 +i=0 ZEi and that −KY = +3E0 −�6 +i=1 Ei. There are only two irreducible effective roots on Y , the strict transforms of the lines ℓ135 and ℓ246 +whose conjugate classes in Cl(Y ) are E0 − E1 − E3 − E5 and E0 − E2 − E4 − E6. +The group Cl(X) identifies with Z(E0−E5−E6)⊥ inside Cl(Y ). We recover the same orthogonal decomposition +as in (9). In particular, we still have −KX = 4E0 − �4 +i=1 Ei − 2E5 − 2E6. On X there are still two (conjugate) +irreducible effective roots, of classes E0 − E1 − E3 − E5 and E0 − E2 − E4 − E6. +We follow the same computation as in section 4.5 and we still put E = (E0 − E5) + (E0 − E6) and F = +E1 + E2 + E3 + E4. Then, for X we have: +− KX = 2F − E, +R = Z(F − E), +and +Cl(X) = Z(F − E) ⊕ ZE, +and for Xs we deduce that: +CaCl(Xs) = Z(2F − E) = Z(−KX) +Cl(Xs) +≃ +−→ +ZE +aF + bE +�−→ +(a + b)E +CaCl(Xs) +≃ +→ +Cl(Xs) +−KX +�→ +E +The canonical embedding induces an isomorphism between the two class groups. +Types of decomposition into irreducible components in |−KX|. — +Since the two class groups are +isomorphic and free of rank one, all the sections are necessarily irreducible. However, they can be absolutely +reducible and we need to review all the possibilities. +As in section 4.5, we are reduced to list all the types of irreducible decompositions of quartics passing through +the six points, the last two with multiplicities at least 2. We follow the same line. +The orbits of lines of degree less than 4 that involve the six points are +ℓ5 ∪ ℓ6, +ℓ1 ∪ ℓ2 ∪ ℓ3 ∪ ℓ4, +ℓ56, +ℓ12 ∪ ℓ23 ∪ ℓ34 ∪ ℓ14, +ℓ16 ∪ ℓ25 ∪ ℓ36 ∪ ℓ45, +and +ℓ135 ∪ ℓ246. +25 + +There are five ways (cases 1 to 5 below) to combine these configurations in order to obtain a curve in the expected +linear system. +The orbits of conics of degree less than 4 that involve the six points are q1234 and q56. Note that compared +to the example of section 4.5, the orbit q1356 ∪ q2456 does not appear since a conic q1356 cannot be irreducible +otherwise it would have three intersection points with the line ℓ135. There are only two ways (cases 6 and 7 below) +to combine the configurations of lines and conics in order to obtain a curve in the expected linear system. +If the decomposition contains a cubic, it must be smooth at p5 and p6 and the complement must be ℓ56; this +is case 8. +This leads to the list below. In Y , the strict transforms �ℓ135 and �ℓ246 are separated from the strict transform �ℓ56. +In X this last curve is contracted to a smooth rational point p. Last, another difference from the example of +section 4.5: the two effective roots of X meet and they are thus contracted by the anticanonical morphism ϕ∗ +onto the same point s. This point is the unique singular point of Xs and it is necessarily a rational point. +���4ℓ − �4 +i=1 pi − 2p5 − 2p6 +��� +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +2ℓ56 ∪ ℓ135 ∪ ℓ246 +�ℓ56 ∪ �ℓ135 ∪ �ℓ246 ∪ E5 ∪ E6 +ϕ∗(E5) ∪ ϕ∗(E6) ∪ {s, p} +2 +2 +ℓ56 ∪ ℓ135 ∪ ℓ246 ∪ ℓ +�ℓ135 ∪ �ℓ246 ∪ �ℓ +ϕ∗(�ℓ) +q + 2 +3 +ℓ16 ∪ ℓ25 ∪ ℓ36 ∪ ℓ45 +�ℓ16 ∪ �ℓ25 ∪ �ℓ36 ∪ �ℓ45 +ϕ∗(�ℓ16) ∪ ϕ∗(�ℓ25) ∪ ϕ∗(�ℓ36) ∪ ϕ∗(�ℓ45) +0 +4 +ℓ5 ∪ ℓ6 ∪ ℓ135 ∪ ℓ246 +�ℓ5 ∪ �ℓ6 ∪ �ℓ135 ∪ �ℓ246 +ϕ∗(�ℓ5) ∪ ϕ∗(�ℓ6) ∪ {s} +2 +5 +2ℓ135 ∪ 2ℓ246 +�ℓ135 ∪ �ℓ246 ∪ E1 ∪ E2 ∪ E3 ∪ E4 +ϕ∗(E1) ∪ ϕ∗(E2) ∪ ϕ∗(E3) ∪ ϕ∗(E4) +1 +6 +ℓ135 ∪ ℓ246 ∪ q56 +�ℓ135 ∪ �ℓ246 ∪ �q56 +ϕ∗(�q56) ∪ {s} +q + 2 +7 +2ℓ56 ∪ q1234 +�q1234 ∪ {p} +ϕ∗(�q1234) ∪ ϕ∗({p}) +q + 2 +8 +c123456 ∪ ℓ56 +�c123456 +ϕ∗(�c123456) +Nq (1) +9 +t123456 singular at p5, p6 +�t123456 +ϕ∗(�t123456) +Nq (1) +Some comments about the numbers of points. +Case 1 & 5. The exceptional curves E5 and E6 do not contain any rational points. The other components +are all contracted to the points p or s. The same is true in case 5, without the point p. +Case 2. The ending curve passes through p but the contraction of �ℓ56 does not add any rational point since ℓ56 +and ℓ meet at a rational point. The ending curve passes through s and the contraction of the roots add a point +if ℓ meet ℓ135 and ℓ246 outside the meeting point of these two curves. +Case 3. All theses lines and ℓ135, ℓ246 and ℓ56 are separated by the blowing ups. Since the four lines cannot +contain any rational point, neither does their image in Xs. +Case 4. The conjugate lines ℓ5 and ℓ6 contain a unique rational point, their intersection point, to which is +added the point s. +Case 6. The curve �q56 no longer meets �ℓ135, �ℓ246 and �ℓ56. The ending curve contains the rational points of q56 +plus the point s. +Case 7. If q1234 meets ℓ56 at two conjugate points then in X, after �ℓ56 being contracted, the strict trans- +form �q1234 passes through p which is an additional rational point. +Necessarily the blowing ups of p1, . . . , p4 +separate the strict transforms �q1234, �ℓ135 and �ℓ246 and the roots contraction does not add any point. +Cases 8 & 9. Same as §4.5. +Finally Nq (−KXs) ≤ Nq(1). +As in the previous example, one has #Xs(Fq) = q2+1, and for q = 2, the evaluation map may not be injective. +Proposition 4.6. Suppose q ̸= 2. Let p1, p2 = pσ +1, p3 = pσ2 +1 , p4 = pσ3 +1 +∈ P2 be four conjugate points in general +position (no three of them are collinear), let p5 be a point of the line (p1p3) inside P2(Fq2) and let p6 = pσ +5 in such +a way that p6 lies on (p2p4). The anticanonical code of the weak del Pezzo surface obtained by blowing up these six +points and then blowing down the strict transform of the line (p5p6) has parameters [q2 + 1, 5, ≥ q2 + 1 − Nq (1)]. +Computation of the global sections from P2. — +This example looks like the previous one and we do not +find a nice explicit basis for the global sections. A slightly modification of the code given for the previous example +leads to a program which permits to compute a generator matrix. +4.7 +Degree 4, singularity of type D5 +This example corresponds to the type number 58 in degree 4 [BH22]. +26 + +Configuration to blow-up. — +In this example, the surfaces Y and X of diagram (8) are equal and we obtain +directly the surface X by blowing up P2 at five rational points p1 ≺ p2 ≺ · · · ≺ p5, with p1, p2, p3 collinear. Let +us denote by π1, . . . , π5 these five blowups at p1, . . . , p5 respectively: +P2 +X1 +X2 +X3 +X4 +X +π1 +π2 +π3 +π4 +π5 +π +The fact that p1, p2, p3 are collinear means that there is a line ℓ123 of P2 whose strict transform by π1 passes +through p2 and whose strict transform by π2 ◦ π1 passes through p3. +The anticanonical model of this weak +del Pezzo surface has a unique singular point of type D5 (since there are five irreducible effective roots whose +intersection graph is D5, see the picture at the end of this example). +Computation of the divisor class groups. — +Since all the blown-up points are rational, there is no need +to work with the base change X. The irreducible effective classes of roots are the strict transform of ℓ123 and +of E1, E2, E3, E4, whose classes in Cl(X) are: +E0 − E1 − E2 − E3, +E1 − E2, +E2 − E3, +E3 − E4, +and +E4 − E5. +The submodule R, generated by these classes, is a direct summand and for example Cl(X) = R ⊕ ZE5; the +projection onto the factor ZE5 leads to an isomorphism Cl(X)/R → ZE5. As for the submodule R⊥, it is defined +by the equations a1 = a2 = · · · = a5 and a0 + a1 + a2 + a3 = 0 and thus R⊥ = ZKX. Since +−KX = 3 (E0 − E1 − E2 − E3) + 2 (E1 − E2) + 4 (E2 − E3) + 6 (E3 − E4) + 5 (E4 − E5) + 4E5, +via the preceding isomorphism, the module R⊥ embeds via −KX �→ 4E5. +In brief, both divisor class groups CaCl(Xs) and Cl(Xs) are free rank one Z-modules, the first one being of +index 4 in the latter via the canonical embedding. +For this example, it makes sense to reverse the order of the paragraphs and we start to compute a basis of the +global sections. +Computation of the global sections from P2. — +We need to compute a basis of the sublinear system +on P2 +|3ℓ − p1 − · · · − p5| . +So we consider a cubic of P2 +X,Y,Z whose restriction to the affine space A2 +x1,y1 (x1 = X +Z , y1 = Y +Z and Z ̸= 0) is +defined by the equation: +C1(x1, y1) = a30x3 +1 + a21x2 +1y1 + a20x2 +1 + a12x1y2 +1 + a11x1y1 + a10x1 + a03y3 +1 + a02y2 +1 + a01y1 + a00 = 0 +We choose p1 = (0, 0) ∈ A2 +x1,y1. The cubic passes through the point p1 if and only if a00 = 0. +Let x2, y2 be the coordinates of the affine chart of the blowing up of A2 +x1,y1 at p1 defined by x1 = x2 +and y1 = x2y2. In this chart, the exceptional divisor E1 has equation x2 = 0 and the strict transform of C1 is +defined by: +C2(x2, y2) = 1 +x2 +C1(x2, x2y2) = a30x2 +2 +a21x2 +2y2 +a20x2 +a12x2 +2y2 +2 +a11x2y2 +a10 +a03x2 +2y3 +2 +a02x2y2 +2 +a01y2 = 0. +We choose p2 = (0, 0) ∈ A2 +x2,y2 which corresponds to the line ℓ123 with affine equation y1 = 0 in A2 +x1,y1 or Y. = 0 +in P2 +X,Y,Z. The cubic passes through the point p2 if and only if a10 = 0. +Let x3, y3 be the coordinates of the affine chart of the blowing up of A2 +x2,y2 at p2 defined by x2 = x3 +and y2 = x3y3. In this chart, the exceptional divisor E2 has equation x3 = 0 and the strict transform of C2 is +defined by: +C3(x3, y3) = 1 +x3 +C2(x3, x3y3) = a30x3 + a21x2 +3y3 + a20 + a12x3 +3y2 +3 + a11x3y3 + a03x4 +3y3 +3 + a02x2 +3y2 +3 + a01y3 = 0. +Since p1, p2, p3 are colinear, we have to choose p3 = (0, 0) ∈ A2 +x3,y3. The cubic passes through the point p3 if and +only if a20 = 0. +Let x4, y4 be the coordinates of the affine chart of the blowing up of A2 +x3,y3 at p3 defined by x3 = x4 +and y3 = x4y4. In this chart, the exceptional divisor E3 has equation x4 = 0 and the strict transform of C3 is +defined by: +C4(x4, y4) = 1 +x4 +C2(x4, x4y4) = a30 + a21x2 +4y4 + a12x4 +4y2 +4 + a11x4y4 + a03x6 +4y3 +4 + a02x3 +4y2 +4 + a01y4 = 0. +27 + +Since p4 ∈ E3, we have to choose p4 = (0, α) ∈ A2 +x4,y4 and since p1, p2, p3, p4 are not colinear, necessarily α ̸= 0. +The cubic passes through the point p4 if and only if a30 + αa01 = 0. +Last, let x5, y5 be the coordinates of the affine chart of the blowing up of A2 +x4,y4 at p4 defined by x4 = x5 +and y4 = α + x5y5. In this chart, the exceptional divisor E4 has equation x5 = 0 and the strict transform of C4 +is defined by: +C5(x5, y5) = 1 +x5 +C2(x5, x5y5) += 1 +x5 +� +a30 + a21x2 +5(α + x5y5) + a12x4 +5(α + x5y5)2 + a11x5(α + x5y5) + a03x6 +5(α + x5y5)3 ++a02x3 +5(α + x5y5)2 + a01(α + x5y5) +� +≡ αa11 + αa21x5 + a01y5 + a11x5y5 mod x2 +5Fq[x5, y5]. +(10) +Since p5 ∈ E4, one can choose p5 = (0, β) ∈ A2 +x5,y5. The cubic passes through the point p5 if and only if αa11 + +βa01 = 0. +To sum up, the global sections are defined by +a00 = a10 = a20 = 0, +a30 = −αa01, +and +a11 = −β +αa01. +The fact that α ̸= 0 is important here. In the projective setting, this leads to the basis +|3ℓ − p1 − · · · − p5| = +� +αY Z2 − βXY Z − α2X3, Y 3, X2Y, XY 2, Y 2Z +� +Fq . +(11) +Types of decomposition into irreducible components in |−KX|. — +Since CaCl(Xs) if of index 4 in- +side Cl(Xs), even if these two groups are free of rank 1, an irreducible Cartier divisor may decompose into Weil +irreducible components. In order to lower bound the minimum distance, we need to review all these kinds of +decompositions into irreducible components for the curves of the anticanonical linear system on Xs. As usual, we +start form P2 and use the one-to-one correspondences: +|3ℓ − p1 − · · · − p5|P2 +−→ +|−KX|X +−→ +|−KXs|Xs +C +�−→ +C♯ +�−→ +ϕ +� +C♯� +where C♯ denotes the virtual transform of C in the composition of the five blowups. +Thanks to the preceding computation, for every curve in |3ℓ − p1 − · · · − p5|P2 there exists α1, . . . , α5 ∈ Fq +such this curves is defined by +α1 +� +αY Z2 − βXY Z − α2X3� ++ α2Y 3 + α3X2Y + α4XY 2 + α5Y 2Z = 0. +We deduce that such a curve can decompose in six different ways, as listed in the tabular below: +• either a cubic c12345 for which p1 is a smooth flex point with tangent line equal to ℓ123, if α1 ̸= 0 (case 1); +• or a cubic singular at p1 which contains ℓ123 as a component, if α1 = 0, the complementary component, of +discriminant α3α2 +5 (up to a constant), being either +– a quadric q12 smooth at p1 with tangent line ℓ123, if α3 ̸= 0 and α5 ̸= 0 (case 2), +– or the union of two lines, if α3 ̸= 0 and α5 = 0, α3 = 0 and α5 ̸= 0, α3 = α5 = 0 and α4 ̸= 0, +α3 = α4 = α5 = 0 (cases 3, 4, 5, 6 respectively). +���3ℓ − �5 +i=1 pi +��� +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +c12345 +�c12345 +ϕ∗(�c12345) +Nq (1) +2 +ℓ123 ∪ q12 +�ℓ123 ∪ �q12 ∪ �E1 ∪ 2 �E2 ∪ 2 �E3 ∪ �E4 +ϕ∗(�q12) +q + 1 +3 +ℓ123 ∪ ℓ1 ∪ ℓ′ +1 +�ℓ123 ∪ 2 �E1 ∪ 2 �E2 ∪ 2 �E3 ∪ �E4 ∪ �ℓ1 ∪ �ℓ′ +1 +ϕ∗(�ℓ1) ∪ ϕ∗(�ℓ′ +1) +2q + 1 +4 +2ℓ123 ∪ ℓ +2�ℓ123 ∪ �E1 ∪ 2 �E2 ∪ 3 �E3 ∪ 2 �E4 ∪ E5 ∪ �ℓ +ϕ∗(E5) ∪ ϕ∗(�ℓ) +2q + 1 +5 +2ℓ123 ∪ ℓ1 +2�ℓ123 ∪ 2 �E1 ∪ 3 �E2 ∪ 4 �E3 ∪ 3 �E4 ∪ 2E5 ∪ �ℓ1 +2ϕ∗(E5) ∪ ϕ∗(�ℓ1) +2q + 1 +6 +3ℓ123 +3�ℓ123 ∪ 2 �E1 ∪ 4 �E2 ∪ 6 �E3 ∪ 5 �E4 ∪ 4E5 +4ϕ∗(E5) +q + 1 +28 + +Let us give details for the computation of the virtual transform π♯(C) if, for example, C = ℓ123 ∪ ℓ1 ∪ ℓ′ +1: +C1 = π♯ +1(C) = �ℓ123 + �ℓ1 + �ℓ′ +1 + 2E1 +[p1 ∈ ℓ123 ∩ ℓ1 ∩ ℓ′ +1 ⇒ mp1(C) = 3] +C2 = π♯ +2(C1) = �ℓ123 + �ℓ1 + �ℓ′ +1 + 2 �E1 + 2E2 +� +p2 ∈ �ℓ123 ∩ E1 ⇒ mp2(C1) = 3 +� +C3 = π♯ +3(C2) = �ℓ123 + �ℓ1 + �ℓ′ +1 + 2 �E1 + 2 �E2 + 2E3 +� +p3 ∈ �ℓ123 ∩ E2 ⇒ mp3(C2) = 3 +� +C4 = π♯ +4(C3) = �ℓ123 + �ℓ1 + �ℓ′ +1 + 2 �E1 + 2 �E2 + 2 �E3 + E4 +[p4 ∈ E3 ⇒ mp4(C3) = 2] +C♯ = π♯ +5(C4) = �ℓ123 + �ℓ1 + �ℓ′ +1 + 2 �E1 + 2 �E2 + 2 �E3 + �E4 +[p5 ∈ E4 ⇒ mp5(C4) = 1] . +This leads to the following decomposition of the canonical class into a sum of effective classes +−KX = (E0 − E1 − E2 − E3) + (E0 − E1) + (E0 − E1) + 2(E1 − E2) + 2(E2 − E3) + 2(E3 − E4) + (E4 − E5) +The intersection graph of the irreducible effective roots in X is connected (see figure below) and all these curves +are contracted by the morphism ϕ to a single rational singular pointy s (of singularity type D5). +on X +�E1(−2) +�E2(−2) +�E3(−2) +�ℓ123(−2) +�E4(−2) +E5(−1) +ϕ∗ +ϕ∗(E5) +•s +on Xs +We comment on the numbers of points. +Cases 2 & 6. All the components in X are roots that are contracted, except �q12 and E5 respectively. These +two strict transforms meet the tree of roots at only one point and by ϕ∗ they are mapped to isomorphic curves +that pass through s. +Case 3. Except �ℓ1 and �ℓ′ +1, all the components on X are irreducible effective roots and they are mapped to +the point s by the morphism ϕ∗. After the contraction, the curves ϕ∗(�ℓ1) and ϕ∗(�ℓ′ +1) meet at this singular point, +thus their union contains 2q + 1 rational points. +Cases 4 & 5. The line E5 does not intersect the lines �ℓ or �ℓ1 in X. However, since ℓ and ℓ123 meet at some +point of P2 (not equal to p1), the lines �ℓ and �ℓ123 intersect in X; in the same way since ℓ1 passes through p1, the +lines �ℓ1 and �E1 intersect in X. Therefore, ϕ∗(E5) and ϕ∗(�ℓ) or ϕ∗(E5) and ϕ∗(�ℓ1) both intersect at s. Thus the +two unions has 2q + 1 rational points. +Finally Nq (−KXs) = 2q + 1. +The surface Xs has a unique singular point s. All the irreducible effective roots of X, that is �ℓ123, �E1, . . . , � +E4 +are contracted to this single point. The last exceptional curve E5 meets E4 and thus ϕ(E5) passes through s. +In conclusion the rational points of Xs(Fq) are in one-to-one correspondence with +� +P2(Fq) \ ℓ123(Fq) +� +∪ E5(Fq), +which counts q2 + q + 1 elements. This number is always strictly greater that 2q + 1 and the evaluation map is +always injective. +Proposition 4.7. Let p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 be infinitely near rational points. Suppose that the first three +ones are collinear. The anticanonical code of the weak del Pezzo surface obtained by blowing up these points has +parameters [q2 + q + 1, 5, q2 − q]. +The construction of a generator matrix of this code is a nice application of proposition 3.3. It consists of +two blocks, the left one, of size 5 × q2, contains the evaluations of the five global sections of (11) at every point +of P2(Fq)\ℓ123(Fq), the right one, of size 5×(q+1) contains the evaluations of the homogeneous parts of degree 1 +of the five global sections of (11) at every point of P1(Fq). Letting y5 = β + z5 in (10), this homogeneous part of +degree 1 equals (αa21 + βa11)x5 + a01z5. Finally, we get the explicit matrix: + + + + + + + + + + +αyz2 − βxyz − α2x3 +... +−β2u + αv +... +y3 +... +0 +... +x2y +(x : y : z) ∈ P2(Fq) | y ̸= 0 +αu +(u : v) ∈ P1(Fq) +xy2 +... +0 +... +y2z +... +0 +... + + + + + + + + + + +, +where α ∈ F∗ +q and β ∈ Fq. +29 + +4.8 +Degree 3, singularity of type A1 +This example corresponds to the type number 11 in degree 3 [BH22]. +Configuration to blow-up and down. — +We blow up six conjugate points p1, . . . , p6 ∈ P2 on a smooth +conic q123456. +p1 +p2 +p3 +p4 +p5 +p6 +q123456 +p2 = pσ +1 +p3 = pσ2 +1 +p4 = pσ3 +1 +p5 = pσ4 +1 +p6 = pσ5 +1 +The resulting surface X is a weak del Pezzo of degree 3, whose anticanonical model Xs has a unique singular +point of type A1. +Computation of the divisor class groups. — +We have: +Cl(X) = +6 +� +i=0 +ZEi +and +Cl(X) = ZE0 ⊕ ZE, +where E = +6 +� +i=1 +Ei. +There is a unique irreducible effective root, the strict transform of the conic q123456, whose class is 2E0 − E. The +root module R, generated this class, is a direct summand, Cl(X) = R ⊕ ZE0. The projection onto the second +factor leads to an isomorphism +Cl(X)/R +−→ +ZE0 +E0 mod R +�−→ +E0 +E mod R +�−→ +2E0 +. +As for the module R⊥, inside Cl(X) it is defined by the single equation 2a0 + a1 + · · · + a6 = 0; after taking +the Galois invariants, we obtain CaCl(Xs) = ZKX, whose image by the previous isomorphism is also ZE0. +Therefore CaCl(Xs) ≃ Cl(Xs) and both of them are free of rank one. +Types of decomposition into irreducible components in |−KX|. — +This proves that all the sections of +the anticanonical divisor are irreducible. As in our previous work [BCH+20], we expect that the curves of the +associated linear system can contain at most Nq (1) rational points. However we need to investigate the types of +irreducible decompositions. Here this is easy since one can check that the only Galois orbits of lines or conics or +cubics that pass through at least one point pi are ℓ14∪ℓ25∪ℓ36 or q123456 or c123456 (all the others lead to Fq-curves +of degree strictly greater than 3). Combining them in order to construct a curve in the expected sub-linear system +leads to very few decompositions: +���3ℓ − �6 +i=1 pi +��� +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +ℓ14 ∪ ℓ25 ∪ ℓ36 +�ℓ14 ∪ �ℓ25 ∪ �ℓ36 +ϕ∗(�ℓ14) ∪ ϕ∗(�ℓ25) ∪ ϕ∗(�ℓ36) +1 +2 +ℓ ∪ q123456 +�ℓ ∪ �q123456 +ϕ∗(�ℓ) ∋ s +q + 2 +3 +c123456 +�c123456 +ϕ∗(�c123456) +Nq (1) +The number of rational point in case 1 is at most 1 if the three lines meet. In case 2, during the process, if the two +meeting points of ℓ and q123456 are not rational, then the singular point s is an additional rational point on ϕ∗(�ℓ). +We deduce that Nq (−KXs) ≤ Nq(1). +Since the blown up points are not rational, the blowing ups do not add point on the surface and #X(Fq) = +q2 + q + 1. Then, the irreducible effective root is contracted and thus #Xs(Fq) = q2 + 1. If q = 2, the evaluation +map may fail to be injective. +Proposition 4.8. Suppose q ̸= 2. Let p1, . . . , p6 ∈ P2 be six conjugate points lying on a smooth conic. The +anticanonical code of the weak del Pezzo surface obtained by blowing up these points has parameters [q2 + 1, 4, ≥ +q2 + 1 − Nq (1)]. +30 + +Computation of the global sections from P2. — +Let Q denote the conic passing through p1, . . . , p6 and +let L14, L25, L36 be the linear forms whose zeros are the lines ℓ14, ℓ25, ℓ36. Then +H0 � +P2, 3ℓ − �6 +i=1 pi +� += ⟨XQ, Y Q, ZQ, L14L25L36⟩Fq +4.9 +Degree 3, singularity of type 3A2 +This example corresponds to the type number 76 in degree 6 [BH22]. As in section 4.5, this example appears in +Koshelev’s work [Kos20, §1.1] but with another point of view. +Configuration to blow-up. — +First we blow up three non-collinear conjugate points p1, p2, p3. This leads +to a degree 6 del Pezzo surface with three exceptional conjugate curves E1, E2, E3, the other exceptional curves +being the strict transforms ℓ12, ℓ13, ℓ23 of the lines joining two of the three points. Then we blow up three other +points p4, p5, p6 with pi+3 ≻ pi, and more precisely p4 is the intersection point of E1 and �ℓ12, p5 is the intersection +point of E2 and �ℓ23 and p6 is the intersection point of E3 and �ℓ13. These points are also conjugate and the +resulting surface X is a weak degree three del Pezzo surface, with three new exceptional curves E4, E5, E6. The +anticanonical model Xs has three conjugate singular points of type A2. +p4 +p5 +p6 +•p1 +•p2 +•p3 +p2 = pσ +1 +p5 = pσ +4 +p3 = pσ2 +1 +p6 = pσ2 +4 . +The point p4 lies on the strict transform of the line (p1p2), which we denote by ℓ124. In the same way we introduce +the lines ℓ235 and ℓ136. +Computation of the divisor class groups. — +There are six irreducible effective roots, the strict transforms +of E1, E2, E3 and the strict transforms of ℓ124, ℓ235, ℓ136; their classes are: +R1 = E1 − E4, +R2 = E2 − E5, +R3 = E3 − E6, +R′ +1 = E0 − E1 − E2 − E4, +R′ +2 = E0 − E2 − E3 − E5, +R′ +3 = E0 − E1 − E3 − E6. +The absolute Galois group acts on this six root classes as (R1R2R3)(R′ +1R′ +2R′ +3) and also on the exceptional curves +as (E1E2E3)(E4E5E6) (the first three exceptional curves are the total transforms of the exceptional curves on +the degree 6 del Pezzo surface, they are no longer irreducible). +We have +Cl(X) = +6 +� +i=0 +ZEi +R = +3 +� +i=1 +ZRi ⊕ ZR′ +i +and +R +⊥ = ZKX. +Let us put: +E = E1 + E2 + E3, +E′ = E4 + E5 + E6, +R = R1 + R2 + R3 = E − E′, +R′ = R′ +1 + R′ +2 + R′ +3 = 3E0 − 2E − E′. +One easily verify that +Cl(X)Γ = ZE0 ⊕ ZE ⊕ ZE′, +R = R +Γ = ZR ⊕ ZR′ = Z(E − E′) ⊕ Z(3E0 − 2E − E′), +and +R⊥ = ZKX. +It turns out that the submodule R is not a direct summand in Cl(X); indeed +R = Z(E − E′) ⊕ Z3(E0 − E) ⊂ Z(E − E′) ⊕ Z(E0 − E) ⊕ ZE′ = Cl(X) +(we have just replaced 3E0 − 2E − E′ by (3E0 − 2E − E′) − (E − E′) in the initial basis). Therefore the projection +onto the two last factors leads to an isomorphism: +Cl(Xs) ≃ Cl(X)/R +−→ +Z/3Z(E0 − E) +⊕ +ZE′ +a0E0 + aE + a′E′ mod R +�−→ +(a0 mod 3) (E0 − E) ++ +(a0 + a + a′)E′ +Via this isomorphism the group CaCl(Xs) = R⊥ = ZKX embeds via −KX �→ E′; this means that CaCl(Xs) is +isomorphic to the free part of Cl(Xs) and these two groups are free of rank one. +31 + +Types of decomposition into irreducible components in |−KX|. — +As in the previous case, the global +sections of the divisor |−KXs| are irreducible but not necessarily absolutely irreducible. As usual, we list the +Galois orbits of lines or conics or cubics of degree less than 3 that pass through at least one of the six points. The +only possibilities are +ℓ1 ∪ ℓ2 ∪ ℓ3, +ℓ124 ∪ ℓ235 ∪ ℓ136, +q123, +c123, +c123456. +(it is important to keep in mind that a curve which passes through p4 necessarily passes through p1). There are +only two combinations that lead to a cubic which passes through the six points: +���3ℓ − �6 +i=1 pi +��� +|−KX| +|−KXs| +Max +on P2 +on X +on Xs +nb. of pts +1 +ℓ124 ∪ ℓ235 ∪ ℓ136 +�ℓ124 ∪ �ℓ235 ∪ �ℓ136 ∪ �E1 ∪ �E2 ∪ �E3 ∪ E4 ∪ E5 ∪ E6 +ϕ∗(E4) ∪ ϕ∗(E5) ∪ ϕ∗(E6) +0 +2 +c123456 +�c123456 +ϕ∗(�c123456) +Nq (1) +The roots of X are �ℓ124, �E2 (mapped to a singular point s ∈ Xs), �ℓ235, �E3 (mapped to a singular point sσ ∈ Xs), +and �ℓ136, �E1 (mapped to a singular point sσ2 ∈ Xs). The curves Ei, i = 4, 5, 6, are not defined over Fq and do +not contain any rational point. In conclusion Nq (−KXs) ≤ Nq(1). +Since the points p1, . . . , p6 are not rational the blowing ups do not add any rational point, and since the +singular points are not rational the contractions do not add any rational point also. Thus #Xs(Fq) = q2 + q + 1, +this number is always strictly greater than Nq(1) and we deduce the parameters given below. +Proposition 4.9. The weak del Pezzo surface of degree 3 associated to the configuration specified at the beginning +of this section has parameters [q2 + q + 1, 4, ≥ q2 + q + 1 − Nq (1)]. +Koshelev [Kos20, §1.1] proves that the minimum distance can be improved by 1 for some q since he shows +that cubics of the considered linear system must have a 3-torsion point. +Computation of the global sections from P2. — +Let L12, L23, L13 be the three conjugate linear forms that +respectively define the lines ℓ124, ℓ235, ℓ136 in P2. The family L12, L23, L13 is a Fq-basis of H0(P2, ℓ), while the +family of degree 3 monomials in L12, L23, L13 is a Fq-basis of H0(P2, 3ℓ). A cubic in this space can be written: +a1L3 +12 + a2L3 +23 + a3L3 +13 + b1L12L2 +23 + c1L13L2 +23 + b2L12L2 +13 + c2L23L2 +13 + b3L13L2 +12 + c3L23L2 +12 + dL12L23L13 +Such a cubic pass through p1 if and only if a2 = 0 (since p1 is a common zero of L12 and L13). In the same way +it passes through p2 and p3 if and only if a3 = 0 and a1 = 0. Now passing through p4 means that if this cubic +is not singular at p1 then its tangent line at this point must be ℓ12. After deshomogenizing by putting L23 = 1 +(this is possible since L23 does not vanish at p1) this means that the linear component b1L12 + c1L13 should +be proportional to L12; necessarily c1 = 0. In the same way passing through p5 (resp. p6) means that b2 = 0 +(resp. c3 = 0). Finally, one has +H0 � +P2, 3ℓ − �6 +i=1 pi +� += +� +L12L2 +23, L23L2 +13, L13L2 +12, L12L23L13 +� +Fq +In order to deduce a Fq-base, we consider θ any primitive element of Fq3 over Fq. The linear independence of +homomorphisms permits to prove that the matrix (σi(θj))1≤i,j≤3 is invertible. Let us put: +C1 = L12L2 +23 + L23L2 +13 + L13L2 +12 +Cθ = θL12L2 +23 + σ(θ)L23L2 +13 + σ2(θ)L13L2 +12 +Cθ2 = θ2L12L2 +23 + σ(θ2)L23L2 +13 + σ2(θ2)L13L2 +12 +then C, Cθ, Cθ2 are defined over Fq, as the product L12L23L13 and one has: +H0 � +P2, 3ℓ − �6 +i=1 pi +� += ⟨C1, Cθ, Cθ2, L12L23L13⟩Fq +The birational morphism +P2 +��� +P4 +(X : Y : Z) +�−→ +(C1 : Cθ : Cθ2 : L12L23L13) +has Xs as image in P4. Thus, if r1, . . . , rq2+q+1 denote the rational points of P2, one of the generating matrix of +this code is nothing else than: + + + + +C1(r1) +· · · +C1(rq2+q+1) +Cθ(r1) +· · · +Cθ(rq2+q+1) +Cθ2(r1) +· · · +Cθ2(rq2+q+1) +L12L23L13(r1) +· · · +L12L23L13(rq2+q+1) + + + + +32 + +References +[AW92] +William A. Adkins and Steven H. 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Manin, Cubic forms. algebra, geometry, arithmetic, North Holland Publishing Compagny, 1974. +[Ser20] +Jean-Pierre Serre, Rational points on curves over finite fields, Documents Math´ematiques (Paris) +[Mathematical Documents (Paris)], vol. 18, Soci´et´e Math´ematique de France, Paris, [2020] ©2020, +With contributions by Everett Howe, Joseph Oesterl´e and Christophe Ritzenthaler, Edited by Alp +Bassa, Elisa Lorenzo Garc´ıa, Christophe Ritzenthaler and Ren´e Schoof. +[Sta18] +The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, 2018. +[Zar07] +Marcos Zarzar, Error-correcting codes on low rank surfaces, Finite Fields Appl. 13 (2007), no. 4, +727–737. +33 + diff --git a/NNFKT4oBgHgl3EQfei69/content/tmp_files/load_file.txt b/NNFKT4oBgHgl3EQfei69/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8c2b84dbc0f45f705385bc92b7d7f49c54388ff --- /dev/null +++ b/NNFKT4oBgHgl3EQfei69/content/tmp_files/load_file.txt @@ -0,0 +1,2222 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf,len=2221 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='11825v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='AG] 27 Jan 2023 Construction of good codes from weak Del Pezzo surfaces R´egis Blache and Emmanuel Hallouin January 30, 2023 Contents 1 Introduction 1 2 Generalities on weak del Pezzo surfaces 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 Ordinary versus non ordinary weak del Pezzo surfaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 Degree 5, singularity of type 2A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4 Degree 4, singularity of type A1 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7 Degree 4, singularity of type D5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8 Degree 3, singularity of type A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9 Degree 3, singularity of type 3A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 31 1 Introduction The geometric construction of error correcting codes goes back to Reed-Solomon and Goppa for curves and to Reed-Muller for affine or projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this work, we focus on evaluation codes from algebraic surfaces whose construction works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Given X an algebraic surface over Fq and D a (Cartier) divisor on X, we denote by X(Fq) the set of rational points of X and by H0(X, D) the space of global sections of the divisor D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The associated evaluation code is the code whose codewords are the evaluations of the functions of H0(X, D) at the points of X(Fq) (see definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The length of such a code is thus #X(Fq), the dimension is dim � H0(X, D) � (at least if the evaluation map is injective) and the third invariant, the minimum distance, is more difficult to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It is related to the maximum of rational points that can contain a curve in the linear system |D| (proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The lower this maximum, the better is the minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The well known Reed-Muller code of degree d for the projective plane P2 over Fq is a nice example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Its codewords are the evaluations of the homogeneous polynomials of degree d at the rational points of P2(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the geometric setting above, this is the evaluation code associated to the algebraic surface P2, the divisor dℓ where ℓ denotes any line, and the whole set of rational points P2(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Its parameters are well known when q > d: the length is the number of rational points #P2(Fq) = q2 + q + 1, the dimension is the dimension of the space of global section dim � H0(P2, dℓ) � = �d+2 2 � , and the minimum distance is q(q − d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This minimum distance can be written (q2 + q + 1) − (1 + dq) and (1 + dq) is nothing else than the maximal number of rational points of a curve lying in the linear system |dℓ| (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' the set of plane curves of degree d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In fact, this is the number of points of the union of d lines of P2 meeting at one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The existence of this kind of extremely reducible curve over Fq impacts negatively the minimum distance, since they contain too many rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Among other things, this is why evaluation codes associated to more general algebraic surfaces have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 1 One can distinguish several strategies in the literature to get rid of reducible curves with many components in the linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The idea of Couvreur [Cou11] is to work with sublinear systems of P2 by adding constraints that remove the very reducible sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In fact by choosing carefully the sublinear system, this kind of sections is no longer defined over the base field, but only over an extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The number of rational points of irreducible curves that are not absolutely irreducible may fail drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the preceding example of the d intersecting lines, if they are not defined over Fq but only conjugate over Fq, then one can easily convince ourselves that their union only contains one rational point, their meeting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Some other examples can be found in Edoukou [Edo08] or Couvreur & Duursma [CD13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Following Zarzar [Zar07], another fairly repeated strategy is to concentrate on surfaces whose (arithmetic) Cartier class group is free of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Indeed, this is a natural way to overcome the difficulty of the existence of (very) reducible sections in the linear system |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Little and Schenck [LS18] have studied anticanonical codes on degree 3 and 4 del Pezzo surfaces having rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In our previous work [BCH+20], we could say that we fill a gap in the study of algebraic geometric codes constructed from del Pezzo surfaces of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us remark that, even if the elements of the linear system |D| are all irreducible, some of them may be absolutely reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As in the example of conjugate lines, it is expected that these configurations do not contain too many points but this requires a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this work, we continue the investigation of codes constructed from del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We do not restrict ourselves to rank one surfaces but above all we consider more general surfaces, that is non ordinary weak del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As ordinary del Pezzo surfaces, non ordinary weak del Pezzo surfaces admit a blowing-up description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' in the ordinary case, the points that are blown up are in general position but in the non ordinary case, they are only in almost general position (three points can be colinear and six points can be conconic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The main consequences of these weaker hypotheses on the configuration of points are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' First, the surface contains −2-curves (and not only −1-curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Secondly, the anticanonical divisor is not ample anymore but only big and nef and the anticanonical model is singular with rational double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In a concomitant work [BH22], we have computed explicit models for all the arithmetic types of weak del Pezzo surfaces of degree at least 3 over a finite field (these types lead to a classification that is coarser than the isomorphism one but that permits to distinguish the main arithmetic properties of the weak del Pezzo surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Taking advantage of this knowledge, we select eight types of (non ordinary) weak del Pezzo that are well suited for coding applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' More precisely, we consider X a (smooth) weak del Pezzo surface of degree d over Fq and we denote by Xs its (singular) anticanonical model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this is the image of the surface X by the morphism ϕ associated to −KX the anticanonical divisor of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since −KX is not ample, the surface Xs is singular with a finite number of rational double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We study the evaluation code associated to the (singular) surface Xs, the Cartier divisor −KXs = ϕ∗(−KX) and the whole set of rational points of Xs (definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Except for small values of q, this code has length n = #Xs(Fq), dimension k = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The last invariant, the minimum distance dmin, is much more subtle to control and requires preparatory calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Before going into details, let us discuss the advantages and disadvantages of considering such weak Del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the process of construction of a del Pezzo surface, there are blowing-up and blowing-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The blowing-up may add rational points and thus may increase the length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The blowing-down permits to contract some lines and thus decreases the types of reducible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the anticanonical model is no longer smooth, besides the exceptional curves some other curves, in fact the effective roots, can be contracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If these curves are components of the most reducible sections of the anticanonical divisor on the weak del Pezzo, the parameters of the code could be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This is the positive aspect of considering anticanonical model of weak Del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' But we should also mention a negative one: because of the singularity of Xs, the notions of Cartier and Weil divisors are not equivalent and this makes it difficult to calculate the minimum distance dmin as we will see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The computation of dmin reduces to compute the number: Nq (−KXs) = max {#C(Fq) | C ∈ |−KXs|} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since every curve of the linear system |−KXs| is of arithmetic genus 1 (adjunction formula), all its absolutely irreducible curves have a number of rational points which is bounded above by the classic: Nq(1) = max {#C(Fq) | C absolutely irreducible, smooth, genus 1, curves over Fq} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' By the Weil-Serre bound, we know that Nq(1) ≤ q + 1 + ⌊2√q⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' in fact, except for very special values of q, the Weil-Serre bound turns to be sharp: Nq(1) = � q + ⌊2√q⌋ if q = pe, e ≥ 5, e odd and p | ⌊2√q⌋, q + 1 + ⌊2√q⌋ otherwise ([Ser20, Chap 2, Th 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Anyway, this bound does not permit to control the number of rational points of reducible or absolutely reducible curves of |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Due to the singularities of Xs or more specifically to the difference between the Cartier or Weil divisors or class groups, the expectation that irreducible, but absolutely 2 reducible curves in the linear system do not contain too many points is more difficult to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Even if the Cartier class group CaCl(Xs) is free of rank 1, generated by −KXs, this does not mean that the curves of the linear system |−KXs| are all irreducible since they can decompose in the Weil class group Cl(Xs), that is into a sum of Weil irreducible divisors that are not Cartier divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' To overcome this difficulty, we took fall advantage of the fact that in the context of weak del Pezzo surfaces, explicit models of all the class groups can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This permits us to accurately measure the difference between the Cartier and the Weil divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This step uses some basic methods on lattices computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then to explicitly compute the maximum Nq (−KXs), we list all the kinds of decompositions into irreducible components that may appear in the linear system |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In general, this can be a difficult issue but in our context this task is greatly facilitated by the fact that all the considered surfaces are blowing-up and down of the projective plane: as explained in Hartshorne’s classic [Har77, Chap V, beginning of §4 & Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1], we are brought back to the study of some sub-linear systems of plane curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We choose examples that illustrate the variety of situations that may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the column CaCl(Xs) ֒→ Cl(X) in the tabular below, we see that the Cartier class group CaCl(Xs) always embeds in the Weil class group Cl(Xs), and via this embedding CaCl(Xs) may be equal to Cl(Xs), or of finite index into Cl(Xs), or of positive co-rank into Cl(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Note also that the lattice CaCl(Xs) is always free, whereas Cl(Xs) may have a torsion subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the column Nq (−KXs), one can see that this is not always the absolutely irreducible curves of the linear system that contains the maximum of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Only a case-by-case proof and a carefully study of all the geometric properties permits to estimate the three invariants [n, k, dmin] that are contained in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' CaCl(Xs) ֒→ Cl(X) Nq (−KXs) [n, k, dmin] §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 6 A1 2Z ֒→ Z 2q + 1 � q2 + 1, 7, q2 − 2q � §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 5 2A1 Z ⊕ 2Z ֒→ Z ⊕ Z 2q + 2 � q2 + q + 1, 6, q2 − q − 1 � §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4 4 A1 Z ≃ Z ≤ Nq(1) � q2 − q + 1, 5, ≥ q2 − q + 1 − Nq(1) � §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 4 4A1 Z ֒→ Z ⊕ Z/2Z ≤ Nq(1) � q2 + 1, 5, ≥ q2 + 1 − Nq(1) � §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6 4 A2 Z ≃ Z ≤ Nq(1) � q2 + 1, 5, ≥ q2 + 1 − Nq(1) � §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7 4 D5 4Z ֒→ Z 2q + 1 � q2 + q + 1, 5, q2 − q � §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8 3 A1 Z ≃ Z ≤ Nq(1) � q2 + 1, 4, ≥ q2 + 1 − Nq(1) � §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9 3 3A2 Z ֒→ Z ⊕ Z/3Z ≤ Nq(1) � q2 + q + 1, 4, ≥ q2 + q + 1 − Nq(1) � In the tabular above, the inequality Nq (−KXs) ≤ Nq(1) means that the curves of the linear system |−KXs| that contain the maximum number of points are the absolutely irreducible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' They are all of arithmetic genus 1, but it may happen that none of these curves is maximal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' has a number of rational points equal to Nq(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This is why in these cases, one can only give an upper bound for Nq (−KXs) and thus a lower bound for dmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It turns out that we recover two examples of Koshelev [Kos20] (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9), where he proved that the linear systems cannot contain a maximal genus one curve for certain finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This permits to increase the lower bound of the minimum distance by one over these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' All the presented codes can be easily constructed using a mathematics software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On the second author’s webpage, we put a magma program that permits to construct all our codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 2 Generalities on weak del Pezzo surfaces Let k be a finite field (most of the results remain true on any field), k its algebraic closure, Γ = Gal(k/k) its absolute Galois group and let σ be the Frobenius automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this section, we recall the classical properties of del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, we focus on the specificities of non ordinary weak del Pezzo surfaces compared to the ordinary ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The essential references for the content of this section are the book of Manin [Man74] or the more recent one of Dolgachev [Dol12, §8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 Ordinary versus non ordinary weak del Pezzo surfaces There are several definitions of a del Pezzo surfaces, even in the Dolgachev’s classic [Dol12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' let us start with the definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='18 of this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A smooth projective surface X is a weak del Pezzo surface if its anticanonical divisor −KX is: (i) big, which means that K·2 X > 0, (ii) and nef, which means that (−KX) · D ≥ 0 for any effective divisor D on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 3 The self-intersection K·2 X is the degree of the del Pezzo surface X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thanks to the Nakai-Moishezon criterion [Har77, Chap V, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='10], these kinds of surfaces are divided into two cases: (i) either the inequalities in (ii) are all strict ((−KX) · D > 0): the anticanonical divisor is thus ample and we say that the del Pezzo surface is an ordinary one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (ii) or there exists an effective divisor D such that (−KX) · D = 0: the anticanonical divisor is not ample and we say that the del Pezzo surface is a non ordinary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These properties have consequences on the negative curves on X, those whose self-intersection is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Indeed, let C be an absolutely irreducible curve on X of arithmetic genus γ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' By adjunction formula, we know that C·2 = 2γ(C)−2+C·(−KX) and since γ(C) ≥ 0 and C·(−KX) ≥ 0, we deduce that C·2 ≥ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus, negative curves on X have self-intersection −2 or −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Moreover C·2 = −2 if and only if γ(C) = 0 and C · (−KX) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this means that only non ordinary del Pezzo surfaces can contain (−2)-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We also prove the same way that (−1)-curves on weak del Pezzo surfaces must have arithmetic genus equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This motivates the following definition which deals with negative curves but also divisor classes of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a weak del Pezzo surface over a field k, let X = X ⊗ k be its extension to the algebraic closure k and let Cl(X) denote the divisor class group of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (i) A divisor class D ∈ Cl(X) is an exceptional class if D·2 = D·KX = −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' an absolutely irreducible curve C on X whose class is exceptional is an exceptional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (ii) A divisor class D ∈ Cl(X) is a root if D·2 = −2 and D · KX = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' a curve C on X whose class is a root is an effective root and if such a curve is absolutely irreducible then C is called a (−2)-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It is well known that the geometry of weak del Pezzo surfaces depends to a large extent of these negative curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For example, if X is a weak ordinary del Pezzo surface then all the exceptional classes are the classes of a (unique) exceptional curve and no root is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On the contrary, if X is weak non ordinary del Pezzo surface then some exceptional classes may be represented by reducible curves and some roots are effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These differences of behaviours appear naturally in the blowup description of the generalized del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 The blow-up model Over k, every del Pezzo surface can be obtained by a sequence of blowing ups starting from the projective plane P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This description makes most of the invariants of the surface very explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Recall that if π : Y → X is the blowing up of a smooth surface X at a point p, with exceptional divisor E, then Cl(Y ) = π∗ Cl(X)⊕ZE, the intersection pairing on Y satisfying E2 = −1, π∗D·E = 0 and π∗D·π∗D′ = D·D′ for all divisors D and D′ of X (the blowing up is an isometry for the intersections pairings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Moreover KY = π∗KX + E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We recall that r ≤ 8 points in P2(k) are said to be in general position if and only if no three lie on a line, no six lie on a conic, and there is no cubic through seven of them having a singular point at the eighth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' in almost general position if and only if no four lie on a line and no seven lie on a conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Del Pezzo surfaces can always be described as follows [Dol12, Th 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a generalized del Pezzo surface over k and let X = X ⊗k k its extension to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If X is of degree d, with 3 ≤ d ≤ 6, then X is the blowing up of P2 at r = 9 − d points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pr in almost general position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' more precisely X results in r successive blowups π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , πr: X πr −→ Xr −→ · · · −→ X2 π1 −→ X1 := P2 k where pi ∈ Xi are in almost general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let E0 be the class of a line in P2 and let E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , Er be the exceptional curves at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then the divisor class group of X with its intersection pairing can be easily described: Cl(X) = ZE0 ⊕ ZE1 ⊕ · · · ⊕ ZEr Mat ( · , (Ei)0≤i≤r) = Diag(1, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=', −1), where Diag denotes the diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This also gives explicitly the canonical class: KX = −3E0 + r � i=1 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (1) The negative classes can be expressed in terms of the basis E0, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , Er [Dol12, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4 Name Exceptional classes E Roots R Conditions E·2 = −1 and E · (−KX) = 1 R·2 = −2 and R · KX = 0 Expression Ei, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=', r} Rij = Ei − Ej, {i, j} ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=', r} in terms of the Ei Eij = E0 − Ei − Ej, {i, j} ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=', r} Rijk = E0 − Ei − Ej − Ek, −Rijk Ei1···i5 = 2E0 − �5 j=1 Eij {i, j, k} ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=', r} {i1, i2, i3, i4, i5} ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=', r} Ri1···i6 = 2E0 − �6 j=1 Eij, −Ri1···i6 {i1, i2, i3, i4, i5, i6} ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=', r} We note that, in the notation Eij, the indices are unordered (which leads to �r 2 � possibilities), whereas they are ordered in the notation Rij since Rji = −Rij (which leads to 2 �r 2 � possibilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Not all these divisor classes are effective and the effectiveness of certain of these classes differentiate some types of Del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the ordinary case, each exceptional class of divisor is represented by a unique irreducible curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Either it is one exceptional curve Ei for some 1 ≤ i ≤ r or the strict transform of the line of P2 passing through pi and pj for the class Eij or the strict transform of the (unique) conic of P2 passing through the pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pi5 for Ei1i2i3i4i5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Their intersection graph is an important invariant of the ordinary Del Pezzo surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' figures of these graphs for 3 ≤ r ≤ 5 can be found in Manin [Man74, §26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9] or in Dolgachev [Dol12, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3, Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As for the root classes, no one is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the non ordinary cases, where the points are no longer in general position but only in almost general position, the exceptional divisors are still effective but not necessarily represented by irreducible curves anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For example, if p1, p2, p3 are collinear then the root R123 becomes effective since it is the class of the strict transform of the line passing through p1, p2, p3 and the four exceptional classes E12, E13, E23, E12345 are represented by reducible curves since E12 = R123 + E3, E13 = R123 + E2, E23 = R123 + E1, E12345 = R123 + E45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this case, all other exceptional divisors are still represented by irreducible curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Another simple example: if p2 is chosen to be on E1, p2 ≻ p1, then the root E1 − E2 becomes effective since it is the strict transform of E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The exceptional classes E1 and E1j j ̸= 1, 2 are no longer represented by irreducible curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In general, a result of Demazure states that exceptional divisors that are represented by irreducible curves are characterized by the fact that they intersect non negatively (≥ 0) all the irreducible roots [CT88, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Another general result states that the set of irreducible roots (the effective classes represented by an irreducible curve) is necessarily a free family in Cl(X⊗Fq) (see loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, there are at most r effective irreducible roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The lattice generated by the effective roots plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a generalized del Pezzo surface over k and let X = X ⊗k k be its extension to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We denote by R the sub-lattice of Cl(X) generated by the effective roots and by R sub-lattice of Cl(X) defined by R = R Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Following Coray and Tsfasman (see loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ), an important invariant of a weak Del Pezzo surface is the graph of negative curves, which is an analog of the intersection graph of the exceptional divisors/curves introduced above in the ordinary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' To take into account the fact that the surface may be non ordinary, the set of vertices is modified: the vertices corresponding to reducible exceptional divisors are cancelled, while vertices corresponding to effective and irreducible roots are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 The anticanonical model Xs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 The morphism induced by the anticanonical divisor −KX Let X be a del Pezzo surface of degree d, with 3 ≤ d ≤ 6 and whose canonical divisor is denoted by KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the ordinary case, the anticanonical divisor −KX is known to be very ample and it induces a projective embedding of X into Pd = P(H0(X, −KX)) (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 for a review about the space of global sections of a divisor) In the non ordinary case, the anticanonical class −KX is no longer ample but its linear system remains base point free and gives a morphism from X to a projective space: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a weak del Pezzo surface of degree d, with 3 ≤ d ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The image ϕ(X), where ϕ : X → P � H0(X, −KX) � = Pd is the projective morphism associated to the anticanonical divisor −KX is called the anticanonical model of X and is denoted by Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We put KXs = ϕ∗(KX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This kind of del Pezzo surface corresponds to the definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 in Dolgachev [Dol12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The fifth talk of Demazure [Dem80, Expos´e V] on del Pezzo surfaces contains all the main properties of this anticanonical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The morphism ϕ satisfies: 5 (i) it is not a projective embedding (since −KX is not ample) but the image Xs is a normal surface whose singularities are rational double points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (ii) it is the minimal desingularization of Xs, it contracts all the irreducible effective roots on X into the singular points and nothing else;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (iii) the Weil divisor KXs = ϕ∗(KX) is a Cartier divisor of Xs which satisfies ϕ∗(KXs) = KX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For each singularity, the exceptional divisor of its minimal resolution is a sum of irreducible effective roots (Ri) with Ri · Rj ∈ {0, 1} for any i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As usual in the ADE classification of rational double points, we describe the type of a singularity by its dual graph: its vertices correspond to the above roots, and there is an edge between the two vertices when the corresponding roots meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the examples below, the types of rational double points that appear correspond to the graphs: A1 A2 D5 As mentioned in the last item, since the anticanonical model of a non ordinary weak del Pezzo surface is not smooth but only normal, a Weil divisor may not be Cartier and the class groups of Cartier or Weil divisor may differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 Cartier versus Weil divisors and class groups Let X be a normal surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' let k(X) be its field of rational functions and OX its structural sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We need to review some general facts about divisors in such surfaces (see Liu [Liu02, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 & 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A prime Weil divisor X is a prime closed sub-variety of codimension 1 and the group of Weil divi- sors WDiv(X) is the free abelian group generated by prime Weil divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A Weil divisor D can be written � i niCi where the Ci’s are irreducible curves on X and where the ni are integers of which only a finite number are non zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Such a divisor is said effective if ni ≥ 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since X is normal, it is regular in codimension 1 and to each rational function f ∈ k(X), one can associate a Weil divisor (f) which is called principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The set of principal divisors is a sub-group of WDiv(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A Cartier divisor, or a locally principal divisor D is a global section of the sheaf k(X)×/O× X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' it consists in a collection (Ui, fi)i∈I where (Ui)i∈I is an open covering of X and where the fi’s are rational functions such that the quotients fi/fj have neither zeroes nor poles on Ui ∩ Uj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' such that fi/fj ∈ O× X(Ui ∩ Uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Two collections (Ui, fi)i∈I and (Vj, gj)j∈I represent the same Cartier divisor if on Ui ∩ Vj, the functions fi and gj differ by a multiplicative factor in O× X(Ui ∩ Vj) for every i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The set of Cartier divisors can be turned into an abelian group which we denote by CDiv(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A Cartier divisor is called effective if it can be represented by a collection (Ui, fi) with fi ∈ OX(Ui) for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A principal Cartier divisor is represented by a collection (X, f), where f ∈ k(X)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The set of principal divisors is also a sub-group of CDiv(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' To each Cartier divisor D one can associate a Weil divisor and this correspondence induces a group ho- morphism CDiv(X) → WDiv(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' since X is supposed to be normal, this morphism is injective [Liu02, Chap 7, Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='14] and it sends an effective Cartier divisor to an effective Weil one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The quotients of the divisor groups CDiv(X) and WDiv(X) by the principal divisors are denoted CaCl(X) and Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The previous correspondence induces an injective homorphism CaCl(X) → Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These are general facts, but in the context of weak del Pezzo surfaces, we are able to be much more explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, one can relate the two groups CaCl(Xs) and Cl(Xs) to the group Cl(X) = CaCl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a weak del Pezzo surface over k and let Xs be its anticanonical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Over k, one has the two exact sequences: 0 −→ R −→ Cl(X) −→ Cl(Xs) −→ 0 =⇒ Cl(Xs) = Cl(X)/R, (2) 0 −→ CaCl(Xs) −→ Cl(X) −→ Hom � R, Z � =⇒ CaCl(Xs) = R ⊥, (3) where the arrow Cl(X) → Hom � R, Z � is given by D �→ [R �→ D · R], and where R ⊥ = {D ∈ Cl(X) | D · R = 0, ∀R ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Over k, one has Cl(X) = CaCl(X) = CaCl(X)Γ = Cl(X)Γ for X, and for its anticanonical model: 0 −→ R −→ Cl(X) −→ Cl(Xs) −→ 0 =⇒ Cl(Xs) = Cl(X)/R, (4) 0 −→ CaCl(Xs) −→ Cl(X) −→ Hom � R, Z �Γ =⇒ CaCl(Xs) = R⊥ (5) Moreover we have an isomorphism Cl(Xs) ≃ Cl(Xs)Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let R be the union of effective roots in X and let U = X \\ R be the open complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' By a result of Hartshorne [Har77, Chap II, Prop 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5], we have the exact sequence: 0 −→ R −→ Cl(X) −→ Cl(U) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let Us be the smooth locus of Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This open set is of codimension 2 in Xs and thus Cl(Xs) ≃ Cl(Us) (see loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the anticanonical map ϕ−KX induces an isomorphism from U to Us one has Cl(Us) ≃ Cl(U) and the sequence (4) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Sequence (2) also follows by extending scalars to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On the other hand, we note that the module R being induced [Man74, Chap IV,§29], we know that H1(Γ, R) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' thus taking the Galois invariants of (2) leads to: 0 −→ R −→ Cl(X)Γ −→ Cl(Xs)Γ −→ 0 Now X is smooth and we have Cl(X) = Cl(X)Γ [Sta18, Tag 0CDS];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' we deduce the last isomorphism Cl(Xs) ≃ Cl(Xs)Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The exact sequence (3) comes from Bright [Bri13, Prop 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We deduce the equality CaCl(Xs) = R ⊥, and from [Sta18, Tag 0CDS], we deduce that CaCl(Xs) = CaCl(Xs)Γ = (R ⊥)Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally, taking the Galois invariants in the sequence (3) gives the exact sequence in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Now since the intersection product is invariant under the Galois action, a divisor in Cl(X) is orthogonal to R if and only if it is orthogonal to R, and we get the isomorphism in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 Lattice computations One of the key step of the study of codes from weak del Pezzo surfaces is the explicit computation of the divisor class groups as in (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Such computations take place in the group Cl(X), which is known to be a free Z- module of finite type endowed with the (non degenerate) intersection bilinear form and involve the root lattices R, which is given by some explicit generators and which satisfies R ∩ R ⊥ = {0} (the orthogonal is relative to the intersection pairing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This is a general issue and let us consider C (for the “class group”) a free Z-module of finite rank with a non degenerate symmetric bilinear form (x, y) �→ x · y (for the intersection product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Recall that a submodule M of C is a direct summand (or is complemented) if there exists a submodule N of C such that C = M ⊕ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' in this case, the submodules M and N are called complementary submodules of C [AW92, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8,§6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let R be a submodule of C such that R ∩ R⊥ = {0} (for the root lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the context of modules over a principal ideal domain, contrary to what is happening in vector spaces over a field, even if R ∩ R⊥ = {0}, the orthogonal submodules R and R⊥ may not be complementary submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There are at least two different kinds of obstructions for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Either the submodule R is not a direct summand or both submodules R and R⊥ are direct summands but they are not complementary submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In any case the smallest submodule containing R which is a direct summand is called the hull of R and is denoted R♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As for the submodule R⊥, since it is the kernel of a morphism of free modules, it is always a direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the same way, even though R♯ and R⊥ are direct summand, they may or may not be complementary submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These phenomenes make the description of the exact sequence: 0 −→ R⊥ −→ C/R −→ C/R ⊕ R⊥ −→ 0 a little bit tricky (ie the comparison between the groups of Weil classes and Cartier classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The main tool for the explicit computation of this sequence is the Invariant factor theorem for submodules that will be used twice (see [AW92, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' First, we apply this result to the submodule R ⊂ C: there exists a Z-basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , en of C and α1 | · · · | αr (r ≤ n) a sequence of positive integers, called invariant factors, such that R = Zα1e1 ⊕ · · · ⊕ Zαrer and R♯ = Ze1 ⊕ · · · ⊕ Zer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The submodule R is a direct summand of C if and only if R♯ = R, if and only if the invariant factors α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , αr are all equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Put M = Zer+1 ⊕ · · · ⊕ Zen then R♯ and M are complementary submodules, C = R♯ ⊕ M, and the projec- tions ιtors and ι onto each factors lead to an isomorphism C/R ≃ −→ R♯/R ⊕ M x mod R �−→ ιtors(x) mod R + ι(x) In other words, the projection ιtors gives an isomorphism from the torsion submodule of C/R to the quotient module R♯/R which is isomorphic to Z/α1Z × · · · × Z/αrZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' the projection ι gives an isomorphism from the torsion-free submodule of C/R to the submodule M of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since R ∩ R⊥ = {0}, we know that R⊥ canonically embeds in the quotient C/R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' being free of torsion it is a submodule of the torsion-free submodule of C/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Via ι it thus embeds in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Using the Invariant factor theorem again, one can choose the basis er+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , en of M, in such a way that there exists βr+1 | · · · | βn such 7 that ι(R⊥) = Zβr+1er+1 ⊕ · · · ⊕ Zβnen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The cokernel C/R ⊕ R⊥ is then isomorphic to Z/βr+1Z × · · · × Z/βnZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, the canonical embedding of R⊥ inside M induces an isomorphism if and only if βr+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , βn are all equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the sequel, we do not give names to the projection morphisms ιtors, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 3 Codes from surfaces: construction and tools for their study In this section k is a finite field Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 Evaluation codes from surfaces Cartier divisors, their spaces of global sections, and the associated complete linear systems are the main ingredients to define and to characterize the parameters of the evaluation codes from an algebraic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us recall the definitions and the basic facts concerning these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We consider X a (not necessarily smooth, but in fact at least normal here) irreducible surface over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We denote by k(X) its function field and by OX its structural sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let D = (Ui, fi)i∈I be a Cartier divisor on this surface X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A global section of D is a function s ∈ k(X) such that for every i ∈ I, the product sfi is regular on Ui, that is sfi ∈ OX(Ui) for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We denote by H0(X, D) the set of these sections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this is a vector space which is known to have finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' By definition, if s ∈ H0(X, D) is a global section of D then the Cartier divisor (Ui, sfi)i∈I is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It can be shown that two global sections of D lead to the same effective Cartier divisor if and only if they differ by a non zero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This means that there is a one-to-one correspondence between the projective space P(H0(X, D)) and the set of effective Cartier divisors linearly equivalent to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This last set is called the complete linear system associated to the divisor D and is currently denoted by |D|, so we have |D| = P(H0(X, D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' An important invariant of the divisor (or linear system) for our purpose is the maximum of rational points that can contain a curve of |D|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' we put: Nq(D) = max{#C(Fq) | C ∈ |D|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (6) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a (not necessarily smooth) irreducible surface over k, let D be a Cartier divisor of X and let P = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pn} be a set of rational points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The evaluation code CX(D, P) is the image of the evaluation map H0(X, D) −→ kn s �−→ (sfip)(p) where for each point p, the index ip is chosen in such a way that p ∈ Uip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the preceding definition, the choice of ip may be not unique but different choices ip, jp of these indices lead to homothetic codes since the quotients fip/fjp are non vanishing regular functions on Uip ∩ Ujp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The usual parameters of the evaluation code are related with some invariants of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If the evaluation map is injective, a CX(D, X(Fq)) has (i) length equal to #X(Fq) the number of rational points of X, (ii) dimension equal to the dimension of the space H0(X, D) of global sections, (iii) minimum distance bounded below by n − Nq(D), where Nq(D) is defined in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thanks to this proposition, it is worth noticing that bounding below the minimum distance of an evaluation code CX(D, X(k)) from a surface X reduces to bounding above Nq(D) the number of points of the curves of the linear system |D| associated to the divisor D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The fewer the number of rational points of the curves in the linear system |D|, the higher the minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 Blowing up, divisors and (non) complete linear systems One of the key tools of the construction of the codes from Del Pezzo surfaces are the blowing-up or the blowing down depending on the sense of the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let π : Y → X be a sequence of blowing ups where all the surfaces involved are supposed to be smooth, except the last one X which is only supposed to be normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Such a morphism leads to two natural maps involving different kinds of divisors and divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' First, starting from a Cartier divisor of X, the pullback π∗D is the Cartier divisor on Y defined locally by (π−1Ui, fi ◦ π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This lead to a morphism π∗ : CDiv(X) −→ CDiv(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Secondly, it can be shown that if C irreducible effective Weil divisor of Y , then π(C) is either a point or an irreducible effective Weil divisor of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then the map π∗ : WDiv(Y ) −→ WDiv(X) defined by π∗(C) = 0 if π(C) is a point and π∗(C) = π(C) otherwise extends to a group homorphism [Liu02, Chap 9, Lem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 8 Moreover, for every Cartier divisor D of X, one has π∗ (π∗D) = D [Liu02, Chap 9, Prop 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='11] where π∗ is applied to the Weil divisor of Y associated to the Cartier divisor π∗D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These two maps induce two homomorphisms π∗ : CaCl(X) → CaCl(Y ) [Liu02, Chap 7, Def 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='34] and π∗ : Cl(Y ) → Cl(X) [Ful98, Chap 1, Th 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' At the level of global sections and linear systems, the map π∗ also induces isomorphisms: H0(X, D) ≃ −→ H0(Y, π∗D) s �−→ s ◦ π |D|X ≃ −→ |π∗D|Y C �−→ π∗C where D is a Cartier divisor on X ([Dem80, Expos´e V, Cor 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since π∗ (π∗C) = C, the inverse of the right isomorphism is nothing else than π∗ |π∗D|Y −→ |D|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We will need to describe the one-to-one correspondence |D|X −→ |π∗D|Y , when the right divisor is replaced by a divisor of the form π∗D − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Some natural sublinear systems appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us go step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let π : Y → X be the blowing-up of a smooth surface X at a point p ∈ X and let E be its exceptional divisor on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the surfaces are supposed to be smooth, we do not have to distinguish Cartier and Weil divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We mainly focus on effective divisors and we call them curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Given C a curve on X, then the pullback π∗C is called the total transform of C, the closure in Y of π−1(C \\ {p}), denoted �C, is called the strict transform of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These two curves on Y are related by the relation: π∗C = �C + mp(C)E, where mp(C) denote the multiplicity of C at the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, for any n ≥ 0, the divisor π∗C − nE is effective if and only if mp(C) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This permits to relate the complete linear system |π∗D−nE| on Y to an uncomplete one on X, that is |D−np| the space of curves of |D| which pass through p with multiplicity at least n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In fact this shows that the map C �→ π∗C − nE leads to a one-to-one correspondence from |D − np| to |π∗D − nE| (the other way around, it says that the blowing-up permits to turn uncomplete linear systems into complete ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The same is true if we blow up several points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let π : Y → X be the blowing-up of a smooth sur- face X at some points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pr, and let E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , Er be the exceptional divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For D a divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let |D − n1p1 − · · · − nrpr| denotes the sub-linear system of the complete linear system |D| consisting of curves of |D| which pass through p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pr with multiplicities at least n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The blowing-up permits to turn this incomplete linear system into a complete one: there is a one-to-one correspondence between ([Har77, loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ], [CA00]), |D − n1p1 − · · · − nrpr| −→ |π∗D − n1E1 − · · · − nrEr| C �−→ C♯ where C♯ def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' = π∗C − n1E1 − · · · − nrEr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (7) This curve C♯ is sometime called the virtual transform of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The total, strict and virtual transforms are thus related by: π∗C = �C + r � i=1 mpi(C)Ei = C♯ + r � i=1 niEi =⇒ C♯ = �C + r � i=1 (mpi(C) − ni) Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular the virtual and the strict transforms coincide when mpi(C) = ni for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This one-to-one correspondence is still true if some points in the sequence of blowing ups are infinitely near points, that is when some pj lies on the exceptional divisor of the blow up of another point pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In order to describe this, we need to carefully define the sub-linear system associated to a family of infinitely near points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us start with only two points: if p1 ≺ p2, that is if p2 lies on the exceptional curve E1 above p1, then for n1, n2 > 0, the sub-linear system of curves passing through p1 and p2 with multiplicities at least n1 and n2 is defined by: |D − n1p1 − n2p2| def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' = {C ∈ |D − n1p1| , mp2(π∗(C) − n1E1) ≥ n2} = � C ∈ |D − n1p1| , mp2(C♯) ≥ n2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular the sub-system |D − p1 − p2| contains all the curves of |D| that pass through p1 with tangent line at p1 equal to p2 union all the curves of |D| singular at p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' indeed, in the last case C♯ = π∗(C)−E1 = �C+(mp1(C) − 1) E1 has E1 as a component and thus passes through p2 (one can check that the conditions p1 ∈ C and p2 ∈ �C are not linear, which is why we choose p2 ∈ C♯ instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the same way, if p1 ≺ p2 ≺ · · · ≺ pr, one can define recursively, the sub-linear system |D − n1p1 − · · · − nrpr|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' With this definition, the one-to-one correspondence (7) is still true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us end by an example: the case X = P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If ℓ denotes the class a line, and if E0 is the pullback of ℓ in Y , then the curves of the complete linear |dE0 −n1E1−· · ·−nrEr|Y on Y corresponds bijectively to |dℓ−n1p1−· · ·−nrpr| the (projective) vector space consisting of plane curves of degree d passing through p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pr with multiplicities at least n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For small degrees, it turns out that the irreducible decompositions of such curves can be easily described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 Blowing up and evaluation codes Let us return to codes and compare the evaluation codes CX(D, X(k)) and CY (π∗D, Y (k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a normal surface, let p be a point of X and let π : Y → X be the blowing-up of X at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We denote by E the divisor sum of the exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (i) If p is of degree > 1, then the codes CX(D, X(k)) and CY (π∗D, Y (k)) are equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' moreover the code CY (π∗D− nE, Y (k)) can be identified with the sub-code of CX(D, X(k)) where only the global section having multiplicity at least n at p are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (ii) If p is rational, then the code CY (π∗D − nE, Y (k)) can be identified with the sub-code of CX(D, X(k) \\ {p}) where only the global sections having multiplicity at least n at p are evaluated and to which we add the following (q + 1) coordinates: the evaluations at rational points of P1 of the homogeneous component of degree n of the local equation at p of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (i) The map s �→ s◦π is a one-to-one correspondence from the spaces of functions H0(X, D) to H0(Y, π∗D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the blown points are not rational, the map π induces a one-to-one correspondence from Y (k) to X(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus the codes CX(D, X(k)) and CY (π∗D, Y (k)) must be equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' By the previous correspondence the global sections of H0(Y, π∗D − nE) are in bijection with the global sections of H0(X, D) that pass through p with multiplicity at least n and the last statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (ii) The set Y (k) is in one-to-one correspondence with (X(k) \\ {p}) ∪ E(k) and we only have to compute the evaluations at the points of E(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We choose an open neighbourhood U ⊂ A2 (x,y) of p in which p = (0, 0) and D has local equation f(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then π−1(U) ⊂ U × P1 (u:v) with equation xv = yu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' there are two affine charts, π−1(U) = V1 ∪ V2, with V1 ⊂ A2 (y,u) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' V2 ⊂ A2 (x,v)) with π(y, u) = (yu, y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' π(x, v) = (x, xv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On V1, the divisor π∗D − nE has local equation f◦π yn = f(yu,y) yn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let s◦ π ∈ H0(Y, π∗D − nE) then sf ∈ OX(U) has multiplicity at least n at p, that is sf(x, y) = pn(x, y) + pn+1(x, y) + · · · , where pn is homogeneous of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus sf◦π yn = pn(yu,y)+pn+1(yu,y)+··· yn = pn(u, 1) + yq(u, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Evaluating at the point (0, u) ∈ E ∩ V1, the section- has value pn(u, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The same is true on V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The examples below provide many examples of this blowing-up operation, especially the one in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4 Anticanonical codes from weak del Pezzo surfaces In this section we describe some evaluation codes from weak del Pezzo surfaces, we compute their parameters and for some of them a generator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The base field is a finite field Fq without any other hypothesis excepts sporadically not being too small (F2 or F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the first subsection, the general construction is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We also fix many notations that will be used until the end of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 General description of the codes and of the main steps of their studies The evaluation codes (definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1) studied in the sequel are the ones corresponding to the following choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X be a weak del Pezzo surface over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We call anticanonical code associated to X the evaluation code CXs (−KXs, Xs(Fq)), where Xs is the anticanonical model of X, −KXs is the anticanonical (Cartier) divisor on Xs, and where Xs(Fq) denotes the set of rational points of Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Note that we could have considered the evaluation codes CX(−KX, X(Fq)) with the same del Pezzo surfaces, but this leads to worth codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In a concomitant work [BH22], we have computed explicit models for all the arithmetic types of del Pezzo surfaces over a finite field (these types lead to a classification that is coarser than the isomorphism one but that permit to distinguish the main arithmetic properties of the weak del Pezzo surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Taking advantage of this knowledge, we select eight types of weak del Pezzo that are well suited for coding applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For each example, our starting point is a blowing-up model of the weak del Pezzo surface, then we study the parameters length, dimension, minimum distance ([n, k, dmin]q) of the associated anticanonical code and last we give a generator matrix (or a program to compute it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — The explicit description of the surfaces X and Xs always starts from the projective plane P2: we first blow up a family of (possibly infinitely near) points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pr to obtain a smooth surface Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' then we may blow down a family of (non intersecting) exceptional curves on Y to obtain the smooth 10 surface X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last X is mapped to a projective space corresponding to the anticanoncial divisor to lead to the singular surface Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' To sum up, we have the following diagram: Y X P2 Xs ⊂ Pdeg(X) π χ ϕ ε π is a sequence of blowing ups at points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , pr, χ is a sequence of contractions of (−1)-curves F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , Fs, ϕ is the morphism ϕ−KX associated to the anticanonical divisor −KX of X, deg(X) is the degree of the del Pezzo surface X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' K·2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (8) All the surfaces and maps are defined over the base field Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The solid arrows π, χ, ϕ denote maps that are morphisms whereas the dashed arrow ε denotes a map which is a rational one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The need to introduce the auxiliary surface Y is due to the fact that some times, the surface X we want to work with cannot be constructed directly by blowing up the plane at some points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Some contractions may be necessary in order to work with applications that are defined over Fq (and not only over Fq);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' however this detour is not always useful and in some examples, one has X = Y and the map χ is only the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Two of the parameters [n, k, dmin]q of the associated anticanonical code are easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The length is nothing else than #Xs(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Following the process of blowing ups and down above, it is not difficult to compute this number since blowing up a point adds q rational points or does not change the number of rational points depending on whether the point is rational or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The dimension is nothing else than d + 1, where d is the degree of the del Pezzo surface X, unless the evaluation map is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This can only occur if #Xs(Fq) ≤ Nq (−KXs) and we compute last number to estimate the minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It turns out that the evaluation map is always injective except if the base field is F2 or F3 in some cases that are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As usual, the last parameter, the minimum distance, requires much more preparatory works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — For these computations, the general ambient space is the geometric divisor class group of Y , which is known to be equal to Cl(Y ) = ZE0 ⊕ ZE1 ⊕ · · · ⊕ ZEr, where, as usual, Ei denotes the exceptional curve above pi in the sequence of blowing ups π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this lattice, one can easily identify the effective roots in Y , but also in X and we are able to give a basis of the sub-lattice R generated by the effective roots of X over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The other (geometric) Cartier and Weil divisor class groups are then given by: Cl(X) = (ZF1 ⊕ · · · ⊕ ZFr)⊥ , CaCl(Xs) = R ⊥, Cl(Xs) = Cl(X)/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (the left orthogonal is computed in the whole Cl(Y ), the middle one in the sub-lattice Cl(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Using tools of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3, explicit bases and canonical embeddings of these geometric divisor class groups can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Taking into account the Galois action, one can also give bases and explicit canonical embedding bases of all the arithmetic divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Depending on the examples, the computations are carried out in the geometric groups Cl(X) and the Galois invariants are taken in the last step to return in Cl(X) or we start to compute the Galois invariants and then perform all the computations in Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thanks to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7, these two ways lead to the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Types of decomposition into irreducible components in |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — The minimum distance is related to the maximum number of rational points that can contain a (effective) curve in the linear system |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' To bound above this number of rational points, one way is to study how the curves in this linear system decompose into irreducible components and use the exact number of points if known or the Weil bound if not on each components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thanks to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2, and since ϕ∗KXs = KX, we have the following one-to-one correspondences: |−χ∗KX|Y ≃ −→ |−KX|X ≃ −→ |−KXs|Xs C �−→ χ∗(C) �−→ ϕ∗ (χ∗(C)) The first arrow consists in contracting the family of non-meeting exceptional curves Fi, 1 ≤ i ≤ s, the second in contracting the effective roots of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus we are reduced to study the types of decompositions into irreducible components on the smooth surface Y , which is easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Indeed, we know that −χ∗KX = dE0 − �r i=1 niEi for some explicit d and ni’s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' in fact, in all examples, d ∈ {3, 4} and ni ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since Y is the blowing up of P2 at a family of points, thanks to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2, curves of |−KY | are in one-to-one correspondence to the plane curves of a well specified (non complete) linear system of P2: |dℓ − n1p1 − · · · − nrpr| ≃ −→ |−χ∗KX| C �−→ C♯ 11 We are thus reduced to list all the types of decompositions into irreducible components of the plane curves of degree d passing through pi with multiplicity ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since d ≤ 4, these absolutely irreducible components must be plane lines, conics, cubics or quartics and an enumeration case by case can be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' More specifically, we follow the steps: degree by degree, we list all the possible absolutely irreducible curves that pass through some of the points pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' we compute their Galois-orbits since if an absolutely irreducible component not defined over Fq appears in the decomposition with multiplicity m, then the same holds for all its conjugates (this permits to get rid of many curves because of their too high degree);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' we combine all these irreducible curves to obtain plane curves in the expected sub-linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In order to make easier this step, we adopt the following notations and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The letters ℓ, q, c, t respectively denote plane lines, quadrics (or conics), cubics and quartics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The indices below these letters are the numbers of the points through which the curve passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For example, ℓ1 denotes a line that passes through p1 (but not through any other point), ℓ123 a line that passes through p1, p2, p3 (if it exists), q123456 a conic passing through the six points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p6 and ℓ or q a line or quadric that do not pass through any pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The goal is then to combine all these irreducible plane curves to obtain a curve in the expected linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' At the end of this step, we are able to compute the maximum Nq (−KXs) to which the minimum distance is related (proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Comparing with the number #Xs(Fq), this also permits us to exclude some too small values of q for which the evaluation map may fail to be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Last, if we want to explicitly compute a generator matrix of the code, we need to exhibit a basis of the sub linear system |dℓ − n1p1 − · · · − nrpr|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then, by construction we know to which points of P2 these functions have to be evaluated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' in some cases we also need to add some extra evaluation points corresponding to points on some exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In any cases, one can compute a generator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This last (concrete) description turns the code into a code close to a Reed-Muller one: the space of polynomials to be evaluated has been restricted, some of the evaluation points have been deleted, some others have been added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If some readers want to use our code, we put on the second author’s webpage, a magma program that permits to construct all the codes presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 Degree 6, singularity of type A1 This example corresponds to the type number 3 in degree 6 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — We blow up P2 at three collinear points that are conjugate over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ℓ123 p1 p2 p3 p2 = pσ 1, p3 = pσ2 1 The resulting surface is a weak del Pezzo surface X whose anticanonical model is denoted Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It has a unique singular point of type A1 which is necessarily rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Over Fq, one has Cl(X) = ZE0 ⊕ ZE1 ⊕ ZE2 ⊕ ZE3 and − KX = 3E0 − E1 − E2 − E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There is a unique effective root, the strict transform of the line ℓ123 passing through the three points p1, p2, p3, and its class is E0 − E1 − E2 − E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then R = Z(E0 − E1 − E2 − E3) R ⊥ = {a0E0 + a1E1 + a2E2 + a3E3 | a0 + a1 + a2 + a3 = 0} = Z(E0 − E1) ⊕ Z(E0 − E2) ⊕ Z(E0 − E3) Both R and R ⊥ are direct summand but R and R ⊥ are not complementary submodules since R ⊕ R ⊥ is of index 2 in Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For a submodule complement to R, one can choose: Cl(X) = R ⊕ (ZE1 ⊕ ZE2 ⊕ ZE3) a0E0 + a1E1 + a2E2 + a3E3 = a0(E0 − E1 − E2 − E3) + (a1 + a0)E1 + (a2 + a0)E2 + (a3 + a0)E3 This leads to the following isomorphism: Cl(X)/R ≃ −→ ZE1 ⊕ ZE2 ⊕ ZE3 a0E0 + a1E1 + a2E2 + a3E3 mod R �−→ (a0 + a1)E1 + (a0 + a2)E2 + (a0 + a3)E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 12 Via this isomorphism, the submodule CaCl(Xs) = R ⊥ identifies with Z(E1 + E2) ⊕ Z(E2 + E3) ⊕ Z(E1 + E3) of invariant factors 1, 1, 2 in ZE1 ⊕ ZE2 ⊕ ZE3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Over Fq, to recover the class groups Cl(Xs) and CaCl(Xs), we only need to take the invariants under the Galois action (E0)(E1E2E3), what is easy here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' One has CaCl(Xs) = CaCl(Xs)Γ = � R ⊥�Γ ≃ Z(3E0 − E1 − E2 − E3) = Z(−KX), Cl(Xs) = Cl(Xs)Γ = � Cl(X)/R �Γ ≃ Z(E1 + E2 + E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' With these identifications, the canonical embedding of CaCl(Xs) into Cl(Xs) becomes: 0 −→ CaCl(Xs) −→ Cl(Xs) −KX �−→ 2(E1 + E2 + E3) Thus both CaCl(Xs) and Cl(Xs) are free of rank 1, but CaCl(Xs) is of index 2 into Cl(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This index has the following consequence: even if CaCl(Xs) is free of rank one generated by −KXs, a Cartier divisor may decompose into a sum of equivalent Weil irreducible divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This explains why we need to investigate how elements of |−KX| can decompose into irreducible components and how the non ordinary weak del Pezzo surfaces we consider here differ from ordinary ones (compare with [BCH+20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Types of decomposition into irreducible components in |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — In this example, there is no need to introduce an auxiliary surface Y (one has Y = X and χ is the identity with the notation of the beginning of this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since −KX = 3E0 − E1 − E2 − E3, the virtual transform composed with the push forward lead to a one-to-one correspondence: |3ℓ − p1 − p2 − p3| −→ |3E0 − E1 − E2 − E3| −→ |−KXs| C �−→ C♯ �−→ ϕ∗(C♯) (the left linear system is on P2, the middle one on X and the right one on Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then we are reduced to list all the types of decompositions into irreducible components of the curves of |3ℓ − p1 − p2 − p3|, the sub linear system of cubics passing through the points p1, p2, p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The orbits of lines, conics, cubics which have degree at most 3 and pass through some of the points pi’s are ℓ1 ∪ ℓ2 ∪ ℓ3, ℓ123, c123, (of course, implicitly ℓ2 = ℓσ 1, ℓ3 = ℓσ2 1 where σ is a generator of Gal(Fq/Fq)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Indeed an absolutely irreducible conic qi or qij cannot be defined over Fq and they have at least three conjugates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' combining these curves with their conjugates lead to plane curves of degree greater than 6 and thus they cannot appear in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A conic q123 cannot be absolutely irreducible otherwise it would have three intersection points with the line ℓ123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us combine these rational irreducible decompositions in order to construct plane curves in the expected sub-linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' First suppose that the decomposition contains a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If this line is ℓ1, then its conjugates ℓ2, ℓ3 must also be geometric components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' the only possibility is ℓ1∪ℓ2∪ℓ3 (line 1 in the tabular below) which is an element of |3ℓ − p1 − p2 − p3|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A component ℓ cannot be completed by a conic passing through the three points and thus if there is a line in the geometric decomposition, ℓ123 must be one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since ℓ123 already passes through p1, p2, p3 it can be completed by any conic (irreducible or not);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this leads to the decompositions of the lines 3 to 7 in the tabular below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last, if the decomposition does not contain any line, it must be an irreducible cubic which passes through the 13 three points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this cubic can be smooth or not and we recover the two last lines of the tabular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' |3ℓ − p1 − p2 − p3| |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts 1 ℓ1 ∪ ℓ2 ∪ ℓ3 �ℓ1 ∪ �ℓ2 ∪ �ℓ3 �3 i=1 ϕ∗(�ℓi) 1 2 ℓ123 ∪ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ℓ123 ∩ q ̸⊂ P2(Fq) �ℓ123 ∪ �q ϕ∗(�q) q + 2 3 ℓ123 ∪ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ℓ123 ∩ q ⊂ P2(Fq) �ℓ123 ∪ �q ϕ∗(�q) q 4 ℓ123 ∪ ℓ ∪ ℓ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ℓ123 ∩ ℓ ∩ ℓ′ ̸= ∅ �ℓ123 ∪ �ℓ ∪ �ℓ′ ϕ∗(�ℓ) ∪ ϕ∗(�ℓ′) 2q + 1 5 ℓ123 ∪ ℓ ∪ ℓ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ℓ123 ∩ ℓ ∩ ℓ′ = ∅ �ℓ123 ∪ �ℓ ∪ �ℓ′ ϕ∗(�ℓ) ∪ ϕ∗(�ℓ′) 2q 6 2ℓ123 ∪ ℓ 2�ℓ123 ∪ �ℓ ∪ �3 i=1 Ei ϕ∗(�ℓ) ∪ �3 i=1 ϕ∗(Ei) q + 1 7 3ℓ123 3�ℓ123 ∪ �3 i=1 2Ei �3 i=1 2ϕ∗(Ei) 1 8 c123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' singular �c123 ϕ∗(�c123) q + 2 9 c123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' smooth �c123 ϕ∗(�c123) Nq (1) Some comments about the three first columns of the previous tabular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The unique irreducible effective root of X is nothing else than the strict transform ℓ123 and this explains why this curve disappears in the third column: this line on X is mapped by ϕ∗ to the unique singular point s ∈ Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Note also that except in the cases 6 and 7, all the curves have exactly multiplicities 1 at the pi and thus their strict or virtual transforms are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On the contrary, in the remaining cases, the curves on X are the virtual transforms of the ones on P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last, in the decomposition ϕ∗(�ℓ) ∪ ϕ∗(�ℓ′), it is worth noticing that irreducible components involves divisors that are not Cartier divisors but only Weil ones on Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Indeed the class of �ℓ in Cl(X) is E0, which is mapped to E1 + E2 + E3 in Cl(Xs), which is not an element of CaCl(Xs) (equivalently E0 ̸∈ R⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Now we make some comments on the numbers of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the lines ℓ1, ℓ2, ℓ3 are conjugate a rational point on their union must be at their intersection which contains at most one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On X the strict transforms �ℓ1, �ℓ2, �ℓ3 do not meet the root ℓ123 and the contraction does not add any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 2, 3, 4 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If the two points of q ∩ ℓ123 are not rational then they are still unrational on �q ∩ �ℓ123 and they are contracted to the singular point s in Xs and thus the image ϕ∗(�q) has one more rational point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' otherwise, if the two points of q ∩ ℓ123 are rational then they are contracted in Xs and thus the image ϕ∗(�q) looses a rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The same is true on lines 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 6 & 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The line �ℓ123 is contracted by ϕ∗ and there are no rational points on the lines Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 8 & 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The starting cubic c123 has (q + 1) or less than Nq (1) rational points depending on whether it is singular or smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On Xs the number of rational points of ϕ∗(�c123) is increased by 1 since the line ℓ123 meets the cubic at three conjugate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The multiplicity of intersection of c123 and ℓ123 at each point pi is one (since otherwise, these two curves would have too many intersection points counting with multiplicities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Therefore, the blowing ups at p1, p2, p3 separate the strict transforms �ℓ123 and �c123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus �c123 and ϕ∗(�c123) are isomorphic and have the same number of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally, we remark that for every q, one has q + 1 + ⌊2√q⌋ ≤ 2q + 1 (with equality if and only if q ∈ {2, 3, 4}) and thus: Nq (−KXs) = 2q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last we note that, except for the cases 1 and 9, all the maximum numbers of points are in fact exact numbers of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus we are not far from having the distribution of weights if the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since p1, p2, p3 are not rational, the three blowing ups do not add any rational point and #X(Fq) = q2 +q +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then, the root �ℓ123 is contracted via the anticanonical morphism and thus #Xs(Fq) = q2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Except if q = 2, one has #Xs(Fq) > Nq(−KXs) and the evaluation map is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' With this choice of weak del Pezzo surface, the code of definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 satisfies the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1, p2, p3 be conjugate collinear point in P2 Fq, with q ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical code associated to the weak del Pezzo surface obtained by blowing up these points has parameters [q2 + 1, 7, q2 − 2q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — To construct this kind of codes, one can choose ℓ123 to be the line of equation Y = 0 in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For any ζ ∈ Fq3 \\ Fq, the point p1 = (ζ : 0 : 1) ∈ P2 is a degree 3 point whose conjugates p2 = (ζσ : 0 : 1) and p3 = (ζσ2 : 0 : 1) are also in ℓ123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let X3 + a2X2 + a1X + a0 ∈ Fq[X] be the minimal polynomial of ζ over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then, we easily verify that |3ℓ − p1 − p2 − p3| = � Y 3, Y 2X, Y 2Z, Y X2, Y Z2, Y XZ, X3 + a2X2Z + a1XZ2 + a0Z3� Fq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 14 Last, the evaluation points are nothing else than the points of P2(Fq) \\ ℓ123(Fq), plus one point of ℓ123(Fq) since the strict transform of ℓ123 is contracted via the anticanonical morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us denote (xi : 1 : zi), 1 ≤ i ≤ q2, the first q2 points, and let us choose (0 : 0 : 1) ∈ ℓ123(Fq), then the corresponding generator matrix of the code is: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 · · 1 0 x1 · · xq2 0 z1 · · zq2 0 x2 1 · · x2 q2 0 z2 1 · · z2 q2 0 x1z1 · · xq2zq2 0 P(x1, 1, z1) · · P(xq2, 1, zq2) a0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , P(X, Y, Z) = X3 + a2X2Z + a1XZ2 + a0Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We recover the classical Reed-Muller code on A2 of degree 2, augmented by one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 Degree 5, singularity of type 2A1 This example corresponds to the type number 5 in degree 5 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — We blow up p1 ≺ p2 and p3 ≺ p4, where p1, p3 and p2, p4 are conjugate points of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ℓ12 p2 ℓ34 p4 p1 p3 ℓ13 p3 = pσ 1 p4 = pσ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the points p2, p4 are infinitely near the points p1, p3, they are represented by tangent lines or directions on the picture above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical model Xs has two singular points of type A1 that are conjugate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Over Fq, one has: Cl(X) = ZE0 ⊕ ZE1 ⊕ ZE3 ⊕ ZE2 ⊕ ZE4 and − KX = 3E0 − E1 − E2 − E3 − E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There are two conjugate effective roots, the strict transforms of E1 and E3 in the sequence of blowing ups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' their classes are E1 − E2 and E3 − E4 in such a way that: R = Z(E1 − E2) ⊕ Z(E3 − E4), R ⊥ = {a0E0 + a1E1 + a2E2 + a3E3 + a4E4 | a1 = a2, a3 = a4} = ZE0 ⊕ Z(E1 + E2) ⊕ Z(E3 + E4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The sub-module R is a direct summand, and as a complementary sub-module one can choose: Cl(X) = R ⊕ ZE0 ⊕ ZE2 ⊕ ZE4 �4 i=0 aiEi = a1(E1 − E2) + a3(E3 − E4) + a0E0 + (a1 + a2)E2 + (a3 + a4)E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We deduce the isomorphism: Cl(Xs) ≃ Cl(X)/R ≃ −→ ZE0 ⊕ ZE2 ⊕ ZE4 �4 i=0 aiEi mod R �−→ a0E0 + (a1 + a2)E2 + (a3 + a4)E4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since CaCl(Xs) ≃ R ⊥, this class group is a rank 3 free sub-group of Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Via the previous isomorphism it is mapped to the sub-group ZE0 ⊕ Z2E2 ⊕ Z2E4, of invariant factors 1, 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The arithmetic groups CaCl(Xs) and Cl(Xs) can be computed by taking the invariants under the Galois action which is (E0)(E1E3)(E2E4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Via the previous isomorphism, if we set E := E2 + E4, the canonical embedding 0 → CaCl(Xs) → Cl(Xs) is only: ZE0 ⊕ Z2E � �� � ≃CaCl(Xs) ⊂ ZE0 ⊕ ZE � �� � ≃Cl(Xs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In other terms, CaCl(Xs) and Cl(Xs) are both free of rank 2 and via the canonical embedding, the first one has invariant factors 1, 2 inside the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 15 Types of decomposition into irreducible components in |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Since the two class groups are not rank one, one expects to find a wide variety of possible decompositions into irreducible components for the curves in the linear system |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In order to list all these types, we start form P2 and use the one-to-one correspondences: |3ℓ − p1 − p3 − p2 − p4| −→ |3E0 − E1 − E3 − E2 − E4| −→ |−KXs| C �−→ C♯ �−→ ϕ∗ � C♯� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The curves of the left linear system are nothing else than the plane cubics over Fq passing through p1, p3 that are either smooth at p1, p3 with tangent lines p2, p4 respectively or singular at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Our notations are the same: ℓ13 is the line (p1p3) which is rational, ℓ12 and ℓ34 are the lines (p1p2) respec- tively (p3p4) (that is the lines of P2 passing through p1, respectively p3, whose strict transform pass through p2, respectively p4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' these last two lines are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The orbits of lines, conics, cubics having degree less than 3 and passing through some of the points pi’s are ℓ1 ∪ ℓ3, ℓ13, ℓ12 ∪ ℓ34, q13, q1234, c13, c1234 (of course, implicitly ℓ3 = ℓσ 1, ℓ34 = ℓσ 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We have just to combine these rational irreducible decompositions in order to construct plane curves in the expected sub-linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Suppose that there is at least one line in the absolute irreducible decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If this line is ℓ12, then by rationality, ℓ34 is also an absolute irreducible component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since ℓ12 ∪ ℓ34 already passes through p1, p2, p3, p4, one can complete by any rational line ℓ or by the line ℓ13 (see cases 1 and 2 in the tabular below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If this line is ℓ13, then the two incidence conditions at p1 and p3 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The complement component must be a (maybe reducible) conic whose strict transform passes through p2 and p4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this conic must neces- sarily pass through p1, p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This conditions suffice since the union of ℓ13 with any conic passing through p1, p3 is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The complement can be the union ℓ12 ∪ ℓ34 (same as case 2), or ℓ1 ∪ ℓ3 = ℓσ 1, or ℓ13 itself union any other line, or twice ℓ13, or q13, or q1234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If this line is ℓ a line that does not pass through the pi’s, then the complement conic must be either ℓ12 ∪ℓ34 as in first case, or a conic passing through the four points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last, if there is not any line in the absolute irreducible decomposition, then the cubic must be absolutely irreducible and it has to pass through the four points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' That being, the possible cubics are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The irreducible effective roots of X are the (conjugate) strict transforms �E1 and �E3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' since they do not meet, their contraction lead to two (conjugate) singular points s and sσ on Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ���3ℓ − �4 i1 pi ��� |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ12 ∪ ℓ34 ∪ ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ12 ∪ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ34 ∪ �ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ12) ∪ ϕ∗(�ℓ34) ∪ ϕ∗(�ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ12 ∪ ℓ34 ∪ ℓ13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ12 ∪ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ34 ∪ �ℓ13 ∪ �E1 ∪ �E3 ∪ E2 ∪ E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ12) ∪ ϕ∗(�ℓ34) ∪ ϕ∗(�ℓ13) ∪ ϕ∗(E2) ∪ ϕ∗(E4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ13 ∪ ℓ1 ∪ ℓ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ13 ∪ �ℓ1 ∪ �ℓ3 ∪ �E1 ∪ �E3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ1) ∪ ϕ∗(�ℓ3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2ℓ13 ∪ ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2�ℓ13 ∪ �ℓ ∪ �E1 ∪ �E3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3ℓ13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3�ℓ13 ∪ 2 �E1 ∪ 2 �E3 ∪ E2 ∪ E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3ϕ∗(�ℓ13) ∪ ϕ∗(E2) ∪ ϕ∗( �E2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ13 ∪ q13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ13 ∪ �q13 ∪ �E1 ∪ �E3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ13) ∪ ϕ∗(�q13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ13 ∪ q1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ13 ∪ �q1234 ∪ �E1 ∪ �E3 ∪ E2 ∪ E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ13) ∪ ϕ∗(�q1234) ∪ ϕ∗(E2) ∪ ϕ∗(E4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ ∪ q1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ ∪ �q1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ) ∪ ϕ∗(�q1234) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='c1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�c1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�c1234) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='Nq (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='We draw all the preceding decompositions in order to illustrate what is going on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The blowing up π : X → P2 is decomposed into two blowing ups π = π2 ◦ π1, where π1 : X1 → P2 is the blowing up at p1 and p3, and where π2 : X → X1 is the blowing up at p2 and p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The left column is the drawing of the starting configuration in P2, the middle one the configuration after having blowing up p1 and p3, the right one the configuration in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The operation from a column to the next one is the virtual transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Curves drawn in gray are not part of virtual transform, curves drawn in red are the effective roots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' these curves are contracted in Xs (but we do not 16 draw this step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In brackets, to the right of the name of a curve, we put its self-intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We draw all the cases of the preceding tabular, except the cubic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In any case, one can verify that the union of the black curves passes through p1, p3, p2, p4 and that the divisor class is equal to −KX = 3E0 − E1 − E3 − E2 − E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p1 p3 • ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ12(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ34(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p2 p4 • ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ12(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ34(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E1(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E3(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ(1) ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ13(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�q13(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�E1(−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�E3(−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E2(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E4(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ13(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q1234(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p1 p3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ13(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�q1234(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E1(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E3(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p2 p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ13(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�q1234(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�E1(−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�E3(−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ13(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E2(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E4(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q1234(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p1 p3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�q1234(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E1(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E3(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='p2 p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�q1234(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�E1(−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�E3(−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E2(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='E4(−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='Some comments about the number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 1, 2, & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The unions of lines ℓ12 ∪ℓ34, or ℓ1 ∪ℓ3 (recall that ℓ3 = ℓσ 1), contain a unique rational point, the intersection point of the two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Except ℓ (cases 1 and 2) or ℓ13 (case 3), all the lines in the decomposition are not defined over Fq and di not contain any rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus to the previous single point we have to add the (q + 1) rational points of the line ℓ or ℓ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The two (black) components have (q +1) rational points but they meet at a rational point, thus their union contains 2q + 1 rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The only component that contains rational points is ϕ∗(�ℓ13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 6, 7, & 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In these cases, they are two disjoint components that contain (q + 1) rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally: Nq (−KXs) = 2q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since none of the points pi is rational, the surfaces X and Xs still have q2 + q + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For every q, one has #Xs(Fq) > Nq (−KXs) and the evaluation map is always injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The parameters of the code are thus given by: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1 ≺ p2 and p3 ≺ p4 be such that p1, p3 and p2, p4 are conjugate points of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical code of the weak del Pezzo surface obtained by blowing up these points has parameters [q2 + q + 1, 6, q2 − q − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Let d ∈ Fq be a non-square and put ζ = √ d ∈ Fq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We choose p1 = (ζ : 0 : 1) and p2 = (ζ : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then p3 = (−ζ : 0 : 1), the line ℓ13 = (p1p3) has equation Y = 0, the line ℓ12 = (p1p2) corresponds to the zeros of the linear form L = X − ζ(Y + Z) and the line ℓ34 = (p3p4) corresponds to the zeros of the linear form Lσ = X + ζ(Y + Z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The linear forms Y, L, Lσ generate the global sections of ℓ and we easily prove that |3ℓ − p1 − p3 − p2 − p4| = � Y 3, Y 2L, Y 2Lσ, Y LLσ, L2Lσ, L(Lσ)2� Fq = � Y 3, Y 2(L + Lσ), Y 2ζ(L − Lσ), Y LLσ, LLσ(L + Lσ), LLσζ(L − Lσ) � Fq = � Y 3, Y 2X, Y 2(Y + Z), Y Π, XΠ, (Y + Z)Π � Fq , where Π = LLσ = X2 − d(Y + Z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The evaluation points are nothing else than all the points of P2(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4 Degree 4, singularity of type A1 This example corresponds to the type number 8 in degree 4 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — We blow up six points on a conic, the first two p1 and p2 being conjugate (or rational), the last four p3, p4, p5, p6 being conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This leads to a degree 3 weak Del Pezzo surface Y π −→ P2 as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this surface, the strict transform of the line ℓ12 passing through p1 and p2 is a 18 rational (−1)-curve that can be contracted: the codomain of the contraction Y χ −→ X is the degree 4 weak Del Pezzo surface we want to work with in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p1 p2 p3 p4 p5 p6 ℓ12 q123456 p2 = pσ 1 p4 = pσ 3 p5 = pσ2 3 p6 = pσ3 3 The anticanonical model of this weak del Pezzo surface X has a unique singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — On Y , one has Cl(Y ) = �6 i=0 ZEi and 27 exceptional classes of divisor, namely the 6 exceptional lines Ei, 1 ≤ i ≤ 6, the 15 strict transforms of the lines passing through two of the six points, Eij = E0 − Ei − Ej, 1 ≤ i < j ≤ 6, and the 6 strict transforms of the quadrics passing through five of the six points, Qi = 2E0 − � j̸=i Ej, 1 ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Due to weaknesses, among these classes, the quadric ones are not represented by irreducible curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Indeed 2E0 − � j̸=i Ej = Ei + � 2E0 − �6 j=1 Ej � and the last class is nothing else than the class of the unique effective root, the strict transform of the quadric q123456 passing through all the six points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The group Cl(X) can be identified with Z(E0 − E1 − E2)⊥ via the orthogonal projection onto this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This projection is given by: Cl(Y ) = Z(E0 − E1 − E2) ⊕ (Z(E0 − E1) ⊕ Z(E0 − E2) ⊕ ZE3 ⊕ · · · ⊕ ZE6) �6 i=0 aiEi = (−a0 − a1 − a2)(E0 − E1 − E2) + � (a0 + a2)(E0 − E1) + (a0 + a1)(E0 − E2) + �6 i=3 aiEi � , and thus Cl(X) = ZL1 ⊕ ZL2 ⊕ ZE3 ⊕ · · · ⊕ ZE6, where Li = E0 − Ei, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, the anticanonical divisors are related by −KY = 3E0 − 6 � i=1 Ei = −(E0 − E1 − E2) + 2L1 + 2L2 − 6 � i=3 Ei � �� � −KX Only the exceptional classes of Y that do not meet E0 − E1 − E2 are mapped to exceptional classes on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' for 3 ≤ i ≤ 6, this leaves the classes: Ei �−→ Ei, E1i �−→ L1 − Ej, E2i �−→ L2 − Ej, Qi �−→ L1 + L2 − � j∈{3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=',6}\\{i} Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As for the unique effective root of Y , it is mapped to the root L1 + L2 − E3 − E4 − E5 − E6 and the last four exceptional classes are not represented by irreducible curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus one has R = Z(L1 + L2 − E3 − · · · − E6), R ⊥ = � a1L1 + a2L2 + 6 � i=3 Ei | a1 + a2 + 6 � i=3 ai = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In order to take into account the Galois action, which acts via (L1L2)(E3E4E5E6), we put L = L1 + L2 and E = �6 i=3 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We easily verify that Cl(X) = CaCl(X) = ZL ⊕ ZE = Z(L − E) ⊕ ZE, R = Z(L − E), R⊥ = Z(2L − E) = ZKX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This leads to the following isomorphism: Cl(Xs) ≃ Cl(X)/R ≃ −→ ZE aL + bE mod R �−→ (a + b)E Via this isomorphism, the sub-module CaCl(Xs) = R⊥ = Z(2L−E) = ZKX is mapped to ZE itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In conclusion both CaCl(Xs) and Cl(Xs) are free Z-module of rank 1 and the canonical embedding turns to be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 19 Types of decomposition into irreducible components in |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Recall that −KX = 2L1 + 2L2 − 6 � i=3 Ei = 4E0 − 2E1 − 2E2 − 6 � i=3 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Global sections of −KX are thus related to quartics of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' More precisely, one has the following one-to-one correspondences: ���4ℓ − 2p1 − 2p2 − �6 i=3 pi ��� Y −→ ���4E0 − 2E1 − 2E2 − �6 i=3 Ei ��� X −→ |−KXs|Xs C �−→ χ∗ � C♯� �−→ ϕ∗ � χ∗ � C♯�� , and we need to list all the quadrics of P2 having multiplicity at least 2 at p1 and p2 and passing through the pi for 3 ≤ i ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Note that in the correspondences above, we skip the surface Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Recall that, as in (8), we have P2 π ←− Y χ −→ X and the morphism χ here is the contraction of the strict transform of the line passing through p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The orbits of lines, respectively conics, having degree less than 4 and passing through some of the points pi’s are ℓ1 ∪ ℓ2, ℓ3 ∪ ℓ4 ∪ ℓ5 ∪ ℓ6, ℓ12, ℓ35 ∪ ℓ46, ℓ13 ∪ ℓ24 ∪ ℓ15 ∪ ℓ26, ℓ14 ∪ ℓ25 ∪ ℓ16 ∪ ℓ23, respectively: q1 ∪ q2, q12, q35 ∪ q46, q3456, q1235 ∪ q1246, q123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The only orbits of cubics or quartics having degree less than 4 that pass through the points pi are c123456 and t123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We now combine the rational irreducible decompositions in order to construct plane curves in the expected sub-linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' First, suppose that the decomposition into absolute irreducible components contains a line If this line joins one of the first two points to one of the last four points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' a line ℓij with i ∈ {1, 2} and j ∈ {3, 4, 5, 6}, then this line has degree 4 and it turns out that its orbit under the Galois action lies in the linear system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (cases 1 and 2 in the tabular below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If this line is ℓ12, which is rational, then this line appears with multiplicity at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If 2ℓ12 is a part of the decomposition then the complementary conic must be rational and pass through the last four points: the conic must be ℓ35 ∪ ℓ46 or q3456 or q123456 (cases 3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If ℓ12 has multiplicity 1, then the complementary cubic passes through the six points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since, except ℓ12, all the lines passing through some pi have even degree, this cubic cannot be a union of three lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The remaining cases are thus q123456∪ℓ or c123456 (cases 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If the line is ℓi for i ≥ 3, then it has degree (at least) 4 and its orbit under Galois has degree 4 (or greater) without passing through p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If the line is ℓ1 then its conjugate ℓσ 1 passes through p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' the complement is a conic passing through the six points, and it must be q123456 (case 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last if this line is ℓ a line passing through none of the six points then the complementary cubic passes through the six points with multiplicity 2 at p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since an irreducible plane cubic has at most one singular point, this cubic must be reducible and it is the union of a line and a conic, whose meeting points are the singular points, that is p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus the line must be ℓ12, and the conic is q123456 and we recover case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Secondly, suppose that there are only two absolutely irreducible conics in the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For the union of these two conics to be singular at p1 and p2, they must pass through p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Taking into account the rationality, there are only three possibilities, cases 9, 10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last if the quartic is absolutely irreducible, then it must pass through the six points with multiplicity 2 at p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the tabular below, we summarize all the possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As noted below, the strict transform �ℓ12 in Y is contracted in X via the morphism Y χ −→ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this explains why the curve disappears in the middle column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then from X to Xs, it is the irreducible effective root �q123456 that is contracted by the morphism X ϕ −→ Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus on Xs, there are two specific rational points, p the image of the contraction of �ℓ12 and s the image of the 20 contraction of �q123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ���4ℓ − 2p1 − 2p2 − �6 i=3 pi ��� |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts 1 ℓ13 ∪ ℓ24 ∪ ℓ15 ∪ ℓ26 �ℓ13 ∪ �ℓ24 ∪ �ℓ15 ∪ �ℓ26 ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ24) ∪ ϕ∗(�ℓ15) ∪ ϕ∗(�ℓ26) 0 2 ℓ14 ∪ ℓ25 ∪ ℓ16 ∪ ℓ23 �ℓ14 ∪ �ℓ25 ∪ �ℓ16 ∪ �ℓ23 ϕ∗(�ℓ14) ∪ ϕ∗(�ℓ25) ∪ ϕ∗(�ℓ16) ∪ ϕ∗(�ℓ23) 0 3 2ℓ12 ∪ ℓ35 ∪ ℓ46 �ℓ35 ∪ �ℓ46 ϕ∗(�ℓ35) ∪ ϕ∗(�ℓ46) 2 4 2ℓ12 ∪ q3456 �q3456 ϕ∗(�q3456) q + 2 5 2ℓ12 ∪ q123456 �q123456 ∪ E1 ∪ E2 {p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' s} ∪ ϕ∗(E1) ∪ ϕ∗(E2) 2 6 ℓ12 ∪ q123456 ∪ ℓ �q123456 ∪ �ℓ ϕ∗(�ℓ) q + 2 7 ℓ12 ∪ c123456 �c123456 ϕ∗(�c123456) Nq (1) 8 ℓ1 ∪ ℓσ 1 ∪ q123456 �ℓ1 ∪ �ℓσ 1 ∪ �q123456 ϕ∗(�ℓ1) ∪ ϕ∗(�ℓσ 1) 2 9 q12 ∪ q123456 �q12 ∪ �q123456 ϕ∗(�q12) q + 2 10 q1235 ∪ q1246 �q1235 ∪ �q1246 ϕ∗(�q1235) ∪ ϕ∗(�q1246) 2 11 2q123456 �q123456 ∪ �6 i=3 Ei {s} ∪ �6 i=3 ϕ∗(Ei) 1 12 t123456 singular at p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p2 �t123456 ϕ∗(�t123456) Nq (1) Some comments about the numbers of points are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 1 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The four lines ℓ13, ℓ24, ℓ15, ℓ26 are conjugate, they do not meet and thus their union in P2 does not contain any rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the blowing-up of P2 at the six points, the strict transforms of the lines ℓ13, ℓ24, ℓ15, ℓ26 no longer meet the strict transform of ℓ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus the contraction of this line does not add any rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since none of the lines ℓ13, ℓ24, ℓ15, ℓ26 can be a tangent line to q123456 at some pi (otherwise the line and the quadric would have too many intersection points by Bezout), blowing-up the pi, 1 ≤ i ≤ 6, separates the strict transforms of the lines and the strict transform of the quadric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The contraction of this quadric neither add rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This proves that these configurations do not contain any rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The lines ℓ35 and ℓ46 are conjugate to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Their intersection point is the unique rational point of their union in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This point is still on ϕ∗(�ℓ35) ∪ ϕ∗(�ℓ46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The added point is p which comes from the contraction of �ℓ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Note that the contraction of �q123456 does not add any point since the lines ℓ35 and ℓ46 are separated from �q123456 in the blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This is because none of these lines can be a tangent to q123456 at one of the six points (otherwise the line and the quadric would have too many intersection points by Bezout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The conics q123456 and q3456 are separated by the blowing up of the last four points and thus the point s does not belong to the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The strict transforms of ℓ12 and of q3456 meet at two points that are mapped to p by the contraction of �ℓ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If these two points are not rational, p is an added rational point of the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 5 & 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The resulting sections contain some exceptional curves Ei in their supports since the multi- plicities at some points pi are strictly greater than the ones expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the points pi are not rational, none of the Ei contain rational points and the only rational points are {p, s} or {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The point p lies on ϕ∗(�ℓ) but it comes from the intersection point between ℓ and ℓ12 (which cannot be one of the pi since it is a rational point) so no points are added in the contraction of ℓ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As for the point s, it lies also on ϕ∗(�ℓ) and it could add one more point if ℓ meet q123456 at two conjugate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The cubic must be smooth at p1 and p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' indeed if it would be singular at one of these points, by Galois conjugation, it would be singular at both of them and it would have too many singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus, to make the multiplicity greater than 2 at p1, p2, the complementary line must be ℓ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This line meets the cubic at p1 and p2 and a third point which must be rational and not on q123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Therefore, the contraction of �ℓ12 pass through p but does not add any rational points to ϕ∗(�ℓ12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As for the contraction of the strict transform of q123456, it does not add points either since blowing up the six points separates the cubic and q123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The meeting point of the curves ℓ1 and ℓσ 1 is necessarily rational and it is the unique rational point of their union in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The strict transforms �ℓ1 and �ℓσ 1 do not meet �ℓ12 and thus the point p does belong to the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The contraction of the root �q123456 add the point s except if the meeting of the curves ℓ1 and ℓσ 1 already belongs to q123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The conics q12 and q123456 are separated by the blowing ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If the conic q12 is chosen in such a way that the tangent line at p1 equals ℓ12, then �q12 and �ℓ12 meet at two unrational points in Y and the contraction of �ℓ12 adds the rational point p to �q12 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This explains why the final number of points is (q + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The two conics are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Besides p1 and p2, they meet at two other points (Bezout) that can 21 be rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If so these points are the only points of P2(Fq) that belong to the union of the two conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Blowing up the six points disconnect the two conics from the strict transforms of ℓ12 and q123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' So no points are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the quartic has at least two singular points, it geometric genus must be at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As predicted by the class group computations, all the curves on Xs in the linear system are irreducible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' they are not necessarily absolutely irreducible but it turns out that curves that are not absolutely irreducible never contain too many rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In any case, one has Nq (−KXs) ≤ Nq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since none of the points pi is rational, the surface Y has q2 + q + 1 rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since X is obtained by contracting �ℓ12, it contains q2 + 1 rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the same way, after contracting �q123456, the surface Xs has q2 − q + 1 rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since q2 − q + 1 ≤ Nq(1) for q ∈ {2, 3}, the evaluation map may be non injective and we do not consider the codes with these two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Suppose that q ̸= 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p6 ∈ P2 Fq be six conconic points, such that p1, p2 and p3, p4, p5, p6 are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical code of the weak del Pezzo surface obtained by blowing up these six points and then blowing down the strict transform of the line (p1p2) has parameters [q2 − q + 1, 5, ≥ q2 − q + 1 − Nq (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Let Q be a quadratic polynomial that defines q123456 and Lij a linear form that defines the line ℓij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then �����4ℓ − 2p1 − 2p2 − 6 � i=3 pi ����� = ⟨QL12X, QL12Y, QL12Z, L2 12L35L46, L13L24L15L26⟩Fq The three first sections are clearly linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The fourth one cannot be a linear combination of the three first ones since otherwise Q would be reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last, the fifth one cannot be a linear combination of the four first ones since otherwise L12 would divide L13L24L15L26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 Degree 4, singularity of type 4A1 This example corresponds to the type number 48 in degree 4 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We recover an example already studied by Koshelev [Kos20, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Our point of view slightly differs from Koshelev’s one, so even if this example appears in the literature, we choose to go into details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — The context is still the one described in (8) with a non trivial map Y χ −→ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1, p2 = pσ 1, p3 = pσ2 1 , p4 = pσ3 1 ∈ P2 be four conjugate points in general position (no three of them are collinear) and, as usual, let ℓ12, ℓ23, ℓ34, ℓ14 denote the lines (p1p2), (p2p3), (p3p4), (p1p4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' they are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p5 be the intersection point of ℓ12 and ℓ34 and p6 be the intersection point of ℓ23 and ℓ14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' They are also conjugate and we denote by ℓ56 the rational line passing through p5, p6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p1 p2 p3 p4 p5 p6 p2 = pσ 1 p3 = pσ2 1 p4 = pσ3 1 p6 = pσ 5 We blow up these six points to obtain a degree 3 weak del Pezzo surface Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The strict transform of the line ℓ56, of class E0 − E5 − E6, is an exceptional curve that can be contracted to obtain the degree four weak del Pezzo surface X we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical model of this surface has four singular points of type A1 (since the four irreducible effective roots do not intersect, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Over Fq, we know that Cl(Y ) = �6 i=0 ZEi and that −KY = 3E0−�6 i=1 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Moreover the surface Y has four irreducible effective roots, the strict transforms of the lines ℓ125, ℓ236, ℓ345, ℓ146 whose conjugate classes in Cl(Y ) are: E0 − E1 − E2 − E5, E0 − E2 − E3 − E6, E0 − E3 − E4 − E5, E0 − E1 − E4 − E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 22 The group Cl(X) identifies with Z(E0 − E5 − E6)⊥ inside Cl(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since Cl(Y ) = Z(E0 − E5 − E6) ⊥⊕ � Z(E0 − E5) ⊕ Z(E0 − E6) ⊕ �4 i=1 ZEi � �6 i=0 aiEi = (−a0 − a5 − a6)(E0 − E5 − E6) + (a0 + a6)(E0 − E5) + (a0 + a5)(E0 − E6) + �4 i=1 aiEi (9) one has Cl(X) = Z(E0 − E5) ⊕ Z(E0 − E6) ⊕ ZE1 ⊕ ZE2 ⊕ ZE3 ⊕ ZE4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, −KX = 2(E0 − E5) + 2(E0 − E6) − E1 − E2 − E3 − E4 = 4E0 − 2E5 − 2E6 − E1 − E2 − E3 − E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On X, there are still four effective roots, the image by the contraction of the effective roots on Y : R = Z(E0 − E1 − E2 − E5) ⊕ Z(E0 − E2 − E3 − E6) ⊕ Z(E0 − E3 − E4 − E5) ⊕ Z(E0 − E1 − E4 − E6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On X there are 16 exceptional classes, (E0 − Ei) − Ej, i ∈ {5, 6}, j ∈ {1, 2, 3, 4}, (E0 − E5) + (E0 − E6) − Ei1 − Ei2 − Ei3, {i1, i2, i3} ⊂ {1, 2, 3, 4}, and E1, E2, E3, E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Only these last four classes are represented by irreducible exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We recover the graph number 9 of the Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 of Coray & Tsfasman [CT88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The Galois group acts as as a 4-cycle on the roots and as (E1E2E3E4) on the (−1)-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us put F = (E0 − E5) + (E0 − E6) and E = E1 + E2 + E3 + E4, in such a way that F·2 = 2, E·2 = −4 and F · E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then one has: � R = Z2(F − E) Cl(X) = Z(F − E) ⊕ ZE =⇒ Cl(Xs) = Cl(X)/R ≃ −→ Z/2Z(F − E) ⊕ ZE aF + bE �−→ a(F − E) mod R + (a + b)E As for the Cartier class group, we find CaCl(Xs) = R⊥ = Z(2F − E) = Z(−KX) which embeds in Cl(Xs) via −KX �→ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus, via the canonical embedding, CaCl(Xs) and the free part of Cl(Xs) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Types of decomposition into irreducible components in |−KXs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — The situation looks like the preceding one except that the multiplicity is at points p5, p6 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' One has: ���4ℓ − �4 i=1 pi − 2p5 − 2p6 ��� Y −→ ���4E0 − 2E5 − 2E6 − �4 i=1 Ei ��� X −→ |−KXs|Xs C �−→ χ∗ � C♯� �−→ ϕ∗ � χ∗ � C♯�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus we are reduced to list all the types of irreducible decompositions of quadrics passing through the six points, the last two with multiplicities at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The orbits of lines of degree less than 4 that involve the six points are ℓ5 ∪ ℓ6, ℓ1 ∪ ℓ2 ∪ ℓ3 ∪ ℓ4, ℓ56, ℓ13 ∪ ℓ24, and ℓ125 ∪ ℓ236 ∪ ℓ345 ∪ ℓ146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There are only two ways (cases 1 and 2 below) to combine these configurations in order to obtain a curve in the expected linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The orbits of conics of degree less than 4 that involve the six points are q1234, q56 and q1356 ∪ q2456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There are only two ways (cases 3 and 4 below) to combine the configurations of lines and conics in order to obtain a curve in the expected linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If the decomposition contains a cubic, it must be smooth at p5 and p6 and the complement must be ℓ56;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this is case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This leads to the list below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us note that on X the curve �ℓ56 in Y is contracted by χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On Xs, this contraction is mapped to a smooth rational point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This surface contains also four singular points si, 1 ≤ i ≤ 4, coming from the contraction of the four roots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' they are conjugate and of degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ���4ℓ − �4 i=1 pi − 2p5 − 2p6 ��� |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts 1 2ℓ56 ∪ ℓ13 ∪ ℓ24 �ℓ13 ∪ �ℓ24 ϕ∗(�ℓ13) ∪ ϕ∗(�ℓ24) 2 2 ℓ125 ∪ ℓ236 ∪ ℓ345 ∪ ℓ146 �ℓ125 ∪ �ℓ236 ∪ �ℓ345 ∪ �ℓ146 ∪ �4 i=1 Ei {si} ∪ �4 i=1 ϕ∗(Ei) 0 3 2ℓ56 ∪ q1234 �q1234 ϕ∗(�q1234) q + 2 4 q1356 ∪ q2456 �q1356 ∪ �q2456 ϕ∗(�q1356) ∪ ϕ∗(�q2456) 2 5 c123456 ∪ ℓ56 �c123456 ϕ∗(�c123456) Nq (1) 6 t123456 singular at p5, p6 �t123456 ϕ∗(�t123456) Nq (1) 23 Some comments on the number of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The lines ℓ13 and ℓ24 are conjugate, their meeting point is the unique rational point of their union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' After the contraction of �ℓ56, these two lines have one more rational point in common, the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If q1234 meets ℓ56 at two conjugate points, the contraction of �ℓ56 adds the point p to the other rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Blowing up p5 and p6 separates the strict transforms �q1356 and �q2456 from ℓ56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' So the contraction of this curve do not add any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On P2 the union of conics �q1356 and �q2456 has at most two rational points: their meeting points that differ from p5, p6 if they are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Necessarily the cubic is smooth at p5, p6 and the tangent lines at these points cannot be equal to ℓ56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Therefore blowing up p5, p6 separates the curves �ℓ56 and �c123456 above p5 and p6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Besides p5, p6 the curves ℓ56 and c123456 meet at a third point (Bezout) which is necessarily rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Via the contraction of �ℓ56, this line concentrates at this third point and no points are added on �c123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Necessarily blowing up p5 and p6 disconnects the strict transforms �t123456 and �ℓ56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Moreover, it desingularizes the quartic t since the singularities at p5 and p6 must be ordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Blowing up p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p4 also disconnects �t from all the effective roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally ϕ∗(t123456) turns to be an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally Nq(−KXs) ≤ Nq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since none of the points pi is rational, #X(Fq) = #P2(Fq) = q2 +q+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then contracting the rational line �ℓ56 decreases the number by q and #Xs(Fq) = q2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Except for q = 2, this number is stricly greater that Nq(1) and the evaluation map is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Suppose q ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1, p2 = pσ 1, p3 = pσ2 1 , p4 = pσ3 1 ∈ P2 Fq be four conjugate points in general position (no three of them are collinear) and let p5 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p6) be the point of intersection of the lines (p1p2) and (p3p4) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (p2p3) and (p1p4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical code of the weak del Pezzo surface obtained by blowing up these six points and then blowing down the strict transform of the line (p5p6) has parameters [q2 + 1, 5, ≥ q2 + 1 − Nq (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As proved by Koshelev [Kos20, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2], for some values of q, the minimum distance can be improved by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the argument is very nice, we choose to briefly sketch it below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The idea is to prove that all the elliptic curves in our linear system must have a rational 2-torsion point and thus an even number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since for some q, the maximum Nq(1) is odd, this means that Nq (−KXs) < Nq(1) and our bound for the minimum distance can be improved by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let c be a cubic passing through the six points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then, for any choice of the origin, the alignments of points permit to show that: p1 + p2 + p5 = p1 + p4 + p6 = p2 + p3 + p6 = p3 + p4 + p5 =⇒ p1 − p3 = p2 − p4 = p4 − p2 = p6 − p5 =⇒ 2(p2 − p4) = 0, and these points must be rational since p2 − p4 = p6 − p5 with p5, p6 conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The case of the quartics in the linear system works in the same way but it is a little bit more technical since we need to know the group law on this kind of curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — In this example, we do not find a nice explicit basis for the global sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Instead, we choose to present a magma code that permits to construct the generator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 1 c l e a r ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' q := 7 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Fq := F i n i t e F i e l d ( q ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' P2 := P r o j e c t i v e S p a c e (Fq , 2) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' XYZ := C oordi nateRi ng ( P2) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 6 phi := map < P2 −> P2 | [Xˆq ,Yˆq , Zˆq ] > ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' // Some b a s i c r o u t i n e s such as v e r i f y i n g i f some p o i n t s are i n g e n e r a l p o s i t i o n l oad ” U t i l i t i e s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' magma” ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 11 // Choose randomly a degree 4 p o i n t i n g e n e r a l p o s i t i o n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' r e p e a t p1 := Random( P2 ( ext< Fq | 4>)) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p2 := phi ( p1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p3 := phi ( p2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p4 := phi ( p3 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cl1234 := C l u s t e r ( p1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 16 u n t i l EnPos i ti onGene r al e ( [ p1 , p2 , p3 , p4 ] ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' //// To compute the p o i n t s p5 , p6 we need the extend the s c a l a r s to Fq4 P2 Fq4 := BaseChange ( P2 , ex t < Fq | 4 >) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 21 p1 Fq4 := P2 Fq4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ElementToSequence ( p1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p2 Fq4 := P2 Fq4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ElementToSequence ( p2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p3 Fq4 := P2 Fq4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ElementToSequence ( p3 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p4 Fq4 := P2 Fq4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ElementToSequence ( p4 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' L12 := Scheme ( P2 Fq4 , S e c t i o n s ( LinearSystem ( LinearSystem ( P2 Fq4 , 1 ) , [ p1 Fq4 , p2 Fq4 ] ) ) [ 1 ] ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' L34 := Scheme ( P2 Fq4 , S e c t i o n s ( LinearSystem ( LinearSystem ( P2 Fq4 , 1 ) , [ p3 Fq4 , p4 Fq4 ] ) ) [ 1 ] ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 26 p5 Fq4 := Poi nts ( L12 meet L34 ) [ 1 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p5 := P2( ex t < Fq | 2 >)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ElementToSequence ( p5 Fq4 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' //// End o f the computation ov er Fq4 31 p6 := phi ( p5 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cl56 := C l u s t e r ( p5 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' C l 5 6 s q u a r e := C l u s t e r ( P2 , I d e a l ( Cl56 ) ˆ 2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 24 L := LinearSystem ( LinearSystem ( P2 , 4 ) , Cl1234 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 36 L := LinearSystem (L , C l 5 6 s q u a r e ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' T heSecti ons := S e c t i o n s (L) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' L56 := Scheme ( P2 , S e c t i o n s ( LinearSystem ( LinearSystem ( P2 , 1 ) , Cl56 ) ) [ 1 ] ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' S := ( Poi nts ( P2 ) d i f f Poi nts ( L56 ) ) j o i n {@ Poi nts ( L56 ) [ 1 ]@} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 41 G := Matrix ( Fq,5 ,1+ q ˆ2 , &cat [ [ Ev al uate ( f , ElementToSequence ( p ) ) : p i n S ] : f i n T heSecti ons ] ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' TheCode := LinearCode (G) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p r i n t f ” Generator matrix G =\\n%o\\n ” , G ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 46 l g := Length ( TheCode ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' dim := Dimension( TheCode ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' d min := MinimumWeight( TheCode ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p r i n t f ”q = %o , \\ n [ n , k , d ] = [%o , %o , %o ] \\ n ” , q , l g , dim , d min ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p r i n t f ”What was prov i ded ( not t a k i n g i n t o account Kas hel ev remark ) = [%o , 5 , %o ] ” , qˆ2+1, qˆ2 − q − Fl oor (2∗ Sq rt ( q ) ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6 Degree 4, singularity of type A2 This example corresponds to the type number 30 in degree 4 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This type works almost as the one described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Let p1, p2 = pσ 1, p3 = pσ2 1 , p4 = pσ3 1 ∈ P2 be four conjugate points in general position (no three of them are collinear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The two lines (p1p3) and (p2p4) are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We choose a degree 2 point p5 on (p1p3) and we let p6 = pσ 5 which lies on (p2p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p1 p2 p3 p4 p5 p6 ℓ135 ℓ246 ℓ56 p2 = pσ 1 p3 = pσ2 1 p4 = pσ3 1 p6 = pσ 5 The surfaces of the diagram (8) are the following: we blow up the six points to obtain the degree 3 weak del Pezzo surface Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On this surface, the strict transform of the line ℓ56, of class E0 − E5 − E6, is an exceptional curve that can be contracted to obtain the weak degree four weak del Pezzo surface X defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical model has a unique singular point of type A2 (since there are only two irreducible effective root that meet, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Over Fq, we know that Cl(Y ) = �6 i=0 ZEi and that −KY = 3E0 −�6 i=1 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There are only two irreducible effective roots on Y , the strict transforms of the lines ℓ135 and ℓ246 whose conjugate classes in Cl(Y ) are E0 − E1 − E3 − E5 and E0 − E2 − E4 − E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The group Cl(X) identifies with Z(E0−E5−E6)⊥ inside Cl(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We recover the same orthogonal decomposition as in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In particular, we still have −KX = 4E0 − �4 i=1 Ei − 2E5 − 2E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' On X there are still two (conjugate) irreducible effective roots, of classes E0 − E1 − E3 − E5 and E0 − E2 − E4 − E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We follow the same computation as in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 and we still put E = (E0 − E5) + (E0 − E6) and F = E1 + E2 + E3 + E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then, for X we have: − KX = 2F − E, R = Z(F − E), and Cl(X) = Z(F − E) ⊕ ZE, and for Xs we deduce that: CaCl(Xs) = Z(2F − E) = Z(−KX) Cl(Xs) ≃ −→ ZE aF + bE �−→ (a + b)E CaCl(Xs) ≃ → Cl(Xs) −KX �→ E The canonical embedding induces an isomorphism between the two class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Types of decomposition into irreducible components in |−KX|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Since the two class groups are isomorphic and free of rank one, all the sections are necessarily irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' However, they can be absolutely reducible and we need to review all the possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5, we are reduced to list all the types of irreducible decompositions of quartics passing through the six points, the last two with multiplicities at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We follow the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The orbits of lines of degree less than 4 that involve the six points are ℓ5 ∪ ℓ6, ℓ1 ∪ ℓ2 ∪ ℓ3 ∪ ℓ4, ℓ56, ℓ12 ∪ ℓ23 ∪ ℓ34 ∪ ℓ14, ℓ16 ∪ ℓ25 ∪ ℓ36 ∪ ℓ45, and ℓ135 ∪ ℓ246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 25 There are five ways (cases 1 to 5 below) to combine these configurations in order to obtain a curve in the expected linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The orbits of conics of degree less than 4 that involve the six points are q1234 and q56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Note that compared to the example of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5, the orbit q1356 ∪ q2456 does not appear since a conic q1356 cannot be irreducible otherwise it would have three intersection points with the line ℓ135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There are only two ways (cases 6 and 7 below) to combine the configurations of lines and conics in order to obtain a curve in the expected linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If the decomposition contains a cubic, it must be smooth at p5 and p6 and the complement must be ℓ56;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this is case 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This leads to the list below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In Y , the strict transforms �ℓ135 and �ℓ246 are separated from the strict transform �ℓ56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In X this last curve is contracted to a smooth rational point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last, another difference from the example of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5: the two effective roots of X meet and they are thus contracted by the anticanonical morphism ϕ∗ onto the same point s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This point is the unique singular point of Xs and it is necessarily a rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ���4ℓ − �4 i=1 pi − 2p5 − 2p6 ��� |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts 1 2ℓ56 ∪ ℓ135 ∪ ℓ246 �ℓ56 ∪ �ℓ135 ∪ �ℓ246 ∪ E5 ∪ E6 ϕ∗(E5) ∪ ϕ∗(E6) ∪ {s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ56 ∪ ℓ135 ∪ ℓ246 ∪ ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ135 ∪ �ℓ246 ∪ �ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ16 ∪ ℓ25 ∪ ℓ36 ∪ ℓ45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ16 ∪ �ℓ25 ∪ �ℓ36 ∪ �ℓ45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ16) ∪ ϕ∗(�ℓ25) ∪ ϕ∗(�ℓ36) ∪ ϕ∗(�ℓ45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ5 ∪ ℓ6 ∪ ℓ135 ∪ ℓ246 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ5 ∪ �ℓ6 ∪ �ℓ135 ∪ �ℓ246 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ5) ∪ ϕ∗(�ℓ6) ∪ {s} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2ℓ135 ∪ 2ℓ246 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ135 ∪ �ℓ246 ∪ E1 ∪ E2 ∪ E3 ∪ E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(E1) ∪ ϕ∗(E2) ∪ ϕ∗(E3) ∪ ϕ∗(E4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ135 ∪ ℓ246 ∪ q56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ135 ∪ �ℓ246 ∪ �q56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�q56) ∪ {s} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2ℓ56 ∪ q1234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�q1234 ∪ {p} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�q1234) ∪ ϕ∗({p}) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='c123456 ∪ ℓ56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�c123456 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�c123456) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='Nq (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='t123456 singular at p5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p6 �t123456 ϕ∗(�t123456) Nq (1) Some comments about the numbers of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 1 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The exceptional curves E5 and E6 do not contain any rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The other components are all contracted to the points p or s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The same is true in case 5, without the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The ending curve passes through p but the contraction of �ℓ56 does not add any rational point since ℓ56 and ℓ meet at a rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The ending curve passes through s and the contraction of the roots add a point if ℓ meet ℓ135 and ℓ246 outside the meeting point of these two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' All theses lines and ℓ135, ℓ246 and ℓ56 are separated by the blowing ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the four lines cannot contain any rational point, neither does their image in Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The conjugate lines ℓ5 and ℓ6 contain a unique rational point, their intersection point, to which is added the point s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The curve �q56 no longer meets �ℓ135, �ℓ246 and �ℓ56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The ending curve contains the rational points of q56 plus the point s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If q1234 meets ℓ56 at two conjugate points then in X, after �ℓ56 being contracted, the strict trans- form �q1234 passes through p which is an additional rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Necessarily the blowing ups of p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p4 separate the strict transforms �q1234, �ℓ135 and �ℓ246 and the roots contraction does not add any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 8 & 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Same as §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally Nq (−KXs) ≤ Nq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As in the previous example, one has #Xs(Fq) = q2+1, and for q = 2, the evaluation map may not be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Suppose q ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1, p2 = pσ 1, p3 = pσ2 1 , p4 = pσ3 1 ∈ P2 be four conjugate points in general position (no three of them are collinear), let p5 be a point of the line (p1p3) inside P2(Fq2) and let p6 = pσ 5 in such a way that p6 lies on (p2p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical code of the weak del Pezzo surface obtained by blowing up these six points and then blowing down the strict transform of the line (p5p6) has parameters [q2 + 1, 5, ≥ q2 + 1 − Nq (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — This example looks like the previous one and we do not find a nice explicit basis for the global sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A slightly modification of the code given for the previous example leads to a program which permits to compute a generator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7 Degree 4, singularity of type D5 This example corresponds to the type number 58 in degree 4 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 26 Configuration to blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — In this example, the surfaces Y and X of diagram (8) are equal and we obtain directly the surface X by blowing up P2 at five rational points p1 ≺ p2 ≺ · · · ≺ p5, with p1, p2, p3 collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us denote by π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , π5 these five blowups at p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p5 respectively: P2 X1 X2 X3 X4 X π1 π2 π3 π4 π5 π The fact that p1, p2, p3 are collinear means that there is a line ℓ123 of P2 whose strict transform by π1 passes through p2 and whose strict transform by π2 ◦ π1 passes through p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical model of this weak del Pezzo surface has a unique singular point of type D5 (since there are five irreducible effective roots whose intersection graph is D5, see the picture at the end of this example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Since all the blown-up points are rational, there is no need to work with the base change X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The irreducible effective classes of roots are the strict transform of ℓ123 and of E1, E2, E3, E4, whose classes in Cl(X) are: E0 − E1 − E2 − E3, E1 − E2, E2 − E3, E3 − E4, and E4 − E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The submodule R, generated by these classes, is a direct summand and for example Cl(X) = R ⊕ ZE5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' the projection onto the factor ZE5 leads to an isomorphism Cl(X)/R → ZE5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As for the submodule R⊥, it is defined by the equations a1 = a2 = · · · = a5 and a0 + a1 + a2 + a3 = 0 and thus R⊥ = ZKX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since −KX = 3 (E0 − E1 − E2 − E3) + 2 (E1 − E2) + 4 (E2 − E3) + 6 (E3 − E4) + 5 (E4 − E5) + 4E5, via the preceding isomorphism, the module R⊥ embeds via −KX �→ 4E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In brief, both divisor class groups CaCl(Xs) and Cl(Xs) are free rank one Z-modules, the first one being of index 4 in the latter via the canonical embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' For this example, it makes sense to reverse the order of the paragraphs and we start to compute a basis of the global sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — We need to compute a basis of the sublinear system on P2 |3ℓ − p1 − · · · − p5| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' So we consider a cubic of P2 X,Y,Z whose restriction to the affine space A2 x1,y1 (x1 = X Z , y1 = Y Z and Z ̸= 0) is defined by the equation: C1(x1, y1) = a30x3 1 + a21x2 1y1 + a20x2 1 + a12x1y2 1 + a11x1y1 + a10x1 + a03y3 1 + a02y2 1 + a01y1 + a00 = 0 We choose p1 = (0, 0) ∈ A2 x1,y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The cubic passes through the point p1 if and only if a00 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let x2, y2 be the coordinates of the affine chart of the blowing up of A2 x1,y1 at p1 defined by x1 = x2 and y1 = x2y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this chart, the exceptional divisor E1 has equation x2 = 0 and the strict transform of C1 is defined by: C2(x2, y2) = 1 x2 C1(x2, x2y2) = a30x2 2 +a21x2 2y2 +a20x2 +a12x2 2y2 2 +a11x2y2 +a10 +a03x2 2y3 2 +a02x2y2 2 +a01y2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We choose p2 = (0, 0) ∈ A2 x2,y2 which corresponds to the line ℓ123 with affine equation y1 = 0 in A2 x1,y1 or Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' = 0 in P2 X,Y,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The cubic passes through the point p2 if and only if a10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let x3, y3 be the coordinates of the affine chart of the blowing up of A2 x2,y2 at p2 defined by x2 = x3 and y2 = x3y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this chart, the exceptional divisor E2 has equation x3 = 0 and the strict transform of C2 is defined by: C3(x3, y3) = 1 x3 C2(x3, x3y3) = a30x3 + a21x2 3y3 + a20 + a12x3 3y2 3 + a11x3y3 + a03x4 3y3 3 + a02x2 3y2 3 + a01y3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since p1, p2, p3 are colinear, we have to choose p3 = (0, 0) ∈ A2 x3,y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The cubic passes through the point p3 if and only if a20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let x4, y4 be the coordinates of the affine chart of the blowing up of A2 x3,y3 at p3 defined by x3 = x4 and y3 = x4y4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this chart, the exceptional divisor E3 has equation x4 = 0 and the strict transform of C3 is defined by: C4(x4, y4) = 1 x4 C2(x4, x4y4) = a30 + a21x2 4y4 + a12x4 4y2 4 + a11x4y4 + a03x6 4y3 4 + a02x3 4y2 4 + a01y4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 27 Since p4 ∈ E3, we have to choose p4 = (0, α) ∈ A2 x4,y4 and since p1, p2, p3, p4 are not colinear, necessarily α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The cubic passes through the point p4 if and only if a30 + αa01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Last, let x5, y5 be the coordinates of the affine chart of the blowing up of A2 x4,y4 at p4 defined by x4 = x5 and y4 = α + x5y5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In this chart, the exceptional divisor E4 has equation x5 = 0 and the strict transform of C4 is defined by: C5(x5, y5) = 1 x5 C2(x5, x5y5) = 1 x5 � a30 + a21x2 5(α + x5y5) + a12x4 5(α + x5y5)2 + a11x5(α + x5y5) + a03x6 5(α + x5y5)3 +a02x3 5(α + x5y5)2 + a01(α + x5y5) � ≡ αa11 + αa21x5 + a01y5 + a11x5y5 mod x2 5Fq[x5, y5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (10) Since p5 ∈ E4, one can choose p5 = (0, β) ∈ A2 x5,y5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The cubic passes through the point p5 if and only if αa11 + βa01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' To sum up, the global sections are defined by a00 = a10 = a20 = 0, a30 = −αa01, and a11 = −β αa01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The fact that α ̸= 0 is important here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the projective setting, this leads to the basis |3ℓ − p1 − · · · − p5| = � αY Z2 − βXY Z − α2X3, Y 3, X2Y, XY 2, Y 2Z � Fq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (11) Types of decomposition into irreducible components in |−KX|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Since CaCl(Xs) if of index 4 in- side Cl(Xs), even if these two groups are free of rank 1, an irreducible Cartier divisor may decompose into Weil irreducible components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In order to lower bound the minimum distance, we need to review all these kinds of decompositions into irreducible components for the curves of the anticanonical linear system on Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As usual, we start form P2 and use the one-to-one correspondences: |3ℓ − p1 − · · · − p5|P2 −→ |−KX|X −→ |−KXs|Xs C �−→ C♯ �−→ ϕ � C♯� where C♯ denotes the virtual transform of C in the composition of the five blowups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thanks to the preceding computation, for every curve in |3ℓ − p1 − · · · − p5|P2 there exists α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , α5 ∈ Fq such this curves is defined by α1 � αY Z2 − βXY Z − α2X3� + α2Y 3 + α3X2Y + α4XY 2 + α5Y 2Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We deduce that such a curve can decompose in six different ways, as listed in the tabular below: either a cubic c12345 for which p1 is a smooth flex point with tangent line equal to ℓ123, if α1 ̸= 0 (case 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' or a cubic singular at p1 which contains ℓ123 as a component, if α1 = 0, the complementary component, of discriminant α3α2 5 (up to a constant), being either – a quadric q12 smooth at p1 with tangent line ℓ123, if α3 ̸= 0 and α5 ̸= 0 (case 2), – or the union of two lines, if α3 ̸= 0 and α5 = 0, α3 = 0 and α5 ̸= 0, α3 = α5 = 0 and α4 ̸= 0, α3 = α4 = α5 = 0 (cases 3, 4, 5, 6 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' ���3ℓ − �5 i=1 pi ��� |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='c12345 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�c12345 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�c12345) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='Nq (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ123 ∪ q12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ123 ∪ �q12 ∪ �E1 ∪ 2 �E2 ∪ 2 �E3 ∪ �E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�q12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ℓ123 ∪ ℓ1 ∪ ℓ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='�ℓ123 ∪ 2 �E1 ∪ 2 �E2 ∪ 2 �E3 ∪ �E4 ∪ �ℓ1 ∪ �ℓ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(�ℓ1) ∪ ϕ∗(�ℓ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2ℓ123 ∪ ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2�ℓ123 ∪ �E1 ∪ 2 �E2 ∪ 3 �E3 ∪ 2 �E4 ∪ E5 ∪ �ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='ϕ∗(E5) ∪ ϕ∗(�ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2ℓ123 ∪ ℓ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2�ℓ123 ∪ 2 �E1 ∪ 3 �E2 ∪ 4 �E3 ∪ 3 �E4 ∪ 2E5 ∪ �ℓ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2ϕ∗(E5) ∪ ϕ∗(�ℓ1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='2q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3ℓ123 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3�ℓ123 ∪ 2 �E1 ∪ 4 �E2 ∪ 6 �E3 ∪ 5 �E4 ∪ 4E5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='4ϕ∗(E5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='Let us give details for the computation of the virtual transform π♯(C) if,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' C = ℓ123 ∪ ℓ1 ∪ ℓ′ 1: C1 = π♯ 1(C) = �ℓ123 + �ℓ1 + �ℓ′ 1 + 2E1 [p1 ∈ ℓ123 ∩ ℓ1 ∩ ℓ′ 1 ⇒ mp1(C) = 3] C2 = π♯ 2(C1) = �ℓ123 + �ℓ1 + �ℓ′ 1 + 2 �E1 + 2E2 � p2 ∈ �ℓ123 ∩ E1 ⇒ mp2(C1) = 3 � C3 = π♯ 3(C2) = �ℓ123 + �ℓ1 + �ℓ′ 1 + 2 �E1 + 2 �E2 + 2E3 � p3 ∈ �ℓ123 ∩ E2 ⇒ mp3(C2) = 3 � C4 = π♯ 4(C3) = �ℓ123 + �ℓ1 + �ℓ′ 1 + 2 �E1 + 2 �E2 + 2 �E3 + E4 [p4 ∈ E3 ⇒ mp4(C3) = 2] C♯ = π♯ 5(C4) = �ℓ123 + �ℓ1 + �ℓ′ 1 + 2 �E1 + 2 �E2 + 2 �E3 + �E4 [p5 ∈ E4 ⇒ mp5(C4) = 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This leads to the following decomposition of the canonical class into a sum of effective classes −KX = (E0 − E1 − E2 − E3) + (E0 − E1) + (E0 − E1) + 2(E1 − E2) + 2(E2 − E3) + 2(E3 − E4) + (E4 − E5) The intersection graph of the irreducible effective roots in X is connected (see figure below) and all these curves are contracted by the morphism ϕ to a single rational singular pointy s (of singularity type D5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' on X �E1(−2) �E2(−2) �E3(−2) �ℓ123(−2) �E4(−2) E5(−1) ϕ∗ ϕ∗(E5) s on Xs We comment on the numbers of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 2 & 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' All the components in X are roots that are contracted, except �q12 and E5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These two strict transforms meet the tree of roots at only one point and by ϕ∗ they are mapped to isomorphic curves that pass through s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Except �ℓ1 and �ℓ′ 1, all the components on X are irreducible effective roots and they are mapped to the point s by the morphism ϕ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' After the contraction, the curves ϕ∗(�ℓ1) and ϕ∗(�ℓ′ 1) meet at this singular point, thus their union contains 2q + 1 rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Cases 4 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The line E5 does not intersect the lines �ℓ or �ℓ1 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' However, since ℓ and ℓ123 meet at some point of P2 (not equal to p1), the lines �ℓ and �ℓ123 intersect in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' in the same way since ℓ1 passes through p1, the lines �ℓ1 and �E1 intersect in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Therefore, ϕ∗(E5) and ϕ∗(�ℓ) or ϕ∗(E5) and ϕ∗(�ℓ1) both intersect at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus the two unions has 2q + 1 rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally Nq (−KXs) = 2q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The surface Xs has a unique singular point s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' All the irreducible effective roots of X, that is �ℓ123, �E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , � E4 are contracted to this single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The last exceptional curve E5 meets E4 and thus ϕ(E5) passes through s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In conclusion the rational points of Xs(Fq) are in one-to-one correspondence with � P2(Fq) \\ ℓ123(Fq) � ∪ E5(Fq), which counts q2 + q + 1 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This number is always strictly greater that 2q + 1 and the evaluation map is always injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 be infinitely near rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Suppose that the first three ones are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical code of the weak del Pezzo surface obtained by blowing up these points has parameters [q2 + q + 1, 5, q2 − q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The construction of a generator matrix of this code is a nice application of proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It consists of two blocks, the left one, of size 5 × q2, contains the evaluations of the five global sections of (11) at every point of P2(Fq)\\ℓ123(Fq), the right one, of size 5×(q+1) contains the evaluations of the homogeneous parts of degree 1 of the five global sections of (11) at every point of P1(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Letting y5 = β + z5 in (10), this homogeneous part of degree 1 equals (αa21 + βa11)x5 + a01z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally, we get the explicit matrix: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed αyz2 − βxyz − α2x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' −β2u + αv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' y3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' x2y (x : y : z) ∈ P2(Fq) | y ̸= 0 αu (u : v) ∈ P1(Fq) xy2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' y2z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , where α ∈ F∗ q and β ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8 Degree 3, singularity of type A1 This example corresponds to the type number 11 in degree 3 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — We blow up six conjugate points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p6 ∈ P2 on a smooth conic q123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p1 p2 p3 p4 p5 p6 q123456 p2 = pσ 1 p3 = pσ2 1 p4 = pσ3 1 p5 = pσ4 1 p6 = pσ5 1 The resulting surface X is a weak del Pezzo of degree 3, whose anticanonical model Xs has a unique singular point of type A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — We have: Cl(X) = 6 � i=0 ZEi and Cl(X) = ZE0 ⊕ ZE, where E = 6 � i=1 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There is a unique irreducible effective root, the strict transform of the conic q123456, whose class is 2E0 − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The root module R, generated this class, is a direct summand, Cl(X) = R ⊕ ZE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The projection onto the second factor leads to an isomorphism Cl(X)/R −→ ZE0 E0 mod R �−→ E0 E mod R �−→ 2E0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As for the module R⊥, inside Cl(X) it is defined by the single equation 2a0 + a1 + · · · + a6 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' after taking the Galois invariants, we obtain CaCl(Xs) = ZKX, whose image by the previous isomorphism is also ZE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Therefore CaCl(Xs) ≃ Cl(Xs) and both of them are free of rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Types of decomposition into irreducible components in |−KX|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — This proves that all the sections of the anticanonical divisor are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As in our previous work [BCH+20], we expect that the curves of the associated linear system can contain at most Nq (1) rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' However we need to investigate the types of irreducible decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Here this is easy since one can check that the only Galois orbits of lines or conics or cubics that pass through at least one point pi are ℓ14∪ℓ25∪ℓ36 or q123456 or c123456 (all the others lead to Fq-curves of degree strictly greater than 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Combining them in order to construct a curve in the expected sub-linear system leads to very few decompositions: ���3ℓ − �6 i=1 pi ��� |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts 1 ℓ14 ∪ ℓ25 ∪ ℓ36 �ℓ14 ∪ �ℓ25 ∪ �ℓ36 ϕ∗(�ℓ14) ∪ ϕ∗(�ℓ25) ∪ ϕ∗(�ℓ36) 1 2 ℓ ∪ q123456 �ℓ ∪ �q123456 ϕ∗(�ℓ) ∋ s q + 2 3 c123456 �c123456 ϕ∗(�c123456) Nq (1) The number of rational point in case 1 is at most 1 if the three lines meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In case 2, during the process, if the two meeting points of ℓ and q123456 are not rational, then the singular point s is an additional rational point on ϕ∗(�ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We deduce that Nq (−KXs) ≤ Nq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the blown up points are not rational, the blowing ups do not add point on the surface and #X(Fq) = q2 + q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then, the irreducible effective root is contracted and thus #Xs(Fq) = q2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' If q = 2, the evaluation map may fail to be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Suppose q ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p6 ∈ P2 be six conjugate points lying on a smooth conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical code of the weak del Pezzo surface obtained by blowing up these points has parameters [q2 + 1, 4, ≥ q2 + 1 − Nq (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 30 Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Let Q denote the conic passing through p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p6 and let L14, L25, L36 be the linear forms whose zeros are the lines ℓ14, ℓ25, ℓ36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then H0 � P2, 3ℓ − �6 i=1 pi � = ⟨XQ, Y Q, ZQ, L14L25L36⟩Fq 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9 Degree 3, singularity of type 3A2 This example corresponds to the type number 76 in degree 6 [BH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='5, this example appears in Koshelev’s work [Kos20, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1] but with another point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Configuration to blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — First we blow up three non-collinear conjugate points p1, p2, p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' This leads to a degree 6 del Pezzo surface with three exceptional conjugate curves E1, E2, E3, the other exceptional curves being the strict transforms ℓ12, ℓ13, ℓ23 of the lines joining two of the three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Then we blow up three other points p4, p5, p6 with pi+3 ≻ pi, and more precisely p4 is the intersection point of E1 and �ℓ12, p5 is the intersection point of E2 and �ℓ23 and p6 is the intersection point of E3 and �ℓ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' These points are also conjugate and the resulting surface X is a weak degree three del Pezzo surface, with three new exceptional curves E4, E5, E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The anticanonical model Xs has three conjugate singular points of type A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p4 p5 p6 p1 p2 p3 p2 = pσ 1 p5 = pσ 4 p3 = pσ2 1 p6 = pσ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The point p4 lies on the strict transform of the line (p1p2), which we denote by ℓ124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the same way we introduce the lines ℓ235 and ℓ136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the divisor class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — There are six irreducible effective roots, the strict transforms of E1, E2, E3 and the strict transforms of ℓ124, ℓ235, ℓ136;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' their classes are: R1 = E1 − E4, R2 = E2 − E5, R3 = E3 − E6, R′ 1 = E0 − E1 − E2 − E4, R′ 2 = E0 − E2 − E3 − E5, R′ 3 = E0 − E1 − E3 − E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The absolute Galois group acts on this six root classes as (R1R2R3)(R′ 1R′ 2R′ 3) and also on the exceptional curves as (E1E2E3)(E4E5E6) (the first three exceptional curves are the total transforms of the exceptional curves on the degree 6 del Pezzo surface, they are no longer irreducible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' We have Cl(X) = 6 � i=0 ZEi R = 3 � i=1 ZRi ⊕ ZR′ i and R ⊥ = ZKX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us put: E = E1 + E2 + E3, E′ = E4 + E5 + E6, R = R1 + R2 + R3 = E − E′, R′ = R′ 1 + R′ 2 + R′ 3 = 3E0 − 2E − E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' One easily verify that Cl(X)Γ = ZE0 ⊕ ZE ⊕ ZE′, R = R Γ = ZR ⊕ ZR′ = Z(E − E′) ⊕ Z(3E0 − 2E − E′), and R⊥ = ZKX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' It turns out that the submodule R is not a direct summand in Cl(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' indeed R = Z(E − E′) ⊕ Z3(E0 − E) ⊂ Z(E − E′) ⊕ Z(E0 − E) ⊕ ZE′ = Cl(X) (we have just replaced 3E0 − 2E − E′ by (3E0 − 2E − E′) − (E − E′) in the initial basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Therefore the projection onto the two last factors leads to an isomorphism: Cl(Xs) ≃ Cl(X)/R −→ Z/3Z(E0 − E) ⊕ ZE′ a0E0 + aE + a′E′ mod R �−→ (a0 mod 3) (E0 − E) + (a0 + a + a′)E′ Via this isomorphism the group CaCl(Xs) = R⊥ = ZKX embeds via −KX �→ E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' this means that CaCl(Xs) is isomorphic to the free part of Cl(Xs) and these two groups are free of rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' 31 Types of decomposition into irreducible components in |−KX|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — As in the previous case, the global sections of the divisor |−KXs| are irreducible but not necessarily absolutely irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' As usual, we list the Galois orbits of lines or conics or cubics of degree less than 3 that pass through at least one of the six points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The only possibilities are ℓ1 ∪ ℓ2 ∪ ℓ3, ℓ124 ∪ ℓ235 ∪ ℓ136, q123, c123, c123456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' (it is important to keep in mind that a curve which passes through p4 necessarily passes through p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' There are only two combinations that lead to a cubic which passes through the six points: ���3ℓ − �6 i=1 pi ��� |−KX| |−KXs| Max on P2 on X on Xs nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' of pts 1 ℓ124 ∪ ℓ235 ∪ ℓ136 �ℓ124 ∪ �ℓ235 ∪ �ℓ136 ∪ �E1 ∪ �E2 ∪ �E3 ∪ E4 ∪ E5 ∪ E6 ϕ∗(E4) ∪ ϕ∗(E5) ∪ ϕ∗(E6) 0 2 c123456 �c123456 ϕ∗(�c123456) Nq (1) The roots of X are �ℓ124, �E2 (mapped to a singular point s ∈ Xs), �ℓ235, �E3 (mapped to a singular point sσ ∈ Xs), and �ℓ136, �E1 (mapped to a singular point sσ2 ∈ Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The curves Ei, i = 4, 5, 6, are not defined over Fq and do not contain any rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In conclusion Nq (−KXs) ≤ Nq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Since the points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , p6 are not rational the blowing ups do not add any rational point, and since the singular points are not rational the contractions do not add any rational point also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus #Xs(Fq) = q2 + q + 1, this number is always strictly greater than Nq(1) and we deduce the parameters given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The weak del Pezzo surface of degree 3 associated to the configuration specified at the beginning of this section has parameters [q2 + q + 1, 4, ≥ q2 + q + 1 − Nq (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Koshelev [Kos20, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content='1] proves that the minimum distance can be improved by 1 for some q since he shows that cubics of the considered linear system must have a 3-torsion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Computation of the global sections from P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' — Let L12, L23, L13 be the three conjugate linear forms that respectively define the lines ℓ124, ℓ235, ℓ136 in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The family L12, L23, L13 is a Fq-basis of H0(P2, ℓ), while the family of degree 3 monomials in L12, L23, L13 is a Fq-basis of H0(P2, 3ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' A cubic in this space can be written: a1L3 12 + a2L3 23 + a3L3 13 + b1L12L2 23 + c1L13L2 23 + b2L12L2 13 + c2L23L2 13 + b3L13L2 12 + c3L23L2 12 + dL12L23L13 Such a cubic pass through p1 if and only if a2 = 0 (since p1 is a common zero of L12 and L13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the same way it passes through p2 and p3 if and only if a3 = 0 and a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Now passing through p4 means that if this cubic is not singular at p1 then its tangent line at this point must be ℓ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' After deshomogenizing by putting L23 = 1 (this is possible since L23 does not vanish at p1) this means that the linear component b1L12 + c1L13 should be proportional to L12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' necessarily c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' In the same way passing through p5 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' p6) means that b2 = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' c3 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Finally, one has H0 � P2, 3ℓ − �6 i=1 pi � = � L12L2 23, L23L2 13, L13L2 12, L12L23L13 � Fq In order to deduce a Fq-base, we consider θ any primitive element of Fq3 over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' The linear independence of homomorphisms permits to prove that the matrix (σi(θj))1≤i,j≤3 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Let us put: C1 = L12L2 23 + L23L2 13 + L13L2 12 Cθ = θL12L2 23 + σ(θ)L23L2 13 + σ2(θ)L13L2 12 Cθ2 = θ2L12L2 23 + σ(θ2)L23L2 13 + σ2(θ2)L13L2 12 then C, Cθ, Cθ2 are defined over Fq, as the product L12L23L13 and one has: H0 � P2, 3ℓ − �6 i=1 pi � = ⟨C1, Cθ, Cθ2, L12L23L13⟩Fq The birational morphism P2 ��� P4 (X : Y : Z) �−→ (C1 : Cθ : Cθ2 : L12L23L13) has Xs as image in P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Thus, if r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' , rq2+q+1 denote the rational points of P2, one of the generating matrix of this code is nothing else than: \uf8eb \uf8ec \uf8ec \uf8ed C1(r1) · · C1(rq2+q+1) Cθ(r1) · · Cθ(rq2+q+1) Cθ2(r1) · · Cθ2(rq2+q+1) L12L23L13(r1) · · L12L23L13(rq2+q+1) \uf8f6 \uf8f7 \uf8f7 \uf8f8 32 References [AW92] William A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Adkins and Steven H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFKT4oBgHgl3EQfei69/content/2301.11825v1.pdf'} +page_content=' Weintraub, Algebra an 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/dev/null +++ b/NNFPT4oBgHgl3EQfljVu/content/tmp_files/2301.13122v1.pdf.txt @@ -0,0 +1,908 @@ +Towards Adversarial Realism and Robust Learning for +IoT Intrusion Detection and Classification +João Vitorino[0000-0002-4968-3653], Isabel Praça[0000-0002-2519-9859] +and Eva Maia[0000-0002-8075-531X] +Research Group on Intelligent Engineering and Computing for Advanced Innovation and +Development (GECAD), School of Engineering, Polytechnic of Porto (ISEP/IPP), +4249-015 Porto, Portugal +{jpmvo,icp,egm}@isep.ipp.pt +Abstract. The Internet of Things (IoT) faces tremendous security challenges. +Machine learning models can be used to tackle the growing number of cyber- +attack variations targeting IoT systems, but the increasing threat posed by adver- +sarial attacks restates the need for reliable defense strategies. This work describes +the types of constraints required for an adversarial cyber-attack example to be +realistic and proposes a methodology for a trustworthy adversarial robustness +analysis with a realistic adversarial evasion attack vector. The proposed method- +ology was used to evaluate three supervised algorithms, Random Forest (RF), +Extreme Gradient Boosting (XGB), and Light Gradient Boosting Machine +(LGBM), and one unsupervised algorithm, Isolation Forest (IFOR). Constrained +adversarial examples were generated with the Adaptative Perturbation Pattern +Method (A2PM), and evasion attacks were performed against models created +with regular and adversarial training. Even though RF was the least affected in +binary classification, XGB consistently achieved the highest accuracy in multi- +class classification. The obtained results evidence the inherent susceptibility of +tree-based algorithms and ensembles to adversarial evasion attacks and demon- +strates the benefits of adversarial training and a security by design approach for +a more robust IoT network intrusion detection. +Keywords: adversarial realism, adversarial robustness, machine learning, tabu- +lar data, internet of things, intrusion detection +1 +Introduction +The Internet of Things (IoT) is accelerating the digital transformation. It represents de- +centralized and heterogeneous systems of interconnected devices, which combine wire- +less sensor networks, real-time computing and actuation technologies [1]. Industrial +IoT (IIoT) is a subfield focused on industrial assets and the automation of manufactur- +ing processes. Due to the integration of physical and business processes, as well as +control and information systems, IIoT is bridging the gap between Operational Tech- +nology and Information Technology [2]. However, the convergence of previously iso- +lated systems and technologies faces tremendous security challenges. The utilized + +2 Internet of Things Preprint +devices commonly have software and communication protocol vulnerabilities, in addi- +tion to weak physical security and computational resource constraints [3]. Conse- +quently, a self-propagating malware can compromise a large number of susceptible de- +vices and establish a botnet to launch a wide range of cyber-attacks [4]. +Machine Learning (ML) can be very valuable to tackle the growing number and in- +creasing sophistication of cyber-attacks targeting IoT systems, but it is susceptible to +adversarial examples: cyber-attack variations specifically crafted to exploit ML [5]. For +instance, tree-based algorithms and ensembles are remarkably well-established for net- +work intrusion detection [6], [7]. However, even though the malicious purpose of a +cyber-attack causes it to have distinct characteristics that could be recognized in a thor- +ough analysis by security practitioners, an attacker can create perturbations in IoT net- +work traffic to deceive these algorithms and be misclassified as benign. The increasing +threat posed by adversarial attacks restates the need for better defense strategies for +intelligent IoT network intrusion detection systems [8], [9]. +To ensure that ML is used in a secure way, organizations should proactively search +for vulnerabilities in their intelligent systems. By simulating realistic attack vectors, +ML engineers and security practitioners can anticipate possible threats and use that +knowledge to improve their countermeasures [10]. But throughout the current scientific +literature, various studies apply adversarial evasion attacks to intrusion detection and +provide the examples as direct input to an ML model without questioning if they are +viable for a real deployment scenario [11], which may result in misleading robustness +evaluations where a model seems to be robust because it was tested against examples +that it will not encounter in real IoT network traffic [12]. +This work addresses the challenge of improving the robustness of tree-based algo- +rithms and ensembles for IoT network intrusion detection. The main contributions are: +(i) a description of the types of constraints required for an adversarial cyber-attack ex- +ample to be realistic, (ii) a methodology for a trustworthy robustness analysis with a +realistic adversarial evasion attack vector, and (iii) an analysis of several tree-based +algorithms and ensembles in binary and multi-class classification scenarios, following +the proposed methodology. The initial evaluation carried out in [13] was extended to +include adversarial attacks performed with the Adaptative Perturbation Pattern Method +(A2PM). Three supervised algorithms, Random Forest (RF), Extreme Gradient Boost- +ing (XGB), and Light Gradient Boosting Machine (LGBM), and one unsupervised al- +gorithm, Isolation Forest (IFOR), were evaluated using the IoT-23 and Bot-IoT da- +tasets. In addition to regular training, the effectiveness of performing adversarial train- +ing with realistically perturbed samples was also analyzed. + The present paper is organized into multiple sections. Section 2 provides a survey +of previous work on ML robustness for IoT network intrusion detection. Section 3 de- +scribes the constraints required to achieve adversarial realism and defines a methodol- +ogy for a trustworthy robustness analysis. Section 4 describes the experimental evalu- +ation performed following the proposed methodology, including the scenarios, datasets, +adversarial method, models, and evaluation metrics. Section 5 presents a comparative +analysis of the results obtained by each ML model in each scenario. Finally, Section 6 +addresses the main conclusions and future research topics. + +Internet of Things Preprint 3 +2 +Related Work +In recent years, the susceptibility of tree-based algorithms to adversarial examples has +been drawing attention for network intrusion detection [14], [15]. To better protect +these ML models from adversarial attacks, several defense strategies have been devel- +oped. Some attempt to improve the intrinsic robustness of entire tree ensembles at once +[16], [17], whereas other address each individual decision tree at a time [18], [19]. +Nonetheless, the most effective and widespread defense is adversarial training because +it anticipates the data variations that an ML model may encounter [20]. Augmenting a +training set with examples created by an adversarial evasion attack method enables a +model to learn additional characteristics that the samples of each class can exhibit, so +it becomes harder for an attacker to deceive it [21]. +However, performing adversarial training with unrealistic examples will make a +model learn distorted characteristics that will not be exhibited by real samples during +its inference phase [22]. This raises a major security concern because training with un- +realistic data may not only deteriorate a model’s robustness against adversarial data, +because it will not learn the subtle nuances that an attacker can exploit, but it may also +be significantly detrimental to a model’s generalization to regular data, leading to acci- +dental data poisoning and to the introduction of hidden backdoors that make a model +even more vulnerable to attacks [23]. +Since the focus of adversarial ML has been the computer vision domain, the common +attack vector is to freely exploit the internal gradients of artificial neural networks to +generate random data perturbations in the pixels of an image [24], which can lead to +unrealistic adversarial examples in tabular data. Consequently, most state-of-the-art +evasion attack methods do not support other settings nor models that do not have loss +gradients [25], which severely limits their applicability to the IoT network intrusion +detection domain. To adversarially train a model and improve its robustness with real- +istic cyber-attack examples, a defender will need to resort to methods that support the +specificities of a communication network. +Even though most methods were intended to attack images, a few could be adapted +to tabular data. Both the Jacobian-based Saliency Map Attack (JSMA) [26] and the +OnePixel attack [27] were developed to minimize the number of modified pixels, which +could be used to solely perturb a few features in a network traffic flow. Nonetheless, +the perturbations are still randomly generated, so the resulting values for those few +features are commonly incompatible with the remaining features of a flow [28]. On the +other hand, A2PM [29] was specifically developed for communication networks, as- +signing an independent sequence of adaptative patterns to analyze the characteristics of +each class and create realistic data perturbations that preserve the purpose of a cyber- +attack. Due to its suitability for IoT network traffic, it was selected for this work. +To determine the most adequate ML models for IoT network intrusion detection, it +is important to understand the results and conclusions of previous performance evalu- +ations. A comprehensive survey [30] analyzed studies published until 2018, highlight- +ing the advantages and limitations of each model. Tree-based algorithms and ensembles +obtained good results in the reviewed performance evaluations, although their robust- +ness was not addressed. In more recent studies, the best overall performances were + +4 Internet of Things Preprint +achieved by RF in a testbed replicating an industrial plant [31], XGB with the CIDDS- +001, UNSW-NB15 and NSL-KDD datasets [32], LGBM with an IIoT dataset [33], and +IFOR in a IoT testbed for zero-day attacks [34]. Due to their promising results, RF, +XGB, LGBM and IFOR were selected for this work. +To the best of our knowledge, no previous work has analyzed the adversarial robust- +ness of these four algorithms against realistic adversarial examples of cyber-attacks +targeting IoT systems nor the effectiveness of an adversarial training approach with +realistically perturbed samples. +3 +Adversarial Realism +This section describes the types of constraints required for an adversarial cyber-attack +example to achieve realism and defines a methodology for a trustworthy adversarial +robustness analysis with a realistic evasion attack vector. +3.1 +Data Constraints +In the IoT network intrusion detection domain, cyber-attacks can be identified by ana- +lyzing the characteristics of network traffic flows, which are represented in a tabular +data format. The features of a flow may be required to follow specific data distributions, +according to the specificities of a communication network and the utilized protocols. +Furthermore, due to their distinct malicious purposes, different cyber-attacks may ex- +hibit entirely different feature correlations. Since a data sample must represent a real +traffic flow, either benign activity or a cyber-attack class, it must fulfill all the con- +straints of this complex tabular data domain. +To generate adversarial cyber-attack examples that could evade detection in a real +IoT system, the constraints must be carefully analyzed. For instance, a key characteris- +tic of an IoT network traffic flow is the Inter-Arrival Time (IAT), which represents the +elapsed time between the arrival of two subsequent packets. Its minimum (MinIAT) +and maximum (MaxIAT) values are valuable features for the detection of several cyber- +attack classes, such as Denial-of-Service (DoS). A low MinIAT can indicate a short +DoS that quickly overloads a server with requests, whereas a high MaxIAT can indicate +a lengthy DoS that overwhelms a server by keeping long connections open [35]. +When perturbing these features, validity is essential because a successful adversarial +attack is not necessarily a successful cyber-attack. If MinIAT was increased to a value +higher than MaxIAT, a flow could become an adversarial example that a model would +misclassify as benign. However, that would be an invalid network flow that a model +would never encounter in a real deployment scenario because it could not be transmitted +through a communication network. Therefore, to preserve validity within the network +traffic structure, a domain constraint must be enforced: MinIAT must not be higher than +MaxIAT. These types of constraints, including value ranges and multiple category +membership, have started being investigated in [28] to improve the feasibility of adver- +sarial attacks for network intrusion detection. + +Internet of Things Preprint 5 +Nonetheless, validity is not enough for an adversarial attack to be a successful cyber- +attack. It is imperative to also address class coherence. Even if the previous domain +constraint was fulfilled when increasing MinIAT, the resulting flow could still not be +coherent with the intended purpose of a cyber-attack class. Valid adversarial examples +with increased MinIATs could be misclassified as benign, but not be quick enough to +overload a server in a real scenario. Consequently, those supposed adversarial examples +would not actually belong to the short DoS class. Instead, they would represent just +regular traffic that would not be useful for a cyber-attack, so an ML model would be +correct to label them as benign. Therefore, to preserve coherence, it is necessary to also +enforce a class-specific constraint: MinIAT must not be higher than the highest known +value of that feature for the short DoS class. These types of constraints are based on the +idea initially introduced in [29], where data perturbations were created according to the +correlation between multiple features. +Even though validity and coherence have previously been investigated, sometimes +with different designations, it is pertinent to address them together in a single unifying +concept: adversarial realism. Hence, for an adversarial example to be realistic, it must +be valid within its domain structure and coherent with the characteristics and purposes +of its class, by simultaneously fulfilling all domain and class-specific constraints. Re- +garding cyber-attacks targeting IoT systems, realistic adversarial examples must be +valid traffic capable of being transmitted through a communication network, as well as +coherent cyber-attacks capable of fulfilling their intended malicious purpose. +3.2 +Analysis Methodology +To perform a trustworthy adversarial robustness analysis of multiple ML models, it is +imperative to carry out realistic evasion attack vectors that use valid and coherent ex- +amples. The proposed methodology is meant to enable a security by design approach +during the development of an intelligent system, and to be regularly replicated with +new data recordings to ensure that the models continue to be adversarially robust. +Considering that network intrusion detection systems are developed in a secure en- +vironment and deployed with security measures to encapsulate the utilized models, an +attacker will not likely have access neither to a model’s training set nor to its internal +parameters. Therefore, in addition to fulfilling all domain and class-specific constraints, +an adversarial attack method will have to rely solely on a model’s class predictions in +a black-box or grey-box setting, depending on the available system information about +the utilized models and feature set [36]. This attack vector can be simulated by solely +giving an evasion attack method access to a holdout set with IoT network traffic that a +model has not yet seen. The analysis can be performed in four steps: +1. Preprocess a dataset, splitting it into training and holdout sets. +2. Train and validate an ML model, using the training set. +3. Perform an evasion attack to create a model-specific adversarial holdout set, using +the regular holdout set and the model’s class predictions. +4. Evaluate the model’s performance on the regular and adversarial holdout sets, ana- +lyzing its generalization to regular data and its robustness to adversarial data. + +6 Internet of Things Preprint +In addition to a regularly trained model, an adversarial training approach can be in- +cluded to analyze the trade-off of performance on regular data to improve the perfor- +mance on adversarial data. The complete analysis can be performed in five steps: +1. Preprocess a dataset, splitting it into training and holdout sets. +2. Create a simple data perturbation in a copy of each sample of the regular training +set, creating an augmented adversarial training set with more data variations. +3. Train and validate two ML models, the first using the regular training set and the +second using the adversarial training set. +4. Perform two evasion attacks to create two model-specific adversarial holdout sets, +using the regular holdout set and each model’s class predictions. +5. Evaluate each model’s performance on the regular and adversarial holdout sets, com- +paring their generalization to regular data and their robustness to adversarial data. +From the comparison performed in the last step, the model with the most adversarially +robust generalization can be selected for deployment. Posteriority, if new data is rec- +orded, this methodology can be replicated to anticipate possible threats and use that +knowledge to improve the defense strategy (see Fig. 1). + +Fig. 1. Adversarial robustness analysis methodology. + +Regular +Regular +Adversarial +Holdout Set +Training Set +Training Set +Model 1 +Model 2 +Adversarial +Adversarial +Holdout Set 1 +Holdout Set 2Internet of Things Preprint 7 +4 +Experimental Evaluation +This section describes the experimental evaluation performed following the proposed +methodology, including the considered scenarios and datasets, and the utilized adver- +sarial method, ML models and performance evaluation metrics. The analysis was car- +ried out on a machine with 16 gigabytes of random-access memory, an 8-core central +processing unit, and a 6-gigabyte graphics processing unit. The implementation relied +on the Python 3 programming language and the following libraries: numpy and pandas +for data preparation and manipulation, scikit-learn for the implementation of RF and +IFOR, xgboost for XGB, and lightgbm for LGBM. The previously developed a2pm +library was used to perform a constrained adversarial example generation. +4.1 +Scenarios and Datasets +Two distinct scenarios were considered for IoT network intrusion detection: binary and +multi-class classification. In the former, the aim of a model was to detect that a network +traffic flow was malicious, whereas in the latter, a model had to correctly identify each +cyber-attack class and distinguish between them. +Both scenarios included the IoT-23 [37] and Bot-IoT [38] datasets. These are public +datasets that contain multiple labeled captures of benign and malicious network flows +within IoT systems. The recorded data is extremely valuable because it manifests real +IoT network traffic patterns and includes various classes of common cyber-attacks. The +former was created in the Stratosphere Research Laboratory and contains twenty-three +labeled captures of malware attacks targeting real IoT devices between 2018 and 2019. +Despite the latter also incorporating simulated devices and services, it resulted from a +realistic testbed environment of botnet activity developed at the University of New +South Wales. Table 1 provides an overview of the main characteristics of the datasets. +The class labels were either Benign or the name of a cyber-attack class, such as Dis- +tributed DoS (DDoS) and Command and Control (C&C). +Table 1. Main characteristics of utilized datasets. +Dataset +Selected +Captures +Total +Samples +Class Samples +Class Label +IoT-23 +1-1 +34-1 +1,031,893 +539,587 +POAHPS +471,198 +Benign +14,394 +DDoS +6,714 +C&C +Bot-IoT +Full5pc-4 +668,522 +576,884 +DDoS +91,082 +Recon +477 +Benign +79 +Theft +A preprocessing stage was applied to both datasets, considering their distinct charac- +teristics. First, the features that did not provide valuable information about a flow’s +benign or malicious purpose, such as origin and destination addresses, were discarded. + +8 Internet of Things Preprint +Then, one-hot encoding was employed to convert the categorical features to numeric +values. Due to their high cardinality, low frequency categories were aggregated into a +single category to avoid encoding qualitative values that had a small relevance. Finally, +the data was randomly split into training and holdout sets with 70% and 30% of the +samples, respectively. To preserve the imbalanced class proportions, the split was per- +formed with stratification. The resulting IoT-23 sets were comprised of four classes and +42 features, 8 numerical and 34 categorical. Similarly, the Bot-IoT sets contained four +classes and 35 features, 15 numerical and 20 categorical. +4.2 +Adversarial Method +The realistic data perturbations required for a trustworthy analysis were created with +A2PM [29]. It relies on sequences of adaptative patterns that learn the characteristics +of each class. The patterns record the value intervals of individual features and value +combinations of multiple features of tabular data. The learnt characteristics are then +used to generate constrained adversarial examples that are coherent with the character- +istics of their class and simultaneously remain valid within a domain. +Considering that the benign class represents regular IoT network traffic that is not +part of an attack, A2PM was applied solely to samples of cyber-attack classes. The +method was configured to use independent patterns for specific feature subsets, ac- +counting for the constraints of numerical features and the correlation between encoded +categorical features like the destination port, the communication protocol, and the con- +nection flags. Then, two different functionalities were used to perform a simple pertur- +bation and a full evasion attack. These exhibit distinct behaviors and were adapted to +different data to prevent any bias in the evaluation of the adversarially trained model. +Simple Adversarial Perturbation. The method was adapted solely to the characteris- +tics of the regular training set and then a single perturbation was created in a copy of +each malicious sample of that set. This resulted in an adversarial training set with twice +as many malicious samples as the regular set, so a model could learn not only from a +recorded cyber-attack, but also from a simple variation of it. +A security practitioner could perform these simple perturbations manually by ana- +lyzing the entire dataset and adding modified samples according to the characteristics +of each cyber-attack class. Nonetheless, the automated process of A2PM was preferred. +When compared to the training time of an ML model, the few additional seconds re- +quired to create the simple sample variations were negligible. +Realistic Adversarial Evasion Attack. The method was adapted solely to the charac- +teristics of the regular holdout set and then a full evasion attack was performed, creating +as many data perturbations as necessary in a copy of each malicious sample of that set +until every flow was misclassified or a maximum of 30 misclassification attempts were +performed. This resulted in an adversarial holdout set with the same size as the regular +set, but where each malicious sample was replaced with an adversarial example. + +Internet of Things Preprint 9 +In the multi-class scenario, the performed adversarial evasion attacks could be un- +targeted, causing any misclassification of malicious samples to different classes, as well +as targeted, seeking to misclassify malicious samples as the benign class. In turn, in the +binary scenario, both types of evasion attacks were equivalent because all cyber-attacks +were aggregated into a single class. +4.3 +Models and Fine-tuning +The RF, XGB, LGBM, and IFOR algorithms were used to create distinct models for +each dataset and scenario, which were fine-tuned through a grid search of well-estab- +lished hyperparameter combinations for cyber-attack classification. To determine the +optimal configuration for each model, a 5-fold cross-validation was performed. There- +fore, in each iteration, a model was trained with 4/5 of a training set and validated with +the remaining 1/5. The macro-averaged F1-Score was selected as the validation metric +to be maximized in both regular and adversarial training, which will be detailed in the +next subsections. After being fine-tuned, each model was retrained with a complete +training set and evaluated using the corresponding holdout set. +Random Forest. RF [39] is a supervised ensemble of decision trees, which are decision +support tools that use a tree-like structure. Each individual tree performs a prediction +according to a specific feature subset, and the most voted class is chosen. It is based on +the wisdom of the crowd, the concept that the collective decisions of multiple classifiers +will be better than the decisions of just one. +The default Gini Impurity criterion was used to measure the quality of the possible +node splits, and the maximum number of features selected to build a tree was the square +root of the total number of features of each dataset. The optimized value for the maxi- +mum depth of a tree was 16, and the minimum number of samples required to create a +leaf node was 2 and 4 for the binary and multi-class scenarios, respectively. Table 2 +summarizes the configuration. +Table 2. Summary of RF configuration. +Parameter +Value +Criterion +Gini Impurity +No. of estimators +100 +Max depth of a tree +16 +Max features +√No. of features +Min samples in a leaf +2 to 4 +Extreme Gradient Boosting. XGB [40] performs gradient boosting using a supervised +ensemble of decision trees. A level-wise growth strategy is employed to split nodes +level by level, seeking to minimize a loss function during its training. +The acknowledged Cross-Entropy loss was used for both binary and multi-class sce- +narios, and the Histogram method was selected because it computes fast histogram- + +10 Internet of Things Preprint +based approximations to choose the best splits. The key parameter of this model is the +learning rate, which controls how quickly the model adapts its weights to the training +data. It was optimized to relatively small values for each training set and scenario, rang- +ing from 0.01 to 0.2. Table 3 summarizes the configuration. +Table 3. Summary of XGB configuration. +Parameter +Value +Method +Histogram +Loss function (objective) +Cross-Entropy +No. of estimators +80 to 120 +Learning rate +0.01 to 0.2 +Max depth of a tree +8 +Min loss reduction (gamma) +0.01 +Feature subsample +0.7 to 0.8 +Light Gradient Boosting Machine. LGBM [41] also utilizes a supervised ensemble +of decision trees to perform gradient boosting. Unlike XGB, a leaf-wise strategy is em- +ployed, following a best-first approach. Hence, the leaf with the maximum loss reduc- +tion is directly split in any level. +The key advantage of this model is its ability to use Gradient-based One-Side Sam- +pling (GOSS) to build the decision trees, which is computationally lighter than previous +methods and therefore provides a faster training process. The Cross-Entropy loss was +also used, and the minimum samples required to create a leaf was optimized to 16. To +avoid fast convergences to suboptimal solutions, the learning rate was also kept at small +values for the distinct datasets and scenarios. Table 4 summarizes the configuration. +Table 4. Summary of LGBM configuration. +Parameter +Value +Method +GOSS +Loss function (objective) +Cross-Entropy +No. of estimators +80 to 120 +Learning rate +0.01 to 0.2 +Max depth of a tree +16 +Max leaves in a tree +32 +Min loss reduction (gamma) +0.01 +Min samples in a leaf +16 +Feature subsample +0.7 to 0.8 +Isolation Forest. IFOR [42] isolates anomalies through an unsupervised ensemble of +decision trees. The samples are repeatedly split by random values of random features +until outliers are segregated from normal observations. Unlike the previous algorithms, +IFOR can only perform anomaly detection with unlabeled data. Nonetheless, it can be + +Internet of Things Preprint 11 +compared to the remaining models in the binary scenario, so cross-validation was also +utilized to optimize its configuration. +This model relies on the contamination ratio of a training set, which must not exceed +50%. Hence, the number of samples intended to be anomalies must be lower than the +number of remaining samples, otherwise outliers cannot be detected. To reduce the +contamination of the training data, each cyber-attack class was randomly subsampled +with stratification. The optimized ratios of the total proportion of malicious samples +were 0.4 and 0.5 for IoT-23 and Bot-IoT, respectively. Therefore, the training data con- +tained 40% and 50% of anomalies. Table 5 summarizes the configuration. +Table 5. Summary of IFOR configuration. +Parameter +Value +Nº of estimators +100 +Contamination +0.4 to 0.5 +Max features +0.9 +Max samples +256 +4.4 +Evaluation Metrics +To analyze a model’s robustness, its performance on the regular holdout set was com- +pared to its performance on its respective adversarial holdout set. The considered eval- +uation metrics and their interpretation are briefly described below [43], [44]. +Accuracy is a standard metric for classification tasks that measures the proportion of +correctly classified samples. It uses the True Positives (TP), True Negatives (TN), False +Positives (FP) and False Negatives (FN) reported by the confusion matrix, regarding +the predicted classes. However, its bias towards the majority classes must not be disre- +garded when the minority classes are particularly relevant, which is the case of IoT +network intrusion detection. Since A2PM generated examples solely for the cyber-at- +tack classes, even if all adversarial examples evaded detection, an accuracy as high as +the proportion of benign flows could still be achieved. Therefore, to correctly exhibit +the misclassifications caused by the performed attacks, the accuracy score was calcu- +lated using the samples of all classes except benign. It can be expressed as: + +𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = +𝑇𝑃 + 𝑇𝑁 +𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 +(1) +Despite the reliability of accuracy, there are other suitable metrics. For instance, preci- +sion measures the proportion of predicted attacks that were actual attacks, which indi- +cates the relevance of a model’s predictions. On the other hand, recall, which corre- +sponds to TPR, measures the proportion of actual attacks that were correctly predicted, +reflecting a model’s ability to identify malicious flows. Another valuable metric is the +false positive rate because it measures the proportion of benign flows that was incor- +rectly predicted to be an attack, leading to false alarms. +These metrics are indirectly consolidated in the F1-Score, which calculates the har- +monic mean of precision and recall. A high F1-Score indicates that malicious flows are + +12 Internet of Things Preprint +being correctly identified and there are low false alarms. It can be macro-averaged to +give all classes the same relevance, which is well suited for imbalanced training data. +Due to the consolidation of multiple metrics, the macro-averaged F1-Score was the +preferred metric. It is mathematically defined as: + +𝑀𝑎𝑐𝑟𝑜-𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑑 𝐹1-𝑆𝑐𝑜𝑟𝑒 = 1 +𝐶 ∗ ∑ 2 ∗ 𝑃𝑖 ∗ 𝑅𝑖 +𝑃𝑖 + 𝑅𝑖 +𝐶 +𝑖=1 + +(2) +where 𝑃𝑖 and 𝑅𝑖 are the precision and recall of class 𝑖, and 𝐶 is the number of classes. +5 +Results and Discussion +This section presents the results obtained by the four tree-based algorithms in the binary +and multi-class scenarios, as well as a comparative analysis of their robustness against +adversarial network flow examples, with regular and adversarial training approaches. +5.1 +Binary Classification +In the binary scenario, the models created with regular training exhibited reasonable +performance declines on the IoT-23 dataset. Even though all four models achieved over +99% accuracy on the original holdout set, numerous misclassifications were caused by +the adversarial attacks. The lowest score on an adversarial set, 68.35%, was obtained +by XGB. In contrast, the models created with adversarial training kept significantly +higher scores. By training with one realistically generated example per malicious flow, +all models successfully learnt to detect most cyber-attack variations. IFOR stood out +for preserving the 99.98% accuracy it obtained on the original holdout set throughout +the entire attack, which highlighted its excellent generalization (see Fig. 2). + +Fig. 2. Accuracy on IoT-23 binary classification. + +100% +90% +80% +Score +70% +60% +Accuracy +50% +40% +30% +20% +10% +0% +RF +XGB +LGBM +IFOR +Attacked(withRegularTraining) +Attacked(withAdversarialTraining +Original(withRegularTraining) +Original (withAdversarial Training)Internet of Things Preprint 13 +Regarding the Bot-IoT dataset, the declines were significantly higher. The inability +of these tree-based algorithms to distinguish between the different classes evidenced +their high susceptibility to adversarial examples. The score of LGBM dropped to +26.04%, followed by IFOR, with 34.31%. Regarding the latter, it could not reach 85% +in the original holdout set, possibly due to the occurrence of overfitting. Despite some +examples still deceiving them, the models created with adversarial training were able +to learn the subtle nuances between each cyber-attack class, which mitigated the impact +of the generated examples. Apart from IFOR, the remaining models consistently +achieved scores over 97%, which indicated a good robustness (see Fig. 3). + +Fig. 3. Accuracy on Bot-IoT binary classification. +5.2 +Multi-class Classification +In the multi-class scenario, the targeted and untargeted attacks had different impacts on +a model’s performance. The former caused malicious flows to be solely predicted as +the benign class, whereas the latter caused malicious flows to be predicted as different +classes, including other cyber-attack classes. Both attacks decreased the accuracy of the +three supervised models on IoT-23, with LGBM being significantly more affected. +Nonetheless, it can be observed that its targeted accuracy, 57.78%, was significantly +higher than the untargeted, 32.11%, with more misclassifications occurring between +different cyber-attack classes. Therefore, despite LGBM being susceptible, the benign +class was more difficult to reach in multi-class intrusion detection. Even though per- +forming adversarial training further increased the high scores of XGB, it was surpassed +by RF on the targeted attack, which achieved 99.97% (see Figs. 4 and 5). + +100% +90% +80% +/Score +70% +60% +Accuracy +50% +40% +30% +20% +10% +0% +RF +XGB +LGBM +IFOR +Attacked(withRegularTraining) +Attacked(withAdversarial Training) +Original (with Regular Training) +Original (with Adversarial Training)14 Internet of Things Preprint + +Fig. 4. Untargeted accuracy on IoT-23 multi-class classification. + +Fig. 5. Targeted accuracy on IoT-23 multi-class classification. +As in the previous scenario, higher declines were exhibited for the Bot-IoT dataset. The +untargeted attacks performed by A2PM dropped the accuracy of RF and XGB to near +65%, although the targeted attacks only decreased it to 87.50% and 97.14%. Adversar- +ial training contributed to the creation of more robust models, leading to fewer incorrect +class predictions. Regarding RF, it could even preserve the 99.98% score it obtained on +the holdout set throughout the entire attack. Even though some malicious flows still +evaded detection, the robustness of both XGB and LGBM was also successfully im- +proved. Overall, the adversarial robustness of the analyzed tree-based algorithms was +significantly improved by augmenting their training data with a simple variation of each +cyber-attack (see Figs. 6 and 7). + +100% +90% +80% +/Score +70% +60% +Accuracy +50% +40% +30% +20% +10% +0% +RF +XGB +LGBM +Attacked (with Regular Training) +Attacked (withAdversarial Training) +Original (with Regular Training) +Original (withAdversarial Training)100% +90% +80% +/Score +70% +60% +Accuracy +50% +40% +30% +20% +10% +0% +RF +XGB +LGBM +Attacked (with RegularTraining) +Attacked(withAdversarial Training) +Original (with RegularTraining) +Original (withAdversarial Training)Internet of Things Preprint 15 + +Fig. 6. Untargeted accuracy on Bot-IoT multi-class classification. + +Fig. 7. Targeted accuracy on Bot-IoT multi-class classification. +6 +Conclusions +This work addressed the use of ML for IoT network intrusion detection from an adver- +sarial robustness perspective. The types of constraints required for an adversarial cyber- +attack example to be valid and coherent were described, and a methodology was pro- +posed for a trustworthy adversarial robustness analysis. The methodology was followed +to analyze the robustness of four algorithms, RF, XGB, LGBM, and IFOR, using the +IoT-23 and Bot-IoT datasets. Targeted and untargeted adversarial evasion attacks were +performed with A2PM, and both regular and adversarial training approaches were eval- +uated in binary and multi-class classification scenarios. +The models created with regular training exhibited significant performance declines, +which were more prominent on the Bot-IoT dataset. Even though RF was the least af- +fected in the binary scenario, XGB consistently achieved the highest accuracy on multi- + +100% +90% +80% +/Score +70% +60% +Accuracy +50% +40% +30% +20% +10% +0% +RF +XGB +LGBM +Attacked(withRegularTraining) +Attacked(withAdversarial Training +Original (with Regular Training) +Original (withAdversarial Training)100% +90% +80% +Score +70% +60% +Accuracy +50% +40% +30% +20% +10% +0% +RF +XGB +LGBM +Attacked (with Regular Training) +Attacked (withAdversarial Training) +Original (with Regular Training) +Original (withAdversarial Training)16 Internet of Things Preprint +class classification. Furthermore, when adversarial training was performed, all four +models successfully learnt to detect most cyber-attack variations and kept significant +higher scores when attacked. The adversarially trained IFOR and RF stood out for pre- +serving the highest accuracy throughout entire attacks, on binary IoT-23 and multi-class +Bot-IoT, respectively. Regarding LGBM, the obtained results suggest that it is highly +susceptible to adversarial examples, especially on imbalanced multi-class classifica- +tion. Nonetheless, this vulnerability can be successfully tackled by augmenting its train- +ing data with one realistic adversarial example per malicious flow. +The performed analysis evidenced the inherent susceptibility of tree-based algo- +rithms to adversarial examples and demonstrated that they can benefit from defense +strategies like adversarial training to create more robust models. In the future, it is per- +tinent to further contribute to robustness research by replicating this methodical analy- +sis with novel datasets, ML models, and evasion attack methods. As the threat of ad- +versarial attacks increases, defense strategies must be improved and a security by de- +sign approach must be followed to ensure that ML models can provide a reliable and +robust IoT network intrusion detection and classification. +Author Contributions. Conceptualization, J.V. and I.P.; methodology, J.V.; software, +J.V.; validation, E.M. and I.P.; investigation, J.V. and E.M.; writing, J.V. and E.M.; +supervision, I.P.; project administration, I.P.; funding acquisition, I.P. All authors have +read and agreed to the published version of the manuscript. +Funding. The present work has been supported by UIDP/00760/2020. +Data Availability. Publicly available datasets were analyzed in this work. The data can +be found at: IoT-23 (https://www.stratosphereips.org/datasets-iot23), Bot-IoT +(https://research.unsw.edu.au/projects/bot-iot-dataset). A publicly available method +was utilized in this work. The method can be found at: A2PM (https://github.com/vito- +rinojoao/a2pm). +Conflicts of Interest. The authors declare no conflict of interest. The funders had no +role in the design of the study; in the collection, analyses, or interpretation of data; in +the writing of the manuscript, or in the decision to publish the results. +References +1. I. Butun, P. Osterberg, and H. Song, “Security of the Internet of Things: Vulnerabilities, +Attacks, and Countermeasures,” IEEE Commun. Surv. Tutorials, vol. 22, no. 1, pp. 616– +644, 2020, doi: 10.1109/COMST.2019.2953364. +2. E. Sisinni, A. Saifullah, S. Han, U. Jennehag, and M. Gidlund, “Industrial internet of things: +Challenges, opportunities, and directions,” IEEE Trans. Ind. Informatics, vol. 14, no. 11, pp. +4724–4734, 2018, doi: 10.1109/TII.2018.2852491. +3. N. Neshenko, E. Bou-Harb, J. Crichigno, G. Kaddoum, and N. 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Kamruzzaman, “Survey of intrusion detection +systems: techniques, datasets and challenges,” Cybersecurity, vol. 2, no. 1, 2019, doi: +10.1186/s42400-019-0038-7. + + diff --git a/NNFPT4oBgHgl3EQfljVu/content/tmp_files/load_file.txt b/NNFPT4oBgHgl3EQfljVu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd6ddc648184aa3a0f09d720fa528b9b46b2cd0f --- /dev/null +++ b/NNFPT4oBgHgl3EQfljVu/content/tmp_files/load_file.txt @@ -0,0 +1,1018 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf,len=1017 +page_content='Towards Adversarial Realism and Robust Learning for IoT Intrusion Detection and Classification João Vitorino[0000-0002-4968-3653], Isabel Praça[0000-0002-2519-9859] and Eva Maia[0000-0002-8075-531X] Research Group on Intelligent Engineering and Computing for Advanced Innovation and Development (GECAD), School of Engineering, Polytechnic of Porto (ISEP/IPP), 4249-015 Porto, Portugal {jpmvo,icp,egm}@isep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='ipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='pt Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The Internet of Things (IoT) faces tremendous security challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Machine learning models can be used to tackle the growing number of cyber- attack variations targeting IoT systems, but the increasing threat posed by adver- sarial attacks restates the need for reliable defense strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' This work describes the types of constraints required for an adversarial cyber-attack example to be realistic and proposes a methodology for a trustworthy adversarial robustness analysis with a realistic adversarial evasion attack vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The proposed method- ology was used to evaluate three supervised algorithms, Random Forest (RF), Extreme Gradient Boosting (XGB), and Light Gradient Boosting Machine (LGBM), and one unsupervised algorithm, Isolation Forest (IFOR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Constrained adversarial examples were generated with the Adaptative Perturbation Pattern Method (A2PM), and evasion attacks were performed against models created with regular and adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even though RF was the least affected in binary classification, XGB consistently achieved the highest accuracy in multi- class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The obtained results evidence the inherent susceptibility of tree-based algorithms and ensembles to adversarial evasion attacks and demon- strates the benefits of adversarial training and a security by design approach for a more robust IoT network intrusion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Keywords: adversarial realism, adversarial robustness, machine learning, tabu- lar data, internet of things, intrusion detection 1 Introduction The Internet of Things (IoT) is accelerating the digital transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It represents de- centralized and heterogeneous systems of interconnected devices, which combine wire- less sensor networks, real-time computing and actuation technologies [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Industrial IoT (IIoT) is a subfield focused on industrial assets and the automation of manufactur- ing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Due to the integration of physical and business processes, as well as control and information systems, IIoT is bridging the gap between Operational Tech- nology and Information Technology [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' However, the convergence of previously iso- lated systems and technologies faces tremendous security challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The utilized 2 Internet of Things Preprint devices commonly have software and communication protocol vulnerabilities, in addi- tion to weak physical security and computational resource constraints [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Conse- quently, a self-propagating malware can compromise a large number of susceptible de- vices and establish a botnet to launch a wide range of cyber-attacks [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Machine Learning (ML) can be very valuable to tackle the growing number and in- creasing sophistication of cyber-attacks targeting IoT systems, but it is susceptible to adversarial examples: cyber-attack variations specifically crafted to exploit ML [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' For instance, tree-based algorithms and ensembles are remarkably well-established for net- work intrusion detection [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' However, even though the malicious purpose of a cyber-attack causes it to have distinct characteristics that could be recognized in a thor- ough analysis by security practitioners, an attacker can create perturbations in IoT net- work traffic to deceive these algorithms and be misclassified as benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The increasing threat posed by adversarial attacks restates the need for better defense strategies for intelligent IoT network intrusion detection systems [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To ensure that ML is used in a secure way, organizations should proactively search for vulnerabilities in their intelligent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' By simulating realistic attack vectors, ML engineers and security practitioners can anticipate possible threats and use that knowledge to improve their countermeasures [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' But throughout the current scientific literature, various studies apply adversarial evasion attacks to intrusion detection and provide the examples as direct input to an ML model without questioning if they are viable for a real deployment scenario [11], which may result in misleading robustness evaluations where a model seems to be robust because it was tested against examples that it will not encounter in real IoT network traffic [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' This work addresses the challenge of improving the robustness of tree-based algo- rithms and ensembles for IoT network intrusion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The main contributions are: (i) a description of the types of constraints required for an adversarial cyber-attack ex- ample to be realistic, (ii) a methodology for a trustworthy robustness analysis with a realistic adversarial evasion attack vector, and (iii) an analysis of several tree-based algorithms and ensembles in binary and multi-class classification scenarios, following the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The initial evaluation carried out in [13] was extended to include adversarial attacks performed with the Adaptative Perturbation Pattern Method (A2PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Three supervised algorithms, Random Forest (RF), Extreme Gradient Boost- ing (XGB), and Light Gradient Boosting Machine (LGBM), and one unsupervised al- gorithm, Isolation Forest (IFOR), were evaluated using the IoT-23 and Bot-IoT da- tasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' In addition to regular training, the effectiveness of performing adversarial train- ing with realistically perturbed samples was also analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The present paper is organized into multiple sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Section 2 provides a survey of previous work on ML robustness for IoT network intrusion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Section 3 de- scribes the constraints required to achieve adversarial realism and defines a methodol- ogy for a trustworthy robustness analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Section 4 describes the experimental evalu- ation performed following the proposed methodology, including the scenarios, datasets, adversarial method, models, and evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Section 5 presents a comparative analysis of the results obtained by each ML model in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Finally, Section 6 addresses the main conclusions and future research topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Internet of Things Preprint 3 2 Related Work In recent years, the susceptibility of tree-based algorithms to adversarial examples has been drawing attention for network intrusion detection [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To better protect these ML models from adversarial attacks, several defense strategies have been devel- oped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Some attempt to improve the intrinsic robustness of entire tree ensembles at once [16], [17], whereas other address each individual decision tree at a time [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Nonetheless, the most effective and widespread defense is adversarial training because it anticipates the data variations that an ML model may encounter [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Augmenting a training set with examples created by an adversarial evasion attack method enables a model to learn additional characteristics that the samples of each class can exhibit, so it becomes harder for an attacker to deceive it [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' However, performing adversarial training with unrealistic examples will make a model learn distorted characteristics that will not be exhibited by real samples during its inference phase [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' This raises a major security concern because training with un- realistic data may not only deteriorate a model’s robustness against adversarial data, because it will not learn the subtle nuances that an attacker can exploit, but it may also be significantly detrimental to a model’s generalization to regular data, leading to acci- dental data poisoning and to the introduction of hidden backdoors that make a model even more vulnerable to attacks [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Since the focus of adversarial ML has been the computer vision domain, the common attack vector is to freely exploit the internal gradients of artificial neural networks to generate random data perturbations in the pixels of an image [24], which can lead to unrealistic adversarial examples in tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Consequently, most state-of-the-art evasion attack methods do not support other settings nor models that do not have loss gradients [25], which severely limits their applicability to the IoT network intrusion detection domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To adversarially train a model and improve its robustness with real- istic cyber-attack examples, a defender will need to resort to methods that support the specificities of a communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even though most methods were intended to attack images, a few could be adapted to tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Both the Jacobian-based Saliency Map Attack (JSMA) [26] and the OnePixel attack [27] were developed to minimize the number of modified pixels, which could be used to solely perturb a few features in a network traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Nonetheless, the perturbations are still randomly generated, so the resulting values for those few features are commonly incompatible with the remaining features of a flow [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' On the other hand, A2PM [29] was specifically developed for communication networks, as- signing an independent sequence of adaptative patterns to analyze the characteristics of each class and create realistic data perturbations that preserve the purpose of a cyber- attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Due to its suitability for IoT network traffic, it was selected for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To determine the most adequate ML models for IoT network intrusion detection, it is important to understand the results and conclusions of previous performance evalu- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' A comprehensive survey [30] analyzed studies published until 2018, highlight- ing the advantages and limitations of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Tree-based algorithms and ensembles obtained good results in the reviewed performance evaluations, although their robust- ness was not addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' In more recent studies, the best overall performances were 4 Internet of Things Preprint achieved by RF in a testbed replicating an industrial plant [31], XGB with the CIDDS- 001, UNSW-NB15 and NSL-KDD datasets [32], LGBM with an IIoT dataset [33], and IFOR in a IoT testbed for zero-day attacks [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Due to their promising results, RF, XGB, LGBM and IFOR were selected for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To the best of our knowledge, no previous work has analyzed the adversarial robust- ness of these four algorithms against realistic adversarial examples of cyber-attacks targeting IoT systems nor the effectiveness of an adversarial training approach with realistically perturbed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 3 Adversarial Realism This section describes the types of constraints required for an adversarial cyber-attack example to achieve realism and defines a methodology for a trustworthy adversarial robustness analysis with a realistic evasion attack vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='1 Data Constraints In the IoT network intrusion detection domain, cyber-attacks can be identified by ana- lyzing the characteristics of network traffic flows, which are represented in a tabular data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The features of a flow may be required to follow specific data distributions, according to the specificities of a communication network and the utilized protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Furthermore, due to their distinct malicious purposes, different cyber-attacks may ex- hibit entirely different feature correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Since a data sample must represent a real traffic flow, either benign activity or a cyber-attack class, it must fulfill all the con- straints of this complex tabular data domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To generate adversarial cyber-attack examples that could evade detection in a real IoT system, the constraints must be carefully analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' For instance, a key characteris- tic of an IoT network traffic flow is the Inter-Arrival Time (IAT), which represents the elapsed time between the arrival of two subsequent packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Its minimum (MinIAT) and maximum (MaxIAT) values are valuable features for the detection of several cyber- attack classes, such as Denial-of-Service (DoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' A low MinIAT can indicate a short DoS that quickly overloads a server with requests, whereas a high MaxIAT can indicate a lengthy DoS that overwhelms a server by keeping long connections open [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' When perturbing these features, validity is essential because a successful adversarial attack is not necessarily a successful cyber-attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' If MinIAT was increased to a value higher than MaxIAT, a flow could become an adversarial example that a model would misclassify as benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' However, that would be an invalid network flow that a model would never encounter in a real deployment scenario because it could not be transmitted through a communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Therefore, to preserve validity within the network traffic structure, a domain constraint must be enforced: MinIAT must not be higher than MaxIAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' These types of constraints, including value ranges and multiple category membership, have started being investigated in [28] to improve the feasibility of adver- sarial attacks for network intrusion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Internet of Things Preprint 5 Nonetheless, validity is not enough for an adversarial attack to be a successful cyber- attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It is imperative to also address class coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even if the previous domain constraint was fulfilled when increasing MinIAT, the resulting flow could still not be coherent with the intended purpose of a cyber-attack class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Valid adversarial examples with increased MinIATs could be misclassified as benign, but not be quick enough to overload a server in a real scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Consequently, those supposed adversarial examples would not actually belong to the short DoS class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Instead, they would represent just regular traffic that would not be useful for a cyber-attack, so an ML model would be correct to label them as benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Therefore, to preserve coherence, it is necessary to also enforce a class-specific constraint: MinIAT must not be higher than the highest known value of that feature for the short DoS class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' These types of constraints are based on the idea initially introduced in [29], where data perturbations were created according to the correlation between multiple features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even though validity and coherence have previously been investigated, sometimes with different designations, it is pertinent to address them together in a single unifying concept: adversarial realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Hence, for an adversarial example to be realistic, it must be valid within its domain structure and coherent with the characteristics and purposes of its class, by simultaneously fulfilling all domain and class-specific constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Re- garding cyber-attacks targeting IoT systems, realistic adversarial examples must be valid traffic capable of being transmitted through a communication network, as well as coherent cyber-attacks capable of fulfilling their intended malicious purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='2 Analysis Methodology To perform a trustworthy adversarial robustness analysis of multiple ML models, it is imperative to carry out realistic evasion attack vectors that use valid and coherent ex- amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The proposed methodology is meant to enable a security by design approach during the development of an intelligent system, and to be regularly replicated with new data recordings to ensure that the models continue to be adversarially robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Considering that network intrusion detection systems are developed in a secure en- vironment and deployed with security measures to encapsulate the utilized models, an attacker will not likely have access neither to a model’s training set nor to its internal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Therefore, in addition to fulfilling all domain and class-specific constraints, an adversarial attack method will have to rely solely on a model’s class predictions in a black-box or grey-box setting, depending on the available system information about the utilized models and feature set [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' This attack vector can be simulated by solely giving an evasion attack method access to a holdout set with IoT network traffic that a model has not yet seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The analysis can be performed in four steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Preprocess a dataset, splitting it into training and holdout sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Train and validate an ML model, using the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Perform an evasion attack to create a model-specific adversarial holdout set, using the regular holdout set and the model’s class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Evaluate the model’s performance on the regular and adversarial holdout sets, ana- lyzing its generalization to regular data and its robustness to adversarial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 6 Internet of Things Preprint In addition to a regularly trained model, an adversarial training approach can be in- cluded to analyze the trade-off of performance on regular data to improve the perfor- mance on adversarial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The complete analysis can be performed in five steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Preprocess a dataset, splitting it into training and holdout sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Create a simple data perturbation in a copy of each sample of the regular training set, creating an augmented adversarial training set with more data variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Train and validate two ML models, the first using the regular training set and the second using the adversarial training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Perform two evasion attacks to create two model-specific adversarial holdout sets, using the regular holdout set and each model’s class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Evaluate each model’s performance on the regular and adversarial holdout sets, com- paring their generalization to regular data and their robustness to adversarial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' From the comparison performed in the last step, the model with the most adversarially robust generalization can be selected for deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Posteriority, if new data is rec- orded, this methodology can be replicated to anticipate possible threats and use that knowledge to improve the defense strategy (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Adversarial robustness analysis methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Regular Regular Adversarial Holdout Set Training Set Training Set Model 1 Model 2 Adversarial Adversarial Holdout Set 1 Holdout Set 2Internet of Things Preprint 7 4 Experimental Evaluation This section describes the experimental evaluation performed following the proposed methodology, including the considered scenarios and datasets, and the utilized adver- sarial method, ML models and performance evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The analysis was car- ried out on a machine with 16 gigabytes of random-access memory, an 8-core central processing unit, and a 6-gigabyte graphics processing unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The implementation relied on the Python 3 programming language and the following libraries: numpy and pandas for data preparation and manipulation, scikit-learn for the implementation of RF and IFOR, xgboost for XGB, and lightgbm for LGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The previously developed a2pm library was used to perform a constrained adversarial example generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='1 Scenarios and Datasets Two distinct scenarios were considered for IoT network intrusion detection: binary and multi-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' In the former, the aim of a model was to detect that a network traffic flow was malicious, whereas in the latter, a model had to correctly identify each cyber-attack class and distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Both scenarios included the IoT-23 [37] and Bot-IoT [38] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' These are public datasets that contain multiple labeled captures of benign and malicious network flows within IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The recorded data is extremely valuable because it manifests real IoT network traffic patterns and includes various classes of common cyber-attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The former was created in the Stratosphere Research Laboratory and contains twenty-three labeled captures of malware attacks targeting real IoT devices between 2018 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Despite the latter also incorporating simulated devices and services, it resulted from a realistic testbed environment of botnet activity developed at the University of New South Wales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 1 provides an overview of the main characteristics of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The class labels were either Benign or the name of a cyber-attack class, such as Dis- tributed DoS (DDoS) and Command and Control (C&C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Main characteristics of utilized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Dataset Selected Captures Total Samples Class Samples Class Label IoT-23 1-1 34-1 1,031,893 539,587 POAHPS 471,198 Benign 14,394 DDoS 6,714 C&C Bot-IoT Full5pc-4 668,522 576,884 DDoS 91,082 Recon 477 Benign 79 Theft A preprocessing stage was applied to both datasets, considering their distinct charac- teristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' First, the features that did not provide valuable information about a flow’s benign or malicious purpose, such as origin and destination addresses, were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 8 Internet of Things Preprint Then, one-hot encoding was employed to convert the categorical features to numeric values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Due to their high cardinality, low frequency categories were aggregated into a single category to avoid encoding qualitative values that had a small relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Finally, the data was randomly split into training and holdout sets with 70% and 30% of the samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To preserve the imbalanced class proportions, the split was per- formed with stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The resulting IoT-23 sets were comprised of four classes and 42 features, 8 numerical and 34 categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Similarly, the Bot-IoT sets contained four classes and 35 features, 15 numerical and 20 categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='2 Adversarial Method The realistic data perturbations required for a trustworthy analysis were created with A2PM [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It relies on sequences of adaptative patterns that learn the characteristics of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The patterns record the value intervals of individual features and value combinations of multiple features of tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The learnt characteristics are then used to generate constrained adversarial examples that are coherent with the character- istics of their class and simultaneously remain valid within a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Considering that the benign class represents regular IoT network traffic that is not part of an attack, A2PM was applied solely to samples of cyber-attack classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The method was configured to use independent patterns for specific feature subsets, ac- counting for the constraints of numerical features and the correlation between encoded categorical features like the destination port, the communication protocol, and the con- nection flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Then, two different functionalities were used to perform a simple pertur- bation and a full evasion attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' These exhibit distinct behaviors and were adapted to different data to prevent any bias in the evaluation of the adversarially trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Simple Adversarial Perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The method was adapted solely to the characteris- tics of the regular training set and then a single perturbation was created in a copy of each malicious sample of that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' This resulted in an adversarial training set with twice as many malicious samples as the regular set, so a model could learn not only from a recorded cyber-attack, but also from a simple variation of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' A security practitioner could perform these simple perturbations manually by ana- lyzing the entire dataset and adding modified samples according to the characteristics of each cyber-attack class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Nonetheless, the automated process of A2PM was preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' When compared to the training time of an ML model, the few additional seconds re- quired to create the simple sample variations were negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Realistic Adversarial Evasion Attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The method was adapted solely to the charac- teristics of the regular holdout set and then a full evasion attack was performed, creating as many data perturbations as necessary in a copy of each malicious sample of that set until every flow was misclassified or a maximum of 30 misclassification attempts were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' This resulted in an adversarial holdout set with the same size as the regular set, but where each malicious sample was replaced with an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Internet of Things Preprint 9 In the multi-class scenario, the performed adversarial evasion attacks could be un- targeted, causing any misclassification of malicious samples to different classes, as well as targeted, seeking to misclassify malicious samples as the benign class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' In turn, in the binary scenario, both types of evasion attacks were equivalent because all cyber-attacks were aggregated into a single class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='3 Models and Fine-tuning The RF, XGB, LGBM, and IFOR algorithms were used to create distinct models for each dataset and scenario, which were fine-tuned through a grid search of well-estab- lished hyperparameter combinations for cyber-attack classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To determine the optimal configuration for each model, a 5-fold cross-validation was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' There- fore, in each iteration, a model was trained with 4/5 of a training set and validated with the remaining 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The macro-averaged F1-Score was selected as the validation metric to be maximized in both regular and adversarial training, which will be detailed in the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' After being fine-tuned, each model was retrained with a complete training set and evaluated using the corresponding holdout set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' RF [39] is a supervised ensemble of decision trees, which are decision support tools that use a tree-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Each individual tree performs a prediction according to a specific feature subset, and the most voted class is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It is based on the wisdom of the crowd, the concept that the collective decisions of multiple classifiers will be better than the decisions of just one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The default Gini Impurity criterion was used to measure the quality of the possible node splits, and the maximum number of features selected to build a tree was the square root of the total number of features of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The optimized value for the maxi- mum depth of a tree was 16, and the minimum number of samples required to create a leaf node was 2 and 4 for the binary and multi-class scenarios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 2 summarizes the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Summary of RF configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Parameter Value Criterion Gini Impurity No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' of estimators 100 Max depth of a tree 16 Max features √No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' of features Min samples in a leaf 2 to 4 Extreme Gradient Boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' XGB [40] performs gradient boosting using a supervised ensemble of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' A level-wise growth strategy is employed to split nodes level by level, seeking to minimize a loss function during its training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The acknowledged Cross-Entropy loss was used for both binary and multi-class sce- narios, and the Histogram method was selected because it computes fast histogram- 10 Internet of Things Preprint based approximations to choose the best splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The key parameter of this model is the learning rate, which controls how quickly the model adapts its weights to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It was optimized to relatively small values for each training set and scenario, rang- ing from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 3 summarizes the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Summary of XGB configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Parameter Value Method Histogram Loss function (objective) Cross-Entropy No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' of estimators 80 to 120 Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='2 Max depth of a tree 8 Min loss reduction (gamma) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='01 Feature subsample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='7 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='8 Light Gradient Boosting Machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' LGBM [41] also utilizes a supervised ensemble of decision trees to perform gradient boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Unlike XGB, a leaf-wise strategy is em- ployed, following a best-first approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Hence, the leaf with the maximum loss reduc- tion is directly split in any level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The key advantage of this model is its ability to use Gradient-based One-Side Sam- pling (GOSS) to build the decision trees, which is computationally lighter than previous methods and therefore provides a faster training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The Cross-Entropy loss was also used, and the minimum samples required to create a leaf was optimized to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To avoid fast convergences to suboptimal solutions, the learning rate was also kept at small values for the distinct datasets and scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 4 summarizes the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Summary of LGBM configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Parameter Value Method GOSS Loss function (objective) Cross-Entropy No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' of estimators 80 to 120 Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='2 Max depth of a tree 16 Max leaves in a tree 32 Min loss reduction (gamma) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='01 Min samples in a leaf 16 Feature subsample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='7 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='8 Isolation Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' IFOR [42] isolates anomalies through an unsupervised ensemble of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The samples are repeatedly split by random values of random features until outliers are segregated from normal observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Unlike the previous algorithms, IFOR can only perform anomaly detection with unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Nonetheless, it can be Internet of Things Preprint 11 compared to the remaining models in the binary scenario, so cross-validation was also utilized to optimize its configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' This model relies on the contamination ratio of a training set, which must not exceed 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Hence, the number of samples intended to be anomalies must be lower than the number of remaining samples, otherwise outliers cannot be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' To reduce the contamination of the training data, each cyber-attack class was randomly subsampled with stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The optimized ratios of the total proportion of malicious samples were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='5 for IoT-23 and Bot-IoT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Therefore, the training data con- tained 40% and 50% of anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 5 summarizes the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Summary of IFOR configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Parameter Value Nº of estimators 100 Contamination 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='5 Max features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='9 Max samples 256 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='4 Evaluation Metrics To analyze a model’s robustness, its performance on the regular holdout set was com- pared to its performance on its respective adversarial holdout set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The considered eval- uation metrics and their interpretation are briefly described below [43], [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Accuracy is a standard metric for classification tasks that measures the proportion of correctly classified samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It uses the True Positives (TP), True Negatives (TN), False Positives (FP) and False Negatives (FN) reported by the confusion matrix, regarding the predicted classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' However, its bias towards the majority classes must not be disre- garded when the minority classes are particularly relevant, which is the case of IoT network intrusion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Since A2PM generated examples solely for the cyber-at- tack classes, even if all adversarial examples evaded detection, an accuracy as high as the proportion of benign flows could still be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Therefore, to correctly exhibit the misclassifications caused by the performed attacks, the accuracy score was calcu- lated using the samples of all classes except benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It can be expressed as: 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑇𝑃 + 𝑇𝑁 𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 (1) Despite the reliability of accuracy, there are other suitable metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' For instance, preci- sion measures the proportion of predicted attacks that were actual attacks, which indi- cates the relevance of a model’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' On the other hand, recall, which corre- sponds to TPR, measures the proportion of actual attacks that were correctly predicted, reflecting a model’s ability to identify malicious flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Another valuable metric is the false positive rate because it measures the proportion of benign flows that was incor- rectly predicted to be an attack, leading to false alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' These metrics are indirectly consolidated in the F1-Score, which calculates the har- monic mean of precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' A high F1-Score indicates that malicious flows are 12 Internet of Things Preprint being correctly identified and there are low false alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It can be macro-averaged to give all classes the same relevance, which is well suited for imbalanced training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Due to the consolidation of multiple metrics, the macro-averaged F1-Score was the preferred metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' It is mathematically defined as: 𝑀𝑎𝑐𝑟𝑜-𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑑 𝐹1-𝑆𝑐𝑜𝑟𝑒 = 1 𝐶 ∗ ∑ 2 ∗ 𝑃𝑖 ∗ 𝑅𝑖 𝑃𝑖 + 𝑅𝑖 𝐶 𝑖=1 (2) where 𝑃𝑖 and 𝑅𝑖 are the precision and recall of class 𝑖, and 𝐶 is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 5 Results and Discussion This section presents the results obtained by the four tree-based algorithms in the binary and multi-class scenarios, as well as a comparative analysis of their robustness against adversarial network flow examples, with regular and adversarial training approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='1 Binary Classification In the binary scenario, the models created with regular training exhibited reasonable performance declines on the IoT-23 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even though all four models achieved over 99% accuracy on the original holdout set, numerous misclassifications were caused by the adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The lowest score on an adversarial set, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='35%, was obtained by XGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' In contrast, the models created with adversarial training kept significantly higher scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' By training with one realistically generated example per malicious flow, all models successfully learnt to detect most cyber-attack variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' IFOR stood out for preserving the 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='98% accuracy it obtained on the original holdout set throughout the entire attack, which highlighted its excellent generalization (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Accuracy on IoT-23 binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 100% 90% 80% Score 70% 60% Accuracy 50% 40% 30% 20% 10% 0% RF XGB LGBM IFOR Attacked(withRegularTraining) Attacked(withAdversarialTraining Original(withRegularTraining) Original (withAdversarial Training)Internet of Things Preprint 13 Regarding the Bot-IoT dataset, the declines were significantly higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The inability of these tree-based algorithms to distinguish between the different classes evidenced their high susceptibility to adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The score of LGBM dropped to 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='04%, followed by IFOR, with 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='31%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Regarding the latter, it could not reach 85% in the original holdout set, possibly due to the occurrence of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Despite some examples still deceiving them, the models created with adversarial training were able to learn the subtle nuances between each cyber-attack class, which mitigated the impact of the generated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Apart from IFOR, the remaining models consistently achieved scores over 97%, which indicated a good robustness (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Accuracy on Bot-IoT binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='2 Multi-class Classification In the multi-class scenario, the targeted and untargeted attacks had different impacts on a model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The former caused malicious flows to be solely predicted as the benign class, whereas the latter caused malicious flows to be predicted as different classes, including other cyber-attack classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Both attacks decreased the accuracy of the three supervised models on IoT-23, with LGBM being significantly more affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Nonetheless, it can be observed that its targeted accuracy, 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='78%, was significantly higher than the untargeted, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='11%, with more misclassifications occurring between different cyber-attack classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Therefore, despite LGBM being susceptible, the benign class was more difficult to reach in multi-class intrusion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even though per- forming adversarial training further increased the high scores of XGB, it was surpassed by RF on the targeted attack, which achieved 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='97% (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 100% 90% 80% /Score 70% 60% Accuracy 50% 40% 30% 20% 10% 0% RF XGB LGBM IFOR Attacked(withRegularTraining) Attacked(withAdversarial Training) Original (with Regular Training) Original (with Adversarial Training)14 Internet of Things Preprint Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Untargeted accuracy on IoT-23 multi-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Targeted accuracy on IoT-23 multi-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' As in the previous scenario, higher declines were exhibited for the Bot-IoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The untargeted attacks performed by A2PM dropped the accuracy of RF and XGB to near 65%, although the targeted attacks only decreased it to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='50% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='14%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Adversar- ial training contributed to the creation of more robust models, leading to fewer incorrect class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Regarding RF, it could even preserve the 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='98% score it obtained on the holdout set throughout the entire attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even though some malicious flows still evaded detection, the robustness of both XGB and LGBM was also successfully im- proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Overall, the adversarial robustness of the analyzed tree-based algorithms was significantly improved by augmenting their training data with a simple variation of each cyber-attack (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='90% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='/Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='70% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='RF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='XGB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='LGBM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked (with Regular Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked (withAdversarial Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (with Regular Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (withAdversarial Training)100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='90% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='/Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='70% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='RF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='XGB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='LGBM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked (with RegularTraining) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked(withAdversarial Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (with RegularTraining) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (withAdversarial Training)Internet of Things Preprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Untargeted accuracy on Bot-IoT multi-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Targeted accuracy on Bot-IoT multi-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 6 Conclusions This work addressed the use of ML for IoT network intrusion detection from an adver- sarial robustness perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The types of constraints required for an adversarial cyber- attack example to be valid and coherent were described, and a methodology was pro- posed for a trustworthy adversarial robustness analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The methodology was followed to analyze the robustness of four algorithms, RF, XGB, LGBM, and IFOR, using the IoT-23 and Bot-IoT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Targeted and untargeted adversarial evasion attacks were performed with A2PM, and both regular and adversarial training approaches were eval- uated in binary and multi-class classification scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The models created with regular training exhibited significant performance declines, which were more prominent on the Bot-IoT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Even though RF was the least af- fected in the binary scenario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' XGB consistently achieved the highest accuracy on multi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='90% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='/Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='70% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='RF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='XGB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='LGBM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked(withRegularTraining) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked(withAdversarial Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (with Regular Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (withAdversarial Training)100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='90% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='70% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='RF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='XGB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='LGBM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked (with Regular Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Attacked (withAdversarial Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (with Regular Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Original (withAdversarial Training)16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='Internet of Things Preprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Furthermore, when adversarial training was performed, all four models successfully learnt to detect most cyber-attack variations and kept significant higher scores when attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The adversarially trained IFOR and RF stood out for pre- serving the highest accuracy throughout entire attacks, on binary IoT-23 and multi-class Bot-IoT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Regarding LGBM, the obtained results suggest that it is highly susceptible to adversarial examples, especially on imbalanced multi-class classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Nonetheless, this vulnerability can be successfully tackled by augmenting its train- ing data with one realistic adversarial example per malicious flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The performed analysis evidenced the inherent susceptibility of tree-based algo- rithms to adversarial examples and demonstrated that they can benefit from defense strategies like adversarial training to create more robust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' In the future, it is per- tinent to further contribute to robustness research by replicating this methodical analy- sis with novel datasets, ML models, and evasion attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' As the threat of ad- versarial attacks increases, defense strategies must be improved and a security by de- sign approach must be followed to ensure that ML models can provide a reliable and robust IoT network intrusion detection and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Author Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Conceptualization, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' methodology, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' software, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' validation, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' investigation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' writing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' supervision, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' project administration, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' funding acquisition, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The present work has been supported by UIDP/00760/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Data Availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Publicly available datasets were analyzed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The data can be found at: IoT-23 (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='stratosphereips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='org/datasets-iot23), Bot-IoT (https://research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='unsw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='au/projects/bot-iot-dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' A publicly available method was utilized in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The method can be found at: A2PM (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='com/vito- rinojoao/a2pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Conflicts of Interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' The funders had no 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Martins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFPT4oBgHgl3EQfljVu/content/2301.13122v1.pdf'} +page_content=' Cruz, T.' metadata={'source': 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a/OtE0T4oBgHgl3EQfTgB4/content/tmp_files/2301.02237v1.pdf.txt b/OtE0T4oBgHgl3EQfTgB4/content/tmp_files/2301.02237v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca4b52372cfe9e33a530089573c2c4ef718d0725 --- /dev/null +++ b/OtE0T4oBgHgl3EQfTgB4/content/tmp_files/2301.02237v1.pdf.txt @@ -0,0 +1,983 @@ +Observation of Anomalous Decay of a Polarized Three-Component Fermi Gas +Grant L. Schumacher,1, ∗ Jere T. M¨akinen,1, 2 Yunpeng Ji,1 Gabriel G. T. +Assumpc¸˜ao,1 Jianyi Chen,1 Songtao Huang,1 Franklin J. Vivanco,1 and Nir Navon1, 2 +1Department of Physics, Yale University, New Haven, Connecticut 06520, USA +2Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA +Systems of fermions with multiple internal states, such as quarks in quantum chromodynamics and nucleons +in nuclear matter, are at the heart of some of the most complex quantum many-body problems. The stability of +such many-body multi-component systems is crucial to understanding, for instance, baryon formation and the +structure of nuclei, but these fermionic problems are typically very challenging to tackle theoretically. Versatile +experimental platforms on which to study analogous problems are thus sought after. Here, we report the cre- +ation of a uniform gas of three-component fermions. We characterize the decay of this system across a range +of interaction strengths and observe nontrivial competition between two- and three-body loss processes. We +observe anomalous decay of the polarized (i.e. spin-population imbalanced) gas, in which the loss rates of each +component unexpectedly differ. We introduce a generalized three-body rate equation which captures the decay +dynamics, but the underlying microscopic mechanism is unknown. +Characterizing if - and into what - a system decays has of- +ten been a pathway to discovering new quantum phenomena. +Such stability problems are ubiquitous in many-body physics, +ranging from the onset of turbulence in quantum liquids [1] +to dissipation processes in Josephson junctions [2], to the sta- +bility of nuclei [3]. Ultracold atomic systems have been a +fertile ground for such studies: their (in)stability has revealed +a wide range of interesting and sometimes unexpected phe- +nomena, such as Efimov three-body physics [4], quantum- +fluctuation stabilization against many-body collapse [5], and +prethermalization of metastable phases [6]. In particular, there +has been renewed appreciation that the stability of quantum +gases against inelastic-collision losses [7, 8] (or lack thereof) +is a pristine probe of many-body quantum correlations [9–12]. +Strongly interacting fermions with contact interactions em- +body a stability success story, especially the case of two- +component (‘spin-1/2’) fermions. The observation of unex- +pected shifts of loss features with respect to scattering - Fes- +hbach - resonances [13–17] led to the understanding that sur- +prisingly stable pairs were forming near those resonances, +protected by a Pauli blocking effect. This discovery unlocked +the field of the BEC-BCS crossover [18]. +Three-component (‘spin-1’) fermions give access to even +richer physics, such as quantum chromodynamics-like phe- +nomena and complex pairing patterns [19–25]. Pioneering ex- +periments showed that unpolarized (i.e. spin-population bal- +anced) gases of three-component fermions exhibit interesting +decay behavior related to the Efimov effect [26–30]. How- +ever, density-dependent losses in those spatially inhomoge- +neous gases were unavoidably coupled to particle transport +and heating, making the interpretation of those experiments +challenging. The metastability of the three-component Fermi +gas has yet to be comprehensively understood. +In this work, we create box-trapped, uniform gases of three- +component fermions with controllable spin-population imbal- +∗ Corresponding author. E-mail: grant.schumacher@yale.edu +ance and study their stability (Fig. 1A). Such spatially homo- +geneous gases generally obey simpler dynamics than their in- +homogeneous counterparts (see e.g. [10, 31–33]). This has +enabled us to observe a surprising violation of the generic ex- +pectation that loss rates among spin components should be +equal in a three-body process involving all three components. +We rule out two credible explanations for this effect, indicat- +ing that unexpected physics is at play. +Our experiment begins with a gas of 6Li atoms in a red- +detuned crossed optical dipole trap. The gas is prepared in +an incoherent mixture of two of the three lowest Zeeman sub- +levels, which encode the three (pseudo-)spin components of +our ‘spin-1’ fermions (respectively labelled |1⟩, |2⟩, and |3⟩, +see top panel of Fig. 1B). This two-component mixture is +evaporatively cooled at a bias magnetic field B0 (which de- +pends on the two states used) and then loaded into a blue- +detuned optical box trap of wavelength 639 nm. +We ramp the magnetic field to the field of interest B in +200 ms, which is slow compared to the two-body collision +rate but fast compared to the lifetime of the two-component +mixtures in the range of fields investigated [35]. We let the +magnetic field settle for 100 ms and then prepare the spin mix- +tures by sequentially applying variable radio-frequency (RF) +pulses to drive the |1⟩−|2⟩ and |2⟩−|3⟩ transitions (top panel +of Fig. 1B). For the range of B explored in this work (deep +in the Paschen-Back regime for 6Li), all three spins are nearly +identically levitated against gravity using a magnetic field gra- +dient [36]. The fermions interact via binary contact interac- +tions, characterized by an s-wave scattering length a for each +pair of spin states at a given magnetic field; each pair has a +broad Feshbach resonance (see bottom panel of Fig. 1B). +We hold the gas for a variable time t before measuring the +number of atoms of spin |j⟩, Nj. Typical in-situ absorption +images of each spin population in an imbalanced mixture are +shown in Fig. 1C (top), together with integrated column densi- +ties along the two directions x and y (bottom). The integrated +density profiles of the three components are consistent with +uniform densities. +We first study the stability of spin-balanced samples, N1 ≈ +arXiv:2301.02237v1 [cond-mat.quant-gas] 5 Jan 2023 + +2 +OD +OD +OD +-50 +0 +50 +-50 +0 +50 +~ +RF +~ +700 +800 +900 +-2 +-1 +0 +1 +2 +FIG. 1. Preparation of a homogeneous three-component Fermi gas. (A) Top: Sketch of the optical box. Bottom: The stability of the mixture +is studied by measuring the density of each spin population over time. (B) Top: Breit-Rabi diagram of the lowest hyperfine states of 6Li. The +three (pseudo)-spins are encoded in the three lowest states, respectively |1⟩, |2⟩ and |3⟩. The polarization of the three component mixture is +controlled via radio-frequency (RF) pulses. Bottom: Scattering length a for each pair of spin states in units of 104a0, where a0 is the Bohr +radius (adjacent colors match the corresponding pair of states). Vertical dashed lines show the locations of Feshbach resonances [34]. (C) Top: +In-situ absorption images of a typical polarized three-component mixture (averaged over 10 realizations), taken along the z axis; the color scale +corresponds to the optical density (OD). The length and radii of this conical box are L = 120(2) µm, R1 = 75(1) µm and R2 = 73(1) µm, +and its trap depth is Ubox = kB× 1.6(2) µK, where kB is Boltzmann’s constant. Note that boxes of various sizes were used in this work. +Bottom: Integrated density along the x and y axes for each component with fits to homogeneous density profiles (dotted lines) [35]. +N2 ≈ N3 as a function of magnetic field B. We typically +evaporate a balanced mixture of |1⟩-|3⟩ near its Feshbach res- +onance at B0 ≈ 690 G, ending with typically N1 ≈ N3 ≈ +5 × 105 at T/TF ≈ 0.25 (where TF ≈ 400 nK is the Fermi +temperature) prior to ramping the field and creating the three- +component mixture. In Fig. 2A-B, we show two typical de- +cays, from which we make two key observations: (i) the mix- +tures remain unpolarized during decay [37], and (ii) these +losses involve all three states, as all two-component mixtures +are much more stable [35]. +The density uniformity of our samples enables us to model- +independently characterize the three-component gas decay +dynamics. In the insets of Fig. 2A-B, we plot the total atom +loss rate ˙N ≡ dN/dt as a function of the total atom number +N ≡ N1+N2+N3. For the magnetic fields explored, the data +is well captured by a power law ˙N ∝ −N γ, where the fitted +γ is displayed in Fig. 2C. For a homogeneous unpolarized gas +with a total density n = N/V and a constant volume V , this +implies that ˙n ∝ −nγ. Importantly, the exponent γ encodes +information on the number of independent particles involved +in the loss events [38]. +For both B ≲ 720 G and B ≳ 810 G, we observe +γ ≈ 3. This is consistent with losses dominated by energy- +independent recombination involving three distinguishable +fermions (for which we expect ˙n ∝ −n3). +The intermediate range B ≈ 720 − 810 G is more com- +plicated, and γ deviates from 3. In that region, the scatter- +ing lengths a12 and a23, respectively between states |1⟩ − |2⟩ +and |2⟩ − |3⟩, are large and positive, so that the products of +three-body recombination can remain trapped [32, 39]. This +allows for subsequent inelastic atom-dimer collisions (involv- +ing three distinguishable atoms), an effectively two-body loss +process [40–42]. Our intermediate 2 ≲ γ ≲ 3 is qualita- +tively consistent with this competition of two- and three-body +effects. +Focusing on the regions dominated by three-body recombi- +nation, we now explore polarized three-component mixtures, +i.e. with spin-population imbalance. +In Fig. 3A, we show a typical decay of a polarized sample at +a field B = 845 G, above all three broad Feshbach resonances +(Fig. 1B). The losses ∆nj ≡ nj(t) − ninitial +j +are equal for all +spins (bottom panel of Fig. 3A). This is unsurprising since we +expect the loss rate equation to take the form +˙nj = −L3n1n2n3, +(1) +where L3 is the (density-independent) three-body recombina- +tion loss coefficient [43]. The data at 845 G agrees very well +with the numerical solution of Eq. (1), shown as solid lines +in Fig. 3A; the initial densities nj are the only fit parameters, +L3 being fixed to the value we measure from the decay of the +unpolarized gas (see the caption of Fig. 3). +Surprisingly, the same measurement at the Feshbach res- +onance of the |1⟩ − |3⟩ states yields a qualitatively different +outcome: the loss rates per spin are distinct, with the majority +component showing larger absolute losses (Fig. 3B). +Nonetheless, we observe that the total loss rate still obeys +the expected three-body rate equation ˙n = −3L3n1n2n3 + +?3 +0 +50 +100 +150 +200 +250 +300 +0 +1 +2 +3 +1 +5 +106 +107 +0 +10 +20 +30 +40 +50 +0 +1 +2 +3 +1 +5 +106 +107 +108 +700 +750 +800 +850 +2 +3 +FIG. 2. Stability of the unpolarized three-component Fermi gas. (A- +B) Typical decay at the |1⟩-|3⟩ resonance (A) and at B = 768 G (B). +The red line is a fit to Eq. (1). Early times (open symbols) are ex- +cluded from our analysis, see [35]. Insets: Total loss rate − ˙N versus +N. The solid blue lines are power-law fits yielding γ. (C) Expo- +nent γ versus magnetic field B. The Feshbach resonances are shown +as dot-dashed gray lines. For B larger than the red dotted line, the +|12⟩ and |23⟩ Feshbach dimers can remain trapped (see text). Error +bars on γ displayed are only statistical (estimated by bootstrapping). +For most points, the dominant source of uncertainty is an additional +10% systematic error coming from box volume calibration [35]. The +uncertainty in B is smaller than the point size. +(orange points in Fig. 3C). What’s more, the extracted L3 +matches the unpolarized gas value (blue points). +We systematically extract L3 at 690 G as a function of po- +larization, see Fig. 3D. Unlike its spin-1/2 counterpart [45– +47], the polarization of the three-component gas is character- +ized by two parameters. The spin fractions xj ≡ nj/n pro- +vide a convenient parameterization, so that the polarization +state can be represented on a ternary plot. We find that L3 +agrees with the unpolarized-gas value for nearly all polariza- +tions, provided the spin fraction of |2⟩ is not very high [35]. +With the total loss scaling established, we now investigate +how losses are apportioned among the spins. As shown in +Fig. 4A, we see that ˙nj appear proportional to their respective +spin fractions xj. To characterize this relation, we introduce +the relative loss yj ≡ ˙nj/ ˙n and compare it directly to the xj, +as shown in Fig. 4B. +This measurement suggests a simple relation: +yj = ηxj + 1 − η +3 +, +(2) +where the proportionality of the losses is captured by the +asymmetry η. The constant (1 − η)/3 is constrained by the +definition of the parameters xj and yj; η = 0 corresponds to +symmetric losses as in Eq. (1). +We now explore η at 690 G as a function of polarization. +Following the procedure outlined in Fig. 4A-B, the ternary +plot is populated with the values of η extracted from each time +series. +We observe that the polarization space is separated into two +regions. In the (larger) region where |1⟩ or |3⟩ is the largest +component (unhatched region in Fig. 4C), η is robust to spin +composition, regardless of whether |2⟩ is the median or the +minority component. +In the (smaller) region where |2⟩ is +the largest component (hatched in Fig. 4C), η is smaller, al- +though distinctly non-zero. Note that exchanging spins |1⟩ +and |3⟩ leaves Fig. 4C essentially unchanged, which reflects +the approximate symmetry of the three scattering lengths (see +Fig. 1B). +Gathering the data from the larger (unhatched) region, we +observe that the relative loss law appears indeed to be univer- +sal over a large range of spin fractions. Fitting the data, we +find η = 0.80(3) (see caption of Fig. 4D). +On the other hand, applying the same procedure at B = +845 G we find η = 0.02(3) (Fig. 4D), characteristic of the +usual three-body loss dynamics of Eq. (1). +The behavior of both the total and relative losses suggests +that the three-component mixture at 690 G obeys the follow- +ing decay dynamics: +˙nj = −3L3 +� +ηxj + 1 − η +3 +� +n1n2n3. +(3) +In Fig. 3B, we show as solid lines the solution of Eq. (3) with +fixed L3 and η, and only the initial nj as adjustable parame- +ters; the agreement with the experimental data is excellent. +Gathering decay measurements across a large density +range, we observe that η is weakly density dependent +(Fig. 4E), although this variation is slow enough so that it is +negligible over our typical decay series. +We now turn to hypotheses for explaining this anomalous +loss asymmetry. First, note that the rate equation Eq. (1) is +an approximation; more fundamentally, losses are expressed + +4 +-1 +0 +1 +FIG. 3. Stability of the polarized three-component Fermi gas. (A) Decay of a polarized gas at 845 G. Solid curves are fits to Eq. (1), with +L3 fixed to the value measured in the unpolarized decay: L3(B = 845 G) = 2.4(1) × 10−20 cm6/s (t = 0 corresponds to the end of +the RF pulses.) (B) Decay of a polarized gas at the |1⟩-|3⟩ resonance. Solid curve are fits to Eq. (3), with early time points excluded (open +symbols) [35]. L3 is fixed to the value measured in the unpolarized decay: L0 +3 = 1.1(1) × 10−21 cm6/s. We estimate additional systematic +uncertainties on measurements of L3 of 20% due to box volume calibration [35]. (C) Total loss rate ˙n versus n1n2n3, for both unpolarized +(blue) and polarized (orange) gases at 690 G, averaged over many experiments. Colored bands are fits (with uncertainties) to the three-body +loss equation for ˙n, from which L3 is extracted. (D) Three-body loss coefficient L3 at 690 G as a function of polarization, relative to L0 +3, +plotted on a barycentric ternary plot [44]. +as correlators of quantum fields [7]. +Calculating ˙nj for a +generic model of a gas on a lattice in the Lindblad formalism +yields [35, 48] +˙nj ∝ − +� +α +� +a† +3,αa† +2,αa† +1,αa1,αa2,αa3,α +� +(4) +where aj,α (resp. a† +j,α) is the annihilation (resp. creation) op- +erator for state |j⟩ at lattice site α, under the assumptions that +no spin-changing collisions occur and that losses are due to +local three-body recombination. As the right-hand side does +not depend on the spin state, η = 0 within this model. This +conclusion holds even in the presence of nontrivial correla- +tions that would preclude factoring Eq. (4) into a product of +densities. +Alternatively, this effect could originate from many-particle +scattering processes. The form of Eq. (3) is indeed quali- +tatively reminiscent of so-called avalanche mechanisms, in +which secondary collisions occur between the products of +three-body recombination and other atoms in the gas, result- +ing in excess losses that augment the loss rate prefactor with +spin-fraction-dependent terms. Avalanche losses have been a +debated topic in ultracold few-body dynamics [49–54]. +Under an avalanche process, one would expect that the +number of excess particles of spin |j⟩ lost should be propor- +tional to the fraction of atoms in that spin state, i.e. equal to +cxj. The prefactor c should be independent of the spin state +to ensure that an unpolarized gas remains unpolarized during +decay; c is then the average number of excess particles per +recombination event. In that case, ˙n ∝ −(3 + c)n1n2n3. +In this model, η = c/(3+c) [35], and our largest measured +η would imply c ≳ 12. However, a generic kinetic model [50] +indicates that c ≈ 4 given the binding energies of the |12⟩ +and |23⟩ Feshbach dimers. Furthermore, repeating this ex- +periment in a box whose size is about the mean free path or +smaller, we find the same asymmetry [35]. This further rules +out an avalanche scenario. Finally, evaporation in a collision- +ally dense gas could give rise to asymmetric losses, as sug- +gested in [55], but is also essentially ruled out as varying the +box depth does not affect the loss rate [35]. +Future work should elucidate this puzzle, for example by +measuring the energy dynamics and temperature dependence +of this process, as well as by characterizing the asymme- +try across interaction regimes. This work also opens several +new avenues, such as studying prethermalization dynamics +of the metastable mixture [6, 56] and searching for signa- +tures of three-component pairing correlations at low tempera- +ture [19, 20]. +We thank Chris Greene, Jose d’Incao, Leonid Glazman, +Steve Girvin, Carlos S´a de Melo, Alexander Schuckert, and +Francesca Ferlaino for helpful discussion. We thank Yvan +Castin, F´elix Werner, Fr´ed´eric Chevy, and Zoran Hadzibabic +for critical comments on the manuscript. This work was sup- +ported by the NSF (Grant Nos. +PHY-1945324 and PHY- +2110303), DARPA (Grant No. W911NF2010090), the David +and Lucile Packard Foundation, and the Alfred P. Sloan Foun- +dation. J.T.M. acknowledges support from the Yale Quantum +Institute. G.L.S acknowledges support from the NSF Gradu- +ate Research Fellowship Program. + +5 +0 +0.025 +0.05 +0.075 +0.1 +2 +4 +6 +0 +0.6 +0 +25 +50 +75 +100 +1 +10 +0 +0.6 +0 +0.6 +0 +0.2 +0.4 +0.6 +0.8 +1 +FIG. 4. Asymmetric losses in a polarized three-component Fermi gas. (A) Top: Sketch of the extraction of spin fractions and relative losses +(xj, yj) from the nj(t) series. Inset: xj versus time (see also [35]). Bottom: ˙nj versus time. (B) The pairs (xj, yj) extracted from the bottom +of panel A lie on a line, whose slope defines η. (C) η at 690 G as a function of polarization on a ternary plot. Each point is obtained by +averaging over a single decay, as shown in B. The hatching marks two distinct regions in polarization space (see also [35]). (D) Top: (x, y) +from the large (unhatched) region of the ternary plot. The green solid line is a linear fit, η = 0.80(3). The gray dash-dotted line shows η = 0. +Bottom: The same procedure at 845 G yields η = 0.02(3). We estimate an additional 5% systematic error on η [35]. (E) η versus total density +n at 690 G. The points are obtained by gathering data from several decays into density bins, then extracting η (as in D). 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CHARACTERIZING OPTICAL BOXES +We determine the volume of the optical box of Fig. 1 of the main text by fitting the in-situ density profiles +� +dz nj(x, y, z) +(where z is the imaging line of sight, Fig. S1A) with a conical model of radii R1 and R2, and length L, convolved with a +Gaussian function of standard deviations σx and σy (that capture imperfections from finite resolution of the imaging and box +projection systems). We find R1 = 75(1) µm, R2 = 73(1) µm, L = 120(2) µm. The values of σx = 2.4 µm and σy = 3.8 µm +reflect the slightly different quality of projection of the ‘end caps’ and the ‘tube’ of the box. In Fig. S1B, we display density +profile cuts of the box of Fig. 1C projected along the x and y axes together with the fits (dashed purple lines). +If imperfections were entirely due to the box projection system - a conservative assumption - we estimate that ≈ 90% of +the atoms are within 10% of the average density in the box. Additionally, we observe during a typical decay that the volume +decreases by ≈ 10% when the atom number decreases by a factor of ≈ 5; taking this small effect into account as an uncertainty +on the volume, we estimate an uncertainty of 10% on γ and of 20% on L3. This also contributes to uncertainties on xj, yj of +≈ 1% and ≈ 5% respectively (reflecting the difference between fitting N and ˙N rather than n and ˙n). Ultimately we find that +the uncertainty in η due to this effect is ≈ 5%. +In Fig. S1C, we compare the density profiles of the three spin states by scaling and subtracting the profile of |3⟩ from the +others. We see that the density profiles are essentially identical up to rescaling. +OD +0 +0.5 +1 +1.5 +OD +-0.2 +0 +0.2 +OD +-0.1 +0 +0.1 +FIG. S1. Homogeneity of a polarized three-component mixture in the optical box. (A) Example of an in-situ optical density image of spin |3⟩ +(same data as Fig. 1C, with x1 = 0.16, x2 = 0.35, and x3 = 0.49). (B) Top: Cuts of the density profile (yellow, solid) and the fit (purple, +dashed) along the y (top) and x (bottom) axes (dashed black lines in A). The fit to the OD image is a smoothed cone (see text). Residuals +(normalized to the peak fitted density along each cut) are shown underneath. (C) Top: Subtracting the measured in-situ density of |3⟩ from |2⟩ +(after scaling |2⟩ to have the same mean density as |3⟩). OD scale is the same as A and the dashed red rectangle indicates the region of the +box. Bottom: Likewise subtracting spin |3⟩ from |1⟩. + +8 +II. TWO- VERSUS THREE-COMPONENT MIXTURES DECAY +We show in this section that in the range of B explored in this work, unpolarized three-component mixtures decay much faster +than any two-component ones. In Fig. S2A, we show examples of decays of two-component mixtures (which may include evap- +orative losses in addition to intrinsic recombination processes). We compare two- and three-component decays quantitatively +in Fig. S2B by extracting an (early-time) effective timescale for losses τeff = n/| ˙n|. The black crosses correspond to three- +component mixtures, with a density per spin state in the range nj ≈ 1.5 − 4 × 1011 cm−3. In green, blue and pink, we show +τeff for the two-component mixtures; for each field, the density per spin state is up to a factor of 4 larger than the corresponding +three-component data, but never smaller. Even in that case, τeff is at least an order of magnitude larger than the corresponding +three-component one. +However, this will no longer hold for extremely polarized three-component gases. At B = 690 G, both a12 and a23 are positive +so that the corresponding two-component mixtures are already unstable with respect to three-body recombination (involving two +identical fermions). Furthermore, the corresponding binding energies of the |12⟩ and |23⟩ dimers are such that all recombination +products escape from the box (see section VII). Thus, a mixture of spins |i⟩ − |j⟩ would decay following a recombination law +˙nj = −Lij +3 (2n2 +jni + n2 +i nj)/3 [32, 39]. We note that the two-component recombination rates agree with the universal prediction +of [39] (solid lines of Fig. S2B) at fields below 720 G (red dotted line of Fig. S2B). At higher magnetic fields, the formation of +|12⟩ and |23⟩ dimers does not directly lead to losses, as their binding energies are less than 6Ubox [32]. +To estimate the influence of two-component losses on the main text’s analysis, we calculate the ratio of the total losses due to +three-component recombination ( ˙nthree ≡ −3L3n1n2n3) to two-component ones ( ˙ntwo ≡ −L12 +3 (n2 +1n2 + n2 +2n1) − L23 +3 (n2 +2n3 + +n2 +3n2)): +˙ntwo +˙nthree += L12 +3 +3L3 +(1/x3 − 1) + L23 +3 +3L3 +(1/x1 − 1) +(S1) +Extracting L12 +3 +and L23 +3 +from Fig. S2A, we find ratios L12 +3 /(3L3) < 0.003 and L23 +3 /(3L3) < 0.001. Setting a limit of +˙ntwo/ ˙nthree < 10%, we find the polarization limit depicted in Fig. S2C. For mixtures less polarized than this limit, three- +component losses are dominating, and we ignore losses arising from two-component processes. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +2 +4 +6 +0 +0.5 +1 +1.5 +2 +2.5 +FIG. S2. Stability of two- versus three-component mixtures. (A) Decay of unpolarized two-component gases at 690 G; a three-component +gas decay is shown as black crosses for reference. (B) Effective timescales across magnetic fields. Feshbach resonances are indicated by +vertical dash-dotted lines. At each field shown, the two-component gases have densities (per spin state) equal to or greater than those of the +corresponding three-component gas. Solid pink and blue curves show predictions from universal recombination to Feshbach dimers [32, 39]. +For B larger than the red dotted line, the |12⟩ and |23⟩ Feshbach dimers can remain trapped. (C) L3 versus polarization at 690 G (normalized +to L0 +3 as in the main text), includes additional data at lower total density (n ∼ 5 × 1010 cm−3) not shown in main text. Dashed red curve gives +polarization outside which two-component processes are expected to account for 10% of total losses or more. + +9 +III. EARLY-TIME DECAY +In the main text, we excluded early times from the analysis of decays at B = 690 G. Here we show in more detail that the +early-time behavior at that field is distinct from behavior at later times. +In Fig. S3A, we show a decay of an unpolarized sample at 690 G. We see that for t ≲ 10 ms, the decay is slower than at later +times (see red solid line in Fig. S3). This effect is even more obvious for ˙n versus n (inset of Fig. S3A). Only at later times the +data follows a γ ≈ 3 law. This effect is also present in polarized-gas measurements, but for t ≳ 15 ms the data is well described +by a single γ ≈ 3 for all our polarizations. By contrast, for the faster losses at B ≳ 845 G (see Fig. S3B), the data follows a +simple γ ≈ 3 law, even from the earliest measured times (inset of Fig. S3B). +We speculate that this behavior is due to the protocol used to create the three-component mixtures, via global rotation of the +spins. Indeed, for the slower losses at B = 690 G, spin coherence at early times would affect both elastic and inelastic collision +cross sections. It would be interesting to investigate the interplay between the coherent RF preparation, inelastic losses and +possible Zeno effects in this system. +0 +20 +40 +60 +80 +100 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +0 +5 +10 +15 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +FIG. S3. Early-time decay of unpolarized three-component gases at B = 690 G (A) and B = 845 G (B). Blue points are averages of the three +components, and the solid red lines are three-body fits for t ≥ 15 ms and for the entire time range in A and B respectively. Insets: ˙n versus n. +Dashed lines are guides to the eye showing the n3 scaling on log-log scales. +IV. SPIN FRACTION DYNAMICS +Here we show that under the effective decay Eq. (3) in the main text, the two distinct regions of polarization in the ternary plot +Fig. 4C are closed under time evolution, i.e. time dynamics does not allow for crossing from the hatched area to the unhatched +one (and vice-versa). Indeed Eq. (3) implies +˙x = (1 − η) 3L3n1n2n3 +n1 + n2 + n3 +(x − c) , +(S2) +where x = (x1, x2, x3) and c = 1 +3(1, 1, 1). The spin fraction flow is a ray from the unpolarized state c, shown in Fig. S4A. +In Fig. S4B, we show the spin fractions over time, with the predictions of Eq. (S2) (solid lines). We note that the magnitude +| ˙x|, the ‘speed’ of this flow, is suppressed for large η. Indeed, at B = 690 G (top) the spin fractions barely change, while at +B = 845 G (bottom), they visibly vary. + +10 +0 +50 +100 +150 +200 +0 +0.2 +0.4 +0.6 +2 +4 +6 +0 +0.2 +0.4 +0.6 +FIG. S4. Spin fraction dynamics under asymmetric three-body decay. (A) Map of the polarization flow. (B) Examples of polarization dynamics +at B = 690 G (top) and B = 845 G (bottom). Empty points are neglected due to early-time effects. The solid lines are solutions to Eq. (S2). +This polarization map has an interesting consequence. A two-component gas with a small contaminant of a third state will +purify into a two-component mixture at a large cost in particle loss if η is large. Indeed, considering a small contaminant +x2 ≪ x1 = x3, the loss rate equations simplify, yielding ( ˙n1 + ˙n3)/ ˙n2 ≈ (2 + η)/(1 − η). For instance, for η = 0.8 we have +˙n1 + ˙n3 ≈ 14 ˙n2, an inefficient way to get rid of the impurity state. +V. RF PREPARATION PROTOCOLS +Here, we provide further details on the RF preparation of the three-component mixtures. To cover extensively the polarization +space, we used several RF protocols, depicted in Fig. S5. The ternary plot data in Fig. S5 is populated with the data of Fig. 4C, +with symbols corresponding to the protocol used. We see that the different protocols produce consistent values in the regions +where they overlap. Depending on the spin states used, the initial evaporation of the two-component mixture was carried out at +different magnetic fields B0 as noted in Fig. S5. +VI. A SIMPLE LOSS MODEL ON A LATTICE +In this section we show that a generic model does not account for the loss asymmetry. We consider a model of a gas on a +three-dimensional lattice, of lattice spacing b. + +11 +0 +0.5 +1 +FIG. S5. Robustness of η with respect to RF preparation protocols. Symbols in the ternary plot correspond to the protocols indicated in the +legend. Colored circles indicate initial and final spin populations, and arrows denote the RF pulses. B0 is the magnetic field at which the +two-component gas was evaporated (prior to the RF pulses). +Treating the gas as an open quantum system, we calculate the rate of change of atom number +d +dt ⟨Nj⟩ = tr ( ˙ρNj) using the +Lindblad equation [3, 4]: +˙ρ = 1 +iℏ [Hsys, ρ] + +� +α +� +LαρL† +α − 1/2 {L† +αLα, ρ} +� +, +(S3) +where ρ is the density matrix of the system, and Hsys is the system’s (conservative) Hamiltonian, containing the kinetic, potential +and elastic interaction energies. The atom number operator in state |j⟩ is defined as Nj = � +α b3a† +j,αaj,α [5], where operators +aj,α and a† +j,α respectively annihilate and create a particle of spin |j⟩ at lattice site α. These operators satisfy the anti-commutation +relations: {ai,α, aj,β} = 0 and {ai,α, a† +j,β} = δi,jδα,β/b3. Lα = √κa1,αa2,αa3,α is the non-hermitian jump operator which +represents the inelastic physics: three particles of different spin states scatter inelastically with a probability ∝ κ when they are +located on the same lattice site α, causing them to escape the trap. This yields +d +dt ⟨Nj⟩ = +� +α,β +⟨L† +α(b3a† +j,βaj,β)Lα − 1/2{L† +αLα, b3a† +j,βaj,β}⟩ , +(S4) +Since we have observed no spin changing collisions in any two-component mixture, we have assumed that Hsys com- +mutes with all Nj. +Using ⟨L† +α(b3a† +j,βaj,β)Lα⟩ = ⟨L† +αLα(b3a† +j,βaj,β)⟩ − δα,β ⟨L† +αLα⟩ and ⟨{L† +αLα, (b3a† +j,βaj,β)}⟩ = +2 ⟨L† +αLα(b3a† +j,βaj,β)⟩, we simplify Eq. (S4) into +d +dt ⟨Nj⟩ = − +� +α +⟨L† +αLα⟩ = −κ +� +α +⟨a† +3,αa† +2,αa† +1,αa1,αa2,αa3,α⟩ +(S5) +There is thus no dependence on the spin label j, and each spin component shares the losses equally: η = 0. +VII. AVALANCHE LOSSES +In an avalanche process, subsequent elastic scattering following a recombination event leads to the loss of additional atoms. +In this section, we elaborate on avalanches as a candidate for the origin of the asymmetric losses. We show that avalanches have +two rather generic consequences, neither of which are observed in our experiment. + +12 +Under an avalanche process, one would expect that the number of excess particles of spin |j⟩ lost should be proportional to +the fraction of atoms in that spin state, i.e. equal to cxj. To ensure that an unpolarized gas remains unpolarized during decay, +c should be independent of the spin state; c is then the average number of excess particles lost per recombination event. In that +case, we expect ˙nj = −K3(1 + cxj)n1n2n3, where K3 is the event rate per unit volume. This equation qualitatively matches +the asymmetric loss equation proposed in Eq. (3), with η = c/(3 + c). For our measured value η ≈ 0.8, this implies c ≈ 12. +On the other hand, we can estimate c using a kinetic model [50]. We assume that three-body recombination produces a dimer +of binding energy ϵb and a free atom. If all three incoming participants are at rest, the dimer will have a kinetic energy ϵb/3 after +formation. In a subsequent elastic collision with a free atom at rest, the dimer will retain 5/9 of its pre-collision kinetic energy +on average. If the gas is collisionally opaque to the dimer (though this is unlikely, see section VIII), these collisions will kick +free atoms from the trap until the dimer has less energy than the trap depth. The dimer can thus kick out about log5/9 (3Ubox/ϵb) +particles. +-480 +-460 +-440 +-20 +0 +20 +-20 +0 +20 +290 +310 +330 +FIG. S6. Measurement of Feshbach dimer binding energies in a balanced |1⟩-|3⟩ mixture at B = 690 G. (A) Spectroscopy on the transition +|1⟩ → |2⟩. (B) Spectroscopy on the transition |3⟩ → |2⟩. For both measurements, the RF pulse time is 80 µs, with a transferred fraction of +≈ 25% on the ‘free atom’ peak (green data) and ≈ 10% on the ‘bound’ peak (orange data, vertically rescaled for comparison). The bare values +ν12 +0 += 76.034 MHz and ν23 +0 += 82.707 MHz (vertical dotted lines) are calibrated on (noninteracting) fully polarized samples. The dashed and +solid lines are gaussian fits from which peak frequencies are extracted. The cartoons are energy level diagrams with the transitions used (not +to scale). +To evaluate this quantity, we performed RF spectroscopy of both Feshbach dimers |12⟩ and |23⟩ at B = 690 G, see Fig. S6. +We find that the binding energies are respectively ϵb/h ≈ 320 kHz and 470 kHz [6]. Using the most deeply bound of the two +dimers, log5/9 (3Ubox/ϵb) + 1 ≈ 4 (where we have accounted for an additional loss from a final inelastic atom-dimer event). +Additionally, this model implies that the total loss rate would depend on the asymmetry. Indeed, for ˙n = −3L3n1n2n3, we +would have within this model 3L3 = K3(3 + c) ∝ +� +1 + +η +1−η +� +. Using the data binning of Fig. 4E, we plot L3 vs η in Fig. S7. +The data displays no systematic variation of L3 with η. + +13 +0.5 +0.6 +0.7 +0.8 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +FIG. S7. Loss coefficient L3 versus η. The data bins are the same as in Fig. 4E. The prediction from the avalanche model is shown as the solid +purple line (with an arbitrary scaling factor for comparison with the experimental data). Error bars are standard deviations within bins. +VIII. MEAN FREE PATH +Avalanche processes require high collisional opacity of the product states in the medium. However, we measure significant +asymmetry even at low densities where the mean free path of typical recombination products ℓ is greater than the system size. +For s-wave scattering of distinguishable particles, the collision cross section is σ = 4π/(k2 + a−2), where a is the relevant +scattering length and k = √2µE/ℏ is the relative wavevector for a reduced mass µ and energy E. For a typical dimer |12⟩ and +free atom |3⟩ formed after recombination, k ≈ 1 × 107 m−1. The dimer (which scatters off all spin states) has the shorter mean +free path: +ℓ ≈ (n1σ1 + n2σ2 + n3σ3)−1 ∼ 200 µm +(S6) +where we have used n1 = n2 = n3 = 2 × 1010 cm−3 (corresponding to the lowest density bin of Fig. 4E of the main text), +and σj is the collisional cross section between the free atom |j⟩ and the dimer |12⟩. For this density, even a single secondary +scattering is thus not very likely in a box of size ∼ 150 microns. Yet, even in those conditions we observe a large η ≳ 0.5. The +result is similar for a typical |23⟩ dimer and atom recombination product. +We repeated the experiment in a smaller, near-cubical box of size ≈ 30 µm. In that box, a random particle would travel +an average distance of 14 µm (in the absence of collisions) before reaching the box boundary. Given an initial total density +n ≈ 7×1011 cm−3, ℓ > 21 µm during the entire decay. Even in that case, we find η = 0.8(1), consistent with the measurements +of the main text. We conclude that an avalanche process is an unlikely explanation. +IX. L3 VERSUS BOX DEPTH +Here, we test the influence of the trap depth on losses. We measure the decay of an unpolarized gas, lowering the box depth +from its initial value Ubox to U immediately after RF preparation of the three-component mixture. We see in Fig. S8 that the +extracted L3 shows no noticeable dependence on U down to a depth of Ubox/2. This indicates that evaporation plays essentially +no role in the anomalous loss behavior. +Suppose that inelastic losses deposit energy in the gas, which could have been released by recombination [7] or other mech- +anisms (e.g. [8]). This would lead to enhanced evaporation and thus a modification of the overall loss rate. For a polarized gas, +these evaporative losses may be shared unequally among the spin states [55] which could mimic the asymmetry we observe. We +calculate the rate of losses assuming they are entirely due to evaporation and that the rate of deposited energy (ϵb per event) is +balanced by the evaporation rate, such that ˙E = ϵbK3V n1n2n3 + αU ˙N = 0, where U is the box depth, and αU is the average +energy of a particle that leaves the trap (α ≳ 1). This implies ˙n = − ϵbK3 +αU n1n2n3, and hence L3 ∝ 1/U, but we do not see +evidence of this dependence (see dashed red line in Fig. S8). + +14 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +FIG. S8. L3 versus trap depth U, where Ubox = kB × 1.6 µK. The red dashed line is the 1/U dependence expected for evaporation-dominated +losses, rescaled by an arbitrary factor for comparison with the data. +[1] C. H. Schunck, Y. Shin, A. Schirotzek, W. Ketterle, Determination of the fermion pair size in a resonantly interacting superfluid, Nature +454, 739–743 (2008). +[2] B. Mukherjee, et al., Spectral response and contact of the unitary Fermi gas, Physical Review Letters 122, 203402 (2019). +[3] E. Braaten, H.-W. Hammer, G. P. Lepage, Lindblad equation for the inelastic loss of ultracold atoms, Physical Review A 95, 012708 (2017). +[4] D. Manzano, A short introduction to the Lindblad master equation, AIP Advances 10, 025106 (2020). +[5] In this section only, Nj is defined as an operator. In the rest of the text it is an average value. +[6] For both dimers, the measured binding energy is within ∼ 10% of the universal expression ϵb = ℏ2/(ma2) and consistent with local +measurements in a harmonically trapped gas at 691 G [1]. Additionally, the shift of the free peak from the bare frequencies is consistent +with the measurements of [2] for T/TF ≈ 0.25. +[7] T. Weber, J. Herbig, M. Mark, H.-C. N¨agerl, R. Grimm, Three-body recombination at large scattering lengths in an ultracold atomic gas, +Physical Review Letters 91, 123201 (2003). +[8] I. Bouchoule, L. Dubois, L.-P. Barbier, Losses in interacting quantum gases: Ultraviolet divergence and its regularization, Physical Review +A 104, L031304 (2021). + diff --git a/OtE0T4oBgHgl3EQfTgB4/content/tmp_files/load_file.txt b/OtE0T4oBgHgl3EQfTgB4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ccbd77480ed61bf75585191578c115ea0094f30a --- /dev/null +++ b/OtE0T4oBgHgl3EQfTgB4/content/tmp_files/load_file.txt @@ -0,0 +1,802 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf,len=801 +page_content='Observation of Anomalous Decay of a Polarized Three-Component Fermi Gas Grant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Schumacher,1, ∗ Jere T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' M¨akinen,1, 2 Yunpeng Ji,1 Gabriel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Assumpc¸˜ao,1 Jianyi Chen,1 Songtao Huang,1 Franklin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Vivanco,1 and Nir Navon1, 2 1Department of Physics, Yale University, New Haven, Connecticut 06520, USA 2Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA Systems of fermions with multiple internal states, such as quarks in quantum chromodynamics and nucleons in nuclear matter, are at the heart of some of the most complex quantum many-body problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The stability of such many-body multi-component systems is crucial to understanding, for instance, baryon formation and the structure of nuclei, but these fermionic problems are typically very challenging to tackle theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Versatile experimental platforms on which to study analogous problems are thus sought after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Here, we report the cre- ation of a uniform gas of three-component fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We characterize the decay of this system across a range of interaction strengths and observe nontrivial competition between two- and three-body loss processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We observe anomalous decay of the polarized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' spin-population imbalanced) gas, in which the loss rates of each component unexpectedly differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We introduce a generalized three-body rate equation which captures the decay dynamics, but the underlying microscopic mechanism is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Characterizing if - and into what - a system decays has of- ten been a pathway to discovering new quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Such stability problems are ubiquitous in many-body physics, ranging from the onset of turbulence in quantum liquids [1] to dissipation processes in Josephson junctions [2], to the sta- bility of nuclei [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Ultracold atomic systems have been a fertile ground for such studies: their (in)stability has revealed a wide range of interesting and sometimes unexpected phe- nomena, such as Efimov three-body physics [4], quantum- fluctuation stabilization against many-body collapse [5], and prethermalization of metastable phases [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In particular, there has been renewed appreciation that the stability of quantum gases against inelastic-collision losses [7, 8] (or lack thereof) is a pristine probe of many-body quantum correlations [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Strongly interacting fermions with contact interactions em- body a stability success story, especially the case of two- component (‘spin-1/2’) fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The observation of unex- pected shifts of loss features with respect to scattering - Fes- hbach - resonances [13–17] led to the understanding that sur- prisingly stable pairs were forming near those resonances, protected by a Pauli blocking effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This discovery unlocked the field of the BEC-BCS crossover [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Three-component (‘spin-1’) fermions give access to even richer physics, such as quantum chromodynamics-like phe- nomena and complex pairing patterns [19–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Pioneering ex- periments showed that unpolarized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' spin-population bal- anced) gases of three-component fermions exhibit interesting decay behavior related to the Efimov effect [26–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' How- ever, density-dependent losses in those spatially inhomoge- neous gases were unavoidably coupled to particle transport and heating, making the interpretation of those experiments challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The metastability of the three-component Fermi gas has yet to be comprehensively understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In this work, we create box-trapped, uniform gases of three- component fermions with controllable spin-population imbal- ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' E-mail: grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='schumacher@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='edu ance and study their stability (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Such spatially homo- geneous gases generally obey simpler dynamics than their in- homogeneous counterparts (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' [10, 31–33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This has enabled us to observe a surprising violation of the generic ex- pectation that loss rates among spin components should be equal in a three-body process involving all three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We rule out two credible explanations for this effect, indicat- ing that unexpected physics is at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Our experiment begins with a gas of 6Li atoms in a red- detuned crossed optical dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The gas is prepared in an incoherent mixture of two of the three lowest Zeeman sub- levels, which encode the three (pseudo-)spin components of our ‘spin-1’ fermions (respectively labelled |1⟩, |2⟩, and |3⟩, see top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This two-component mixture is evaporatively cooled at a bias magnetic field B0 (which de- pends on the two states used) and then loaded into a blue- detuned optical box trap of wavelength 639 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We ramp the magnetic field to the field of interest B in 200 ms, which is slow compared to the two-body collision rate but fast compared to the lifetime of the two-component mixtures in the range of fields investigated [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We let the magnetic field settle for 100 ms and then prepare the spin mix- tures by sequentially applying variable radio-frequency (RF) pulses to drive the |1⟩−|2⟩ and |2⟩−|3⟩ transitions (top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For the range of B explored in this work (deep in the Paschen-Back regime for 6Li), all three spins are nearly identically levitated against gravity using a magnetic field gra- dient [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The fermions interact via binary contact interac- tions, characterized by an s-wave scattering length a for each pair of spin states at a given magnetic field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' each pair has a broad Feshbach resonance (see bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We hold the gas for a variable time t before measuring the number of atoms of spin |j⟩, Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Typical in-situ absorption images of each spin population in an imbalanced mixture are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1C (top), together with integrated column densi- ties along the two directions x and y (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The integrated density profiles of the three components are consistent with uniform densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We first study the stability of spin-balanced samples, N1 ≈ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='02237v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='quant-gas] 5 Jan 2023 2 OD OD OD 50 0 50 50 0 50 ~ RF ~ 700 800 900 2 1 0 1 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Preparation of a homogeneous three-component Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A) Top: Sketch of the optical box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Bottom: The stability of the mixture is studied by measuring the density of each spin population over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (B) Top: Breit-Rabi diagram of the lowest hyperfine states of 6Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The three (pseudo)-spins are encoded in the three lowest states, respectively |1⟩, |2⟩ and |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The polarization of the three component mixture is controlled via radio-frequency (RF) pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Bottom: Scattering length a for each pair of spin states in units of 104a0, where a0 is the Bohr radius (adjacent colors match the corresponding pair of states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Vertical dashed lines show the locations of Feshbach resonances [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (C) Top: In-situ absorption images of a typical polarized three-component mixture (averaged over 10 realizations), taken along the z axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' the color scale corresponds to the optical density (OD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The length and radii of this conical box are L = 120(2) µm, R1 = 75(1) µm and R2 = 73(1) µm, and its trap depth is Ubox = kB× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6(2) µK, where kB is Boltzmann’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Note that boxes of various sizes were used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Bottom: Integrated density along the x and y axes for each component with fits to homogeneous density profiles (dotted lines) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' N2 ≈ N3 as a function of magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We typically evaporate a balanced mixture of |1⟩-|3⟩ near its Feshbach res- onance at B0 ≈ 690 G, ending with typically N1 ≈ N3 ≈ 5 × 105 at T/TF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='25 (where TF ≈ 400 nK is the Fermi temperature) prior to ramping the field and creating the three- component mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 2A-B, we show two typical de- cays, from which we make two key observations: (i) the mix- tures remain unpolarized during decay [37], and (ii) these losses involve all three states, as all two-component mixtures are much more stable [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The density uniformity of our samples enables us to model- independently characterize the three-component gas decay dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 2A-B, we plot the total atom loss rate ˙N ≡ dN/dt as a function of the total atom number N ≡ N1+N2+N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For the magnetic fields explored, the data is well captured by a power law ˙N ∝ −N γ, where the fitted γ is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For a homogeneous unpolarized gas with a total density n = N/V and a constant volume V , this implies that ˙n ∝ −nγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Importantly, the exponent γ encodes information on the number of independent particles involved in the loss events [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For both B ≲ 720 G and B ≳ 810 G, we observe γ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This is consistent with losses dominated by energy- independent recombination involving three distinguishable fermions (for which we expect ˙n ∝ −n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The intermediate range B ≈ 720 − 810 G is more com- plicated, and γ deviates from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In that region, the scatter- ing lengths a12 and a23, respectively between states |1⟩ − |2⟩ and |2⟩ − |3⟩, are large and positive, so that the products of three-body recombination can remain trapped [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This allows for subsequent inelastic atom-dimer collisions (involv- ing three distinguishable atoms), an effectively two-body loss process [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Our intermediate 2 ≲ γ ≲ 3 is qualita- tively consistent with this competition of two- and three-body effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Focusing on the regions dominated by three-body recombi- nation, we now explore polarized three-component mixtures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' with spin-population imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3A, we show a typical decay of a polarized sample at a field B = 845 G, above all three broad Feshbach resonances (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The losses ∆nj ≡ nj(t) − ninitial j are equal for all spins (bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This is unsurprising since we expect the loss rate equation to take the form ˙nj = −L3n1n2n3, (1) where L3 is the (density-independent) three-body recombina- tion loss coefficient [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The data at 845 G agrees very well with the numerical solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (1), shown as solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' the initial densities nj are the only fit parameters, L3 being fixed to the value we measure from the decay of the unpolarized gas (see the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Surprisingly, the same measurement at the Feshbach res- onance of the |1⟩ − |3⟩ states yields a qualitatively different outcome: the loss rates per spin are distinct, with the majority component showing larger absolute losses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Nonetheless, we observe that the total loss rate still obeys the expected three-body rate equation ˙n = −3L3n1n2n3 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='3 0 50 100 150 200 250 300 0 1 2 3 1 5 106 107 0 10 20 30 40 50 0 1 2 3 1 5 106 107 108 700 750 800 850 2 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Stability of the unpolarized three-component Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A- B) Typical decay at the |1⟩-|3⟩ resonance (A) and at B = 768 G (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The red line is a fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Early times (open symbols) are ex- cluded from our analysis, see [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Insets: Total loss rate − ˙N versus N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The solid blue lines are power-law fits yielding γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (C) Expo- nent γ versus magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The Feshbach resonances are shown as dot-dashed gray lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For B larger than the red dotted line, the |12⟩ and |23⟩ Feshbach dimers can remain trapped (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Error bars on γ displayed are only statistical (estimated by bootstrapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For most points, the dominant source of uncertainty is an additional 10% systematic error coming from box volume calibration [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The uncertainty in B is smaller than the point size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (orange points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' What’s more, the extracted L3 matches the unpolarized gas value (blue points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We systematically extract L3 at 690 G as a function of po- larization, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Unlike its spin-1/2 counterpart [45– 47], the polarization of the three-component gas is character- ized by two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The spin fractions xj ≡ nj/n pro- vide a convenient parameterization, so that the polarization state can be represented on a ternary plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We find that L3 agrees with the unpolarized-gas value for nearly all polariza- tions, provided the spin fraction of |2⟩ is not very high [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' With the total loss scaling established, we now investigate how losses are apportioned among the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4A, we see that ˙nj appear proportional to their respective spin fractions xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' To characterize this relation, we introduce the relative loss yj ≡ ˙nj/ ˙n and compare it directly to the xj, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This measurement suggests a simple relation: yj = ηxj + 1 − η 3 , (2) where the proportionality of the losses is captured by the asymmetry η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The constant (1 − η)/3 is constrained by the definition of the parameters xj and yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' η = 0 corresponds to symmetric losses as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We now explore η at 690 G as a function of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Following the procedure outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4A-B, the ternary plot is populated with the values of η extracted from each time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We observe that the polarization space is separated into two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In the (larger) region where |1⟩ or |3⟩ is the largest component (unhatched region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4C), η is robust to spin composition, regardless of whether |2⟩ is the median or the minority component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In the (smaller) region where |2⟩ is the largest component (hatched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4C), η is smaller, al- though distinctly non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Note that exchanging spins |1⟩ and |3⟩ leaves Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4C essentially unchanged, which reflects the approximate symmetry of the three scattering lengths (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Gathering the data from the larger (unhatched) region, we observe that the relative loss law appears indeed to be univer- sal over a large range of spin fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Fitting the data, we find η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='80(3) (see caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' On the other hand, applying the same procedure at B = 845 G we find η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='02(3) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4D), characteristic of the usual three-body loss dynamics of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The behavior of both the total and relative losses suggests that the three-component mixture at 690 G obeys the follow- ing decay dynamics: ˙nj = −3L3 � ηxj + 1 − η 3 � n1n2n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (3) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3B, we show as solid lines the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (3) with fixed L3 and η, and only the initial nj as adjustable parame- ters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' the agreement with the experimental data is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Gathering decay measurements across a large density range, we observe that η is weakly density dependent (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4E), although this variation is slow enough so that it is negligible over our typical decay series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We now turn to hypotheses for explaining this anomalous loss asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' First, note that the rate equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (1) is an approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' more fundamentally, losses are expressed 4 1 0 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Stability of the polarized three-component Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A) Decay of a polarized gas at 845 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Solid curves are fits to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (1), with L3 fixed to the value measured in the unpolarized decay: L3(B = 845 G) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4(1) × 10−20 cm6/s (t = 0 corresponds to the end of the RF pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=') (B) Decay of a polarized gas at the |1⟩-|3⟩ resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Solid curve are fits to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (3), with early time points excluded (open symbols) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' L3 is fixed to the value measured in the unpolarized decay: L0 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='1(1) × 10−21 cm6/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We estimate additional systematic uncertainties on measurements of L3 of 20% due to box volume calibration [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (C) Total loss rate ˙n versus n1n2n3, for both unpolarized (blue) and polarized (orange) gases at 690 G, averaged over many experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Colored bands are fits (with uncertainties) to the three-body loss equation for ˙n, from which L3 is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (D) Three-body loss coefficient L3 at 690 G as a function of polarization, relative to L0 3, plotted on a barycentric ternary plot [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' as correlators of quantum fields [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Calculating ˙nj for a generic model of a gas on a lattice in the Lindblad formalism yields [35, 48] ˙nj ∝ − � α � a† 3,αa† 2,αa† 1,αa1,αa2,αa3,α � (4) where aj,α (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' a† j,α) is the annihilation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' creation) op- erator for state |j⟩ at lattice site α, under the assumptions that no spin-changing collisions occur and that losses are due to local three-body recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' As the right-hand side does not depend on the spin state, η = 0 within this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This conclusion holds even in the presence of nontrivial correla- tions that would preclude factoring Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (4) into a product of densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Alternatively, this effect could originate from many-particle scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (3) is indeed quali- tatively reminiscent of so-called avalanche mechanisms, in which secondary collisions occur between the products of three-body recombination and other atoms in the gas, result- ing in excess losses that augment the loss rate prefactor with spin-fraction-dependent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Avalanche losses have been a debated topic in ultracold few-body dynamics [49–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Under an avalanche process, one would expect that the number of excess particles of spin |j⟩ lost should be propor- tional to the fraction of atoms in that spin state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' equal to cxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The prefactor c should be independent of the spin state to ensure that an unpolarized gas remains unpolarized during decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' c is then the average number of excess particles per recombination event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In that case, ˙n ∝ −(3 + c)n1n2n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In this model, η = c/(3+c) [35], and our largest measured η would imply c ≳ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' However, a generic kinetic model [50] indicates that c ≈ 4 given the binding energies of the |12⟩ and |23⟩ Feshbach dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Furthermore, repeating this ex- periment in a box whose size is about the mean free path or smaller, we find the same asymmetry [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This further rules out an avalanche scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Finally, evaporation in a collision- ally dense gas could give rise to asymmetric losses, as sug- gested in [55], but is also essentially ruled out as varying the box depth does not affect the loss rate [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Future work should elucidate this puzzle, for example by measuring the energy dynamics and temperature dependence of this process, as well as by characterizing the asymme- try across interaction regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This work also opens several new avenues, such as studying prethermalization dynamics of the metastable mixture [6, 56] and searching for signa- tures of three-component pairing correlations at low tempera- ture [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We thank Chris Greene, Jose d’Incao, Leonid Glazman, Steve Girvin, Carlos S´a de Melo, Alexander Schuckert, and Francesca Ferlaino for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We thank Yvan Castin, F´elix Werner, Fr´ed´eric Chevy, and Zoran Hadzibabic for critical comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This work was sup- ported by the NSF (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' PHY-1945324 and PHY- 2110303), DARPA (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' W911NF2010090), the David and Lucile Packard Foundation, and the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Sloan Foun- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' acknowledges support from the Yale Quantum Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='S acknowledges support from the NSF Gradu- ate Research Fellowship Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='1 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0 25 50 75 100 1 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Asymmetric losses in a polarized three-component Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A) Top: Sketch of the extraction of spin fractions and relative losses (xj, yj) from the nj(t) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Inset: xj versus time (see also [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Bottom: ˙nj versus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (B) The pairs (xj, yj) extracted from the bottom of panel A lie on a line, whose slope defines η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (C) η at 690 G as a function of polarization on a ternary plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Each point is obtained by averaging over a single decay, as shown in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The hatching marks two distinct regions in polarization space (see also [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (D) Top: (x, y) from the large (unhatched) region of the ternary plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The green solid line is a linear fit, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='80(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The gray dash-dotted line shows η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Bottom: The same procedure at 845 G yields η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='02(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We estimate an additional 5% systematic error on η [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (E) η versus total density n at 690 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The points are obtained by gathering data from several decays into density bins, then extracting η (as in D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The highest two bins correspond to the data of D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' this plot includes additional data at lower density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The variation of η(n) is slow enough that it can be neglected over the density range considered during a typical decay (see A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Vinen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Niemela, Quantum turbulence, Journal of Low Tem- perature Physics 128, 167–231 (2002).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=', Resonant atom-dimer collisions in cesium: Testing universality at positive scattering lengths, Physical Re- view A 90, 022704 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Parish, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Huse, Evaporative depolarization and spin transport in a unitary trapped Fermi gas, Physical Review A 80, 063605 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' [56] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Takasu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Takahashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Cazalilla, Sup- pression and control of prethermalization in multicomponent Fermi gases following a quantum quench, Physical Review A 101, 053620 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 7 Supplementary Material Observation of Anomalous Decay of a Polarized Three-Component Fermi Gas I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' CHARACTERIZING OPTICAL BOXES We determine the volume of the optical box of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1 of the main text by fitting the in-situ density profiles � dz nj(x, y, z) (where z is the imaging line of sight, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S1A) with a conical model of radii R1 and R2, and length L, convolved with a Gaussian function of standard deviations σx and σy (that capture imperfections from finite resolution of the imaging and box projection systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We find R1 = 75(1) µm, R2 = 73(1) µm, L = 120(2) µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The values of σx = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 µm and σy = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 µm reflect the slightly different quality of projection of the ‘end caps’ and the ‘tube’ of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S1B, we display density profile cuts of the box of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1C projected along the x and y axes together with the fits (dashed purple lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' If imperfections were entirely due to the box projection system - a conservative assumption - we estimate that ≈ 90% of the atoms are within 10% of the average density in the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Additionally, we observe during a typical decay that the volume decreases by ≈ 10% when the atom number decreases by a factor of ≈ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' taking this small effect into account as an uncertainty on the volume, we estimate an uncertainty of 10% on γ and of 20% on L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This also contributes to uncertainties on xj, yj of ≈ 1% and ≈ 5% respectively (reflecting the difference between fitting N and ˙N rather than n and ˙n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Ultimately we find that the uncertainty in η due to this effect is ≈ 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S1C, we compare the density profiles of the three spin states by scaling and subtracting the profile of |3⟩ from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We see that the density profiles are essentially identical up to rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' OD 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 OD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 OD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Homogeneity of a polarized three-component mixture in the optical box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A) Example of an in-situ optical density image of spin |3⟩ (same data as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 1C, with x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='16, x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='35, and x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (B) Top: Cuts of the density profile (yellow, solid) and the fit (purple, dashed) along the y (top) and x (bottom) axes (dashed black lines in A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The fit to the OD image is a smoothed cone (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Residuals (normalized to the peak fitted density along each cut) are shown underneath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (C) Top: Subtracting the measured in-situ density of |3⟩ from |2⟩ (after scaling |2⟩ to have the same mean density as |3⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' OD scale is the same as A and the dashed red rectangle indicates the region of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Bottom: Likewise subtracting spin |3⟩ from |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 8 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' TWO- VERSUS THREE-COMPONENT MIXTURES DECAY We show in this section that in the range of B explored in this work, unpolarized three-component mixtures decay much faster than any two-component ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S2A, we show examples of decays of two-component mixtures (which may include evap- orative losses in addition to intrinsic recombination processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We compare two- and three-component decays quantitatively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S2B by extracting an (early-time) effective timescale for losses τeff = n/| ˙n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The black crosses correspond to three- component mixtures, with a density per spin state in the range nj ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 − 4 × 1011 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In green, blue and pink, we show τeff for the two-component mixtures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' for each field, the density per spin state is up to a factor of 4 larger than the corresponding three-component data, but never smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Even in that case, τeff is at least an order of magnitude larger than the corresponding three-component one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' However, this will no longer hold for extremely polarized three-component gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' At B = 690 G, both a12 and a23 are positive so that the corresponding two-component mixtures are already unstable with respect to three-body recombination (involving two identical fermions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Furthermore, the corresponding binding energies of the |12⟩ and |23⟩ dimers are such that all recombination products escape from the box (see section VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Thus, a mixture of spins |i⟩ − |j⟩ would decay following a recombination law ˙nj = −Lij 3 (2n2 jni + n2 i nj)/3 [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We note that the two-component recombination rates agree with the universal prediction of [39] (solid lines of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S2B) at fields below 720 G (red dotted line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' At higher magnetic fields, the formation of |12⟩ and |23⟩ dimers does not directly lead to losses, as their binding energies are less than 6Ubox [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' To estimate the influence of two-component losses on the main text’s analysis, we calculate the ratio of the total losses due to three-component recombination ( ˙nthree ≡ −3L3n1n2n3) to two-component ones ( ˙ntwo ≡ −L12 3 (n2 1n2 + n2 2n1) − L23 3 (n2 2n3 + n2 3n2)): ˙ntwo ˙nthree = L12 3 3L3 (1/x3 − 1) + L23 3 3L3 (1/x1 − 1) (S1) Extracting L12 3 and L23 3 from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S2A, we find ratios L12 3 /(3L3) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='003 and L23 3 /(3L3) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Setting a limit of ˙ntwo/ ˙nthree < 10%, we find the polarization limit depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For mixtures less polarized than this limit, three- component losses are dominating, and we ignore losses arising from two-component processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 1 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Stability of two- versus three-component mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A) Decay of unpolarized two-component gases at 690 G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' a three-component gas decay is shown as black crosses for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (B) Effective timescales across magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Feshbach resonances are indicated by vertical dash-dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' At each field shown, the two-component gases have densities (per spin state) equal to or greater than those of the corresponding three-component gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Solid pink and blue curves show predictions from universal recombination to Feshbach dimers [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For B larger than the red dotted line, the |12⟩ and |23⟩ Feshbach dimers can remain trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (C) L3 versus polarization at 690 G (normalized to L0 3 as in the main text), includes additional data at lower total density (n ∼ 5 × 1010 cm−3) not shown in main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Dashed red curve gives polarization outside which two-component processes are expected to account for 10% of total losses or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 9 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' EARLY-TIME DECAY In the main text, we excluded early times from the analysis of decays at B = 690 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Here we show in more detail that the early-time behavior at that field is distinct from behavior at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S3A, we show a decay of an unpolarized sample at 690 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We see that for t ≲ 10 ms, the decay is slower than at later times (see red solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This effect is even more obvious for ˙n versus n (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Only at later times the data follows a γ ≈ 3 law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This effect is also present in polarized-gas measurements, but for t ≳ 15 ms the data is well described by a single γ ≈ 3 for all our polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' By contrast, for the faster losses at B ≳ 845 G (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S3B), the data follows a simple γ ≈ 3 law, even from the earliest measured times (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We speculate that this behavior is due to the protocol used to create the three-component mixtures, via global rotation of the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Indeed, for the slower losses at B = 690 G, spin coherence at early times would affect both elastic and inelastic collision cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' It would be interesting to investigate the interplay between the coherent RF preparation, inelastic losses and possible Zeno effects in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Early-time decay of unpolarized three-component gases at B = 690 G (A) and B = 845 G (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Blue points are averages of the three components, and the solid red lines are three-body fits for t ≥ 15 ms and for the entire time range in A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Insets: ˙n versus n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Dashed lines are guides to the eye showing the n3 scaling on log-log scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' SPIN FRACTION DYNAMICS Here we show that under the effective decay Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (3) in the main text, the two distinct regions of polarization in the ternary plot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4C are closed under time evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' time dynamics does not allow for crossing from the hatched area to the unhatched one (and vice-versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Indeed Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (3) implies ˙x = (1 − η) 3L3n1n2n3 n1 + n2 + n3 (x − c) , (S2) where x = (x1, x2, x3) and c = 1 3(1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The spin fraction flow is a ray from the unpolarized state c, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S4B, we show the spin fractions over time, with the predictions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (S2) (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We note that the magnitude | ˙x|, the ‘speed’ of this flow, is suppressed for large η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Indeed, at B = 690 G (top) the spin fractions barely change, while at B = 845 G (bottom), they visibly vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 10 0 50 100 150 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Spin fraction dynamics under asymmetric three-body decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A) Map of the polarization flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (B) Examples of polarization dynamics at B = 690 G (top) and B = 845 G (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Empty points are neglected due to early-time effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The solid lines are solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This polarization map has an interesting consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' A two-component gas with a small contaminant of a third state will purify into a two-component mixture at a large cost in particle loss if η is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Indeed, considering a small contaminant x2 ≪ x1 = x3, the loss rate equations simplify, yielding ( ˙n1 + ˙n3)/ ˙n2 ≈ (2 + η)/(1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For instance, for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 we have ˙n1 + ˙n3 ≈ 14 ˙n2, an inefficient way to get rid of the impurity state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' RF PREPARATION PROTOCOLS Here, we provide further details on the RF preparation of the three-component mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' To cover extensively the polarization space, we used several RF protocols, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The ternary plot data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S5 is populated with the data of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4C, with symbols corresponding to the protocol used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We see that the different protocols produce consistent values in the regions where they overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Depending on the spin states used, the initial evaporation of the two-component mixture was carried out at different magnetic fields B0 as noted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' A SIMPLE LOSS MODEL ON A LATTICE In this section we show that a generic model does not account for the loss asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We consider a model of a gas on a three-dimensional lattice, of lattice spacing b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Robustness of η with respect to RF preparation protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Symbols in the ternary plot correspond to the protocols indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Colored circles indicate initial and final spin populations, and arrows denote the RF pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' B0 is the magnetic field at which the two-component gas was evaporated (prior to the RF pulses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Treating the gas as an open quantum system, we calculate the rate of change of atom number d dt ⟨Nj⟩ = tr ( ˙ρNj) using the Lindblad equation [3, 4]: ˙ρ = 1 iℏ [Hsys, ρ] + � α � LαρL† α − 1/2 {L† αLα, ρ} � , (S3) where ρ is the density matrix of the system, and Hsys is the system’s (conservative) Hamiltonian, containing the kinetic, potential and elastic interaction energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The atom number operator in state |j⟩ is defined as Nj = � α b3a† j,αaj,α [5], where operators aj,α and a† j,α respectively annihilate and create a particle of spin |j⟩ at lattice site α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' These operators satisfy the anti-commutation relations: {ai,α, aj,β} = 0 and {ai,α, a† j,β} = δi,jδα,β/b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Lα = √κa1,αa2,αa3,α is the non-hermitian jump operator which represents the inelastic physics: three particles of different spin states scatter inelastically with a probability ∝ κ when they are located on the same lattice site α, causing them to escape the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This yields d dt ⟨Nj⟩ = � α,β ⟨L† α(b3a† j,βaj,β)Lα − 1/2{L† αLα, b3a† j,βaj,β}⟩ , (S4) Since we have observed no spin changing collisions in any two-component mixture, we have assumed that Hsys com- mutes with all Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Using ⟨L† α(b3a† j,βaj,β)Lα⟩ = ⟨L† αLα(b3a† j,βaj,β)⟩ − δα,β ⟨L† αLα⟩ and ⟨{L† αLα, (b3a† j,βaj,β)}⟩ = 2 ⟨L† αLα(b3a† j,βaj,β)⟩, we simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (S4) into d dt ⟨Nj⟩ = − � α ⟨L† αLα⟩ = −κ � α ⟨a† 3,αa† 2,αa† 1,αa1,αa2,αa3,α⟩ (S5) There is thus no dependence on the spin label j, and each spin component shares the losses equally: η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' AVALANCHE LOSSES In an avalanche process, subsequent elastic scattering following a recombination event leads to the loss of additional atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In this section, we elaborate on avalanches as a candidate for the origin of the asymmetric losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We show that avalanches have two rather generic consequences, neither of which are observed in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 12 Under an avalanche process, one would expect that the number of excess particles of spin |j⟩ lost should be proportional to the fraction of atoms in that spin state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' equal to cxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' To ensure that an unpolarized gas remains unpolarized during decay, c should be independent of the spin state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' c is then the average number of excess particles lost per recombination event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In that case, we expect ˙nj = −K3(1 + cxj)n1n2n3, where K3 is the event rate per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This equation qualitatively matches the asymmetric loss equation proposed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (3), with η = c/(3 + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For our measured value η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8, this implies c ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' On the other hand, we can estimate c using a kinetic model [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We assume that three-body recombination produces a dimer of binding energy ϵb and a free atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' If all three incoming participants are at rest, the dimer will have a kinetic energy ϵb/3 after formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In a subsequent elastic collision with a free atom at rest, the dimer will retain 5/9 of its pre-collision kinetic energy on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' If the gas is collisionally opaque to the dimer (though this is unlikely, see section VIII), these collisions will kick free atoms from the trap until the dimer has less energy than the trap depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The dimer can thus kick out about log5/9 (3Ubox/ϵb) particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 480 460 440 20 0 20 20 0 20 290 310 330 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Measurement of Feshbach dimer binding energies in a balanced |1⟩-|3⟩ mixture at B = 690 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (A) Spectroscopy on the transition |1⟩ → |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' (B) Spectroscopy on the transition |3⟩ → |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For both measurements, the RF pulse time is 80 µs, with a transferred fraction of ≈ 25% on the ‘free atom’ peak (green data) and ≈ 10% on the ‘bound’ peak (orange data, vertically rescaled for comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The bare values ν12 0 = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='034 MHz and ν23 0 = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='707 MHz (vertical dotted lines) are calibrated on (noninteracting) fully polarized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The dashed and solid lines are gaussian fits from which peak frequencies are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The cartoons are energy level diagrams with the transitions used (not to scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' To evaluate this quantity, we performed RF spectroscopy of both Feshbach dimers |12⟩ and |23⟩ at B = 690 G, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We find that the binding energies are respectively ϵb/h ≈ 320 kHz and 470 kHz [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Using the most deeply bound of the two dimers, log5/9 (3Ubox/ϵb) + 1 ≈ 4 (where we have accounted for an additional loss from a final inelastic atom-dimer event).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Additionally, this model implies that the total loss rate would depend on the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Indeed, for ˙n = −3L3n1n2n3, we would have within this model 3L3 = K3(3 + c) ∝ � 1 + η 1−η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Using the data binning of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4E, we plot L3 vs η in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The data displays no systematic variation of L3 with η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Loss coefficient L3 versus η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The data bins are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The prediction from the avalanche model is shown as the solid purple line (with an arbitrary scaling factor for comparison with the experimental data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Error bars are standard deviations within bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' MEAN FREE PATH Avalanche processes require high collisional opacity of the product states in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' However, we measure significant asymmetry even at low densities where the mean free path of typical recombination products ℓ is greater than the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For s-wave scattering of distinguishable particles, the collision cross section is σ = 4π/(k2 + a−2), where a is the relevant scattering length and k = √2µE/ℏ is the relative wavevector for a reduced mass µ and energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For a typical dimer |12⟩ and free atom |3⟩ formed after recombination, k ≈ 1 × 107 m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The dimer (which scatters off all spin states) has the shorter mean free path: ℓ ≈ (n1σ1 + n2σ2 + n3σ3)−1 ∼ 200 µm (S6) where we have used n1 = n2 = n3 = 2 × 1010 cm−3 (corresponding to the lowest density bin of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 4E of the main text), and σj is the collisional cross section between the free atom |j⟩ and the dimer |12⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For this density, even a single secondary scattering is thus not very likely in a box of size ∼ 150 microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Yet, even in those conditions we observe a large η ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The result is similar for a typical |23⟩ dimer and atom recombination product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We repeated the experiment in a smaller, near-cubical box of size ≈ 30 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' In that box, a random particle would travel an average distance of 14 µm (in the absence of collisions) before reaching the box boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Given an initial total density n ≈ 7×1011 cm−3, ℓ > 21 µm during the entire decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Even in that case, we find η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8(1), consistent with the measurements of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We conclude that an avalanche process is an unlikely explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' L3 VERSUS BOX DEPTH Here, we test the influence of the trap depth on losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We measure the decay of an unpolarized gas, lowering the box depth from its initial value Ubox to U immediately after RF preparation of the three-component mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S8 that the extracted L3 shows no noticeable dependence on U down to a depth of Ubox/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This indicates that evaporation plays essentially no role in the anomalous loss behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Suppose that inelastic losses deposit energy in the gas, which could have been released by recombination [7] or other mech- anisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This would lead to enhanced evaporation and thus a modification of the overall loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' For a polarized gas, these evaporative losses may be shared unequally among the spin states [55] which could mimic the asymmetry we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' We calculate the rate of losses assuming they are entirely due to evaporation and that the rate of deposited energy (ϵb per event) is balanced by the evaporation rate, such that ˙E = ϵbK3V n1n2n3 + αU ˙N = 0, where U is the box depth, and αU is the average energy of a particle that leaves the trap (α ≳ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' This implies ˙n = − ϵbK3 αU n1n2n3, and hence L3 ∝ 1/U, but we do not see evidence of this dependence (see dashed red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' L3 versus trap depth U, where Ubox = kB × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content='6 µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' The red dashed line is the 1/U dependence expected for evaporation-dominated losses, rescaled by an arbitrary factor for comparison with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE0T4oBgHgl3EQfTgB4/content/2301.02237v1.pdf'} +page_content=' Schunck, 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Ganguly1 +1 Northeastern University, Boston, MA, USA +2 Hitachi Industrial AI Lab, Santa Clara, CA, USA +Abstract. Recent advances in domain adaptation reveal that adversar- +ial learning on deep neural networks can learn domain invariant features +to reduce the shift between source and target domains. While such adver- +sarial approaches achieve domain-level alignment, they ignore the class +(label) shift. When class-conditional data distributions are significantly +different between the source and target domain, it can generate ambigu- +ous features near class boundaries that are more likely to be misclassi- +fied. In this work, we propose a two-stage model for domain adaptation +called Contrastive-adversarial Domain Adaptation (CDA). While the +adversarial component facilitates domain-level alignment, two-stage con- +trastive learning exploits class information to achieve higher intra-class +compactness across domains resulting in well-separated decision bound- +aries. Furthermore, the proposed contrastive framework is designed as a +plug-and-play module that can be easily embedded with existing adver- +sarial methods for domain adaptation. We conduct experiments on two +widely used benchmark datasets for domain adaptation, namely, Office- +31 and Digits-5, and demonstrate that CDA achieves state-of-the-art +results on both datasets. +Keywords: Adversarial Domain Adaptation, Contrastive Learning +1 +Introduction +Deep neural networks (DNNs) have significantly improved the state-of-the-art +in many machine learning problems [14]. When trained on large-scale labeled +datasets, DNNs can learn semantically meaningful features that can be used to +solve various downstream tasks such as object classification, detection and lan- +guage processing. [36][41]. However, DNNs need to be qualified with caveats [1] - +they are understood to be brittle and tend to generalize poorly to new datasets +[24][34]. Even a small shift compared to the training data can cause the deep +network to make spurious predictions on the target domain. This phenomenon is +known as domain shift [37][2], where the marginal probability distribution of the +underlying data changes across different datasets or domains. A typical solution +is to fine-tune a model trained on a sufficiently labeled dataset by leveraging +the limited number of labeled samples from the target dataset [10][23]. However, +arXiv:2301.03826v1 [cs.CV] 10 Jan 2023 + +2 +Yadav, N. et al. +A +B +(i) +(ii) +(i) +(ii) +(iv) +(iii) +misclassification near +decision boundary +intra-class compactness +~ well-separated +decision boundary +Previous Methods +Proposed Method +Source Domain: +Target Domain: +Fig. 1. Illustration of the improvements proposed by CDA for unsupervised domain +adaptation (UDA).(A) Existing adversarial methods for UDA align the source and +target domain only at the domain level ignoring class boundaries. (B) In comparison, +CDA achieves both domain and class-level alignment in a multi-step training regime. +In step 1, CDA performs supervised contrastive learning on the labeled source domain, +resulting in better intra-class compactness and well-separated decision boundaries for +the target domain to align. In the next step, adversarial learning leads to domain-level +alignment, while cross-domain contrastive learning pulls target samples to align with +similar samples from the source domain and pushes away dissimilar clusters. +in real-world problems it might be expensive, or in some instances impossible +[30], to collect sufficient labeled data in the intended (target) domain leaving the +fine-tuning or transferring process challenging to execute. +Learning a model that reduces the dataset shift between training and testing +distribution is known as domain adaptation [3]. When no labeled data is avail- +able in the target domain, it is called unsupervised domain adaptation (UDA) +[15][35], which is the focus of this work. While the earliest domain adaptation +methods worked with fixed feature representations, recent advances in deep do- +main adaptation (DDA) embed domain adaptation modules within deep learn- +ing architectures. Thus, domain adaptation and features learning are achieved +simultaneously (end-to-end) in a single training process. One of the most well- +known approaches to DDA is the use of adversarial learning for reducing the +discrepancy between the source and target domain [16][31][25][22]. Adversarial +domain adaptation (ADA) approaches domain adaptation as a minimax game +similar to how Generative Adversarial Networks (GANs) [12] work. An auxiliary +domain discriminator is trained to distinguish latent feature embeddings from +source and target domains. At the same time, a deep neural network learns fea- +ture representations that are indistinguishable by the domain discriminator. In +other words, the deep network, comprising a generator and a dense head, and the +domain discriminator try to fool each other, resulting in latent features that can- + +CDA: Contrastive-adversarial Domain Adaptation +3 +G +D +C +Generator +Discriminator +Contrastive +Module +LCE +LSupCL +Source +Domain +Class 1 S1 +Class 1 S2 +Class 2 S1 +Class 2 S2 +Target +Domain +G +D +C +Discriminator +LCE +LAdv +LCrossCL +Source +Target +Class 1 +Class 2 +Class 1 +Class 2 +A ++ve ++ve ++ve ++ve +B +Stage I +Stage II +Fig. 2. An overview of the two-stage CDA framework. In stage-I (A), we perform +supervised contrastive learning (CL) using the labeled source dataset. The motivation +is to achieve better intra-class compactness and well-separated decision boundaries to +make class-level alignment in stage-II (B) easier to perform. Stage-II is where the actual +domain adaptation (DA) occurs using a combination of adversarial and cross-domain +contrastive loss. The overall CDA objective function comprises multiple losses that are +optimized in tandem to achieve DA. For a detailed explanation, see section 3.(figure +best viewed in color). +not be distinguished by which domain they come from. Although ADA achieves +domain-level alignment, it fails to capture the multimodal structure within a +specific domain’s data distribution [33][40]. Even if a domain discriminator is +fully confused, there is no guarantee for class-level alignment. In scenarios where +class-conditional distributions across domains are significantly different, ADA +can generate ambiguous features near class boundaries that are more likely to +be misclassified (see Figure 1) [7] . Some of the recent works have tried to tackle +the problem of class-level alignment via training separate domain discriminators +[25] [32]; however, it gives rise to convergence issues amidst a lack of equilib- +rium guarantee. Other works directly encode class information in the domain +adaptation module [22][9]. +In this work, we propose a novel two-stage domain adaptation mechanism +called Contrastive-adversarial Domain Adaptation (CDA). CDA leverages the +mechanism of contrastive learning [20][26] for achieving class-level alignment +in tandem with adversarial learning which focuses on domain-level alignment. +The idea of contrastive learning is to learn an embedding space where simi- +lar data samples - and corresponding features - lie close to each other while +dissimilar samples are pushed away. Although contrastive learning has been +most successfully used in self-supervised learning [8][17][6] tasks, the underly- +ing idea can be exploited to solve domain adaptation. The contrastive mod- +ule improves intra-class compactness (stage-I) and class-conditioned alignment +(stage-II), while ADA focuses on the overall domain-level alignment. The ex- +pected outcome is a more tightly coupled domain alignment that is class-aware. +We conduct experiments on two benchmark datasets for UDA (Office-31 and +Digits-5) to demonstrate that CDA achieves state-of-the-art results. + +4 +Yadav, N. et al. +1.1 +Contributions +The key contributions of this work can be summarized as follows: +– We propose a novel two-stage deep domain adaptation method (CDA) that +combines contrastive and adversarial approaches for unsupervised domain +adaptation (UDA). +– Experiments show the efficacy of our proposed methods by achieving state- +of-the-art results on well-known benchmarks datasets for UDA. +– The proposed contrastive module can be easily embedded within existing +adversarial domain adaptation methods for improved performance. +2 +Related Work +2.1 +Unsupervised Domain Adaptation (UDA) +The central idea of UDA is to learn domain-invariant feature representations. +While the earliest (shallow) approaches worked with fixed features, the current +methods combine the expressiveness of deep neural networks with domains adap- +tation for end-to-end learning [15][23][7]. There is extensive literature on deep +domain adaptation methods ranging from moment matching to more recent ad- +versarial approaches. Both approaches aim to minimize the discrepancy between +the source and target domain. While moment matching methods explicitly min- +imize the difference using a loss function such as Maximum Mean Discrepancy +(MMD) [21][23], adversarial methods seek to reduce the discrepancy using an ad- +versarial objective which pits two networks against each other - a generator and +a discriminator. For domain adaptation, the generator’s goal is to produce latent +features the domain discriminator cannot classify correctly. Doing so generates +domain-invariant feature representation, i.e., the target domain gets aligned with +the source domain. A common criticism of the earliest ADA methods was that +they only result in domain-level alignment and ignore class-specific distributions. +Recent works have built on the seminal work of Ganin et al. [15] in the context +of ADA - they attempt to incorporate class-level information in the model for +achieving a more tightly-coupled alignment across domains [22][9][25]. +2.2 +Contrastive Learning +Contrastive learning (CL) has achieved state-of-the-art results in self-supervised +representation learning [8][17]. The goal of CL is to learn a model where features +representations of similar samples lie close to each other in the latent space, and +dissimilar samples lie further apart. In the absence of labels, an augmented ver- +sion corresponding to a sample is generated to create a positive (similar) pair. +The other samples in the training minibatch become negative pairs. Entropy- +based loss functions that simultaneously maximize the similarity of positive pairs +and minimize the similarity of negative pairs are used. Recent works [6] have + +CDA: Contrastive-adversarial Domain Adaptation +5 +shown how contrastive learning can learn semantically meaningful feature rep- +resentations that can be used to solve various downstream tasks, and can even +outperform supervised tasks solved in supervised settings [5]. +2.3 +Contrastive Learning for UDA +Recent works have applied the core principle of CL to domain adaptation tasks. +Carlucci et al. [4] used a pretext task (solving jigsaw puzzle) for self-supervision +to solve domain adaptation. Kim et al. [19] proposed cross-domain self-supervised +learning and extended by Yue et al.[38] to align cluster-based class prototypes +across domains for few-shot learning. Singh et al. [29] used CL with strongly +augmented pairs to reduce the intra-domain discrepancy. Picking the appropri- +ate augmentations for CL is heuristic and may not generalize to other datasets +with the same model. We avoid data augmentation using a two-stage CL ap- +proach. To the best of our knowledge, this is the first work that systematically +integrates CL with adversarial methods for the problem of unsupervised domain +adaptation. +3 +Contrastive-Adversarial Domain Adaptation +3.1 +Problem Formulation +In UDA, we aim to transfer a model learned on a labeled source domain to an +unlabeled target domain. We assume that the marginal probability distributions +of the two domains are not equal, i.e., P(Xs) ̸= P(Xt). We are given a labeled +source dataset Ds = (Xs, Ys) = {(xi +s, yi +s)}ns +i=1 and an unlabeled dataset in the +target domain Dt = Xt = {xi +t}nt +i=1 with ns and nt samples, respectively. Both +{xi +s} and {xi +t} belong to the same set of N classes with P(Xs) ̸= P(Xt). The +goal is to predict labels for test samples in the target domain using the model +(G, C) : Xt → Yt trained on Ds ∪ Dt. The trained model includes a feature +generator G : Xt → Rd and a classifier C : Rd → RN, where d is the dimension +of the intermediate features produced by the generator. +3.2 +Model Overview +CDA is a two-stage model for with three major components - a feature generator +G, a classifier C, and an auxiliary domain classifier D (Figure 2). Further, a +contrastive module is spaced between G and C. Broadly, there are two objectives +achieved by the CDA model: 1) domain-level alignment using adversarial learning +and 2) class-level alignment using contrastive learning. The following sections +describe the mechanism of each objective in detail. + +6 +Yadav, N. et al. +Algorithm 1: Contrastive-adversarial Domain Adaptation +Input +: labeled source dataset Ds = {Xs, Ys}, unlabeled target dataset +Dt = {Xt}, max epochs E, iterations per epoch K, model (C, D, G) +Output: trained model (G, C) +for e = 1 to E do +for k = 1 to K do +Sample batch {xs, ys} from Ds and compute LSupCL + LCE using +Eqn. 3 +if e ≥ E′ then +Sample batch {xt} from Dt and compute LAdv using Eqn. 1 +if e ≥ E′′ then +LSupCL = 0 +Generate pseudo-labels yt and compute LCrossCL using Eqn. 5 +end +end +Compute LT otal using Eqn. 7 +Backpropagate and update C, D and G +end +end +3.3 +Domain-Level Adversarial Learning +Adversarial learning aims to learn domain-invariant features by training the +feature generator G and domain discriminator D with competing (minimax) ob- +jectives. The adversarial component is adapted from the seminal work of Ganin +et al. (DANN) [15] that originally proposed the idea. As a first step in the zero- +sum game, G takes the labeled source and unlabeled target domain inputs and +generates feature embeddings zs and zt. In the next step, D takes the feature +embeddings and attempts to classify them as either coming from the source or +target domain. The goal of G is to fool the discriminator such that output feature +embeddings cannot be classified correctly by D. It is achieved by training D and +G with an adversarial loss LAdv with gradient reversal (for G). For a given source +sample xs ∼ Xs and target sample xt ∼ Xt, LAdv can be formulated as a binary +cross-entropy loss: +LAdv(Xs, Xt) = +� +xs∼Xs +xt∼Xt +(log (D (G (xt))) + log (1 − D (G (xs)))) +(1) +with the following objective, +min +G max +D (LAdv) +(2) +In other words, G tries to minimize LAdv while D learns to maximize it. The +theoretical argument is that convergence will result in domain-invariant feature + +CDA: Contrastive-adversarial Domain Adaptation +7 +embeddings. However, such an adversarial approach only results in domain-level +alignment without considering the complex multi-mode class distribution present +in the source and target domain. Even when the domain discriminator is fully +confused, there is no guarantee the classifier can successfully discriminate target +samples based on the class labels. The absence of class-level alignment results in +under-transfer or negative transfer when the class-conditional distributions are +significantly different across the two domains. +3.4 +Class-Discriminative Contrastive Learning +To generate feature embeddings that are not domain-invariant but also class- +discriminative across the two domains, CDA proposes a constrastive learning- +based (CL) module. For clarification, the CL module is not a neural network +per se. It is an intermediary component that links G, D, and C and where the +proposed two-stage contrastive objective is optimized. +Stage I: The CL module performs supervised contrastive learning on the +source domain. In every batch, samples from the same class are considered posi- +tive pairs, while samples from different classes are automatically assigned as neg- +ative pairs. Training progresses by optimizing a modified InfoNCE loss [8] where +NCE stands for Noise-contrastive Estimation (see Eq. ). Although CL is best as- +sociated with self-supervised representation learning, recent works (Khosla et al. +[18]) have shown that minimizing a contrastive loss can outperform the standard +cross-entropy loss for supervised classification tasks. The idea is that clusters of +samples belonging to the same class are pulled together in the embedding space +while simultaneously pushing apart clusters of samples from different classes cre- +ating well-separated decision boundaries for better aligning the target domain +samples in the next step. The combined objective function during stage-I is as +follows: +LStageI = LSupCL + LCE +(3) +LSupCL(Xs, Ys) = − +� +z,z+∈Ds +log +exp(z⊺z+/τ) +exp(z⊺z+/τ) + � +z−∈Ds exp(z⊺z−/τ) +(4) +where, LCE is the standard cross-entropy loss for multiclass classification. +LSupCL is the supervised contrastive loss applied to samples from the labeled +source domain. The variable zs denote the l2 normalized latent embedding gen- +erated by G corresponding to the input sample xs. The variable τ refers to the +temperature scaling (hyperparameter) which affects how the model learns from +hard negatives [11]. +Stage II: For class-level alignment, CDA performs cross-domain contrastive +learning. It is based on the understanding that samples belonging to the same + +8 +Yadav, N. et al. +class across the two domains should cluster together in the latent embedding +space. Unlike supervised CL in stage-I, samples from the same class across do- +mains are considered positive pairs, and samples from different classes become +the negative pairs. However, we need labels for the target domain which are not +available. Some of the current methods in this space generate pseudo-labels us- +ing k-means clustering [29]. Clustering on the source domain is either performed +once during preprocessing or performed every few epochs during training, and +target labels are assigned based on the nearest cluster centroid. We argue that +both approaches are sub-optimal and propose making target label generation +part of the training process itself without the need to perform clustering. +LCrossCL(Xs, Ys, Xt) = − +N +� +i=1 +zs∈Ds +zt∈Dt +log +exp(zi +s +⊺zi +t/τ) +exp(zis +⊺zi +t/τ) + �N +i̸=k=1 exp(zis +⊺zk +t /τ) +(5) +where, LCrossCL is the cross-domain contrastive loss in stage-II. zs and zt +are the l2 normalized embeddings from the source and target, respectively. The +superscript i and k are used to identify the class labels (pesudo labels in case of +target domain). +3.5 +CDA: Overall Framework +In CDA, we take a multi-step approach to optimize multiple objective functions +during training. In the first stage, we train only on the source domain for the +first E′ epochs (hyperparameter) to ensure the model reaches a certain level of +classification accuracy. +Next, we initiate the process for domain-level alignment as described above. +We add LAdv to the overall objective function using a time-varying weighting +scheme lambda. Once we have achieved well-separated clustering in the source +domain and some level of domain alignment, we gradually introduce the last +loss function LCrossCL. The (pseudo) target labels are obtained by executing a +forward pass on the model (G, C): yt = argmax(C(G(xt))). Some target samples +are expected to be misclassified initially, but as the training continues and target +samples get aligned, decision boundaries will get updated accordingly, and model +performance will improve with each iteration. LCrossCL pulls same-class clusters +in the two domains closer to each other and pushes different clusters further +apart. Finally, we also employ a standard cross-entropy loss function LCE during +the entire training process to keep track of the classification task. The overall +training objective can be formulated as follows: +LT otal = LStage1 + LStage2 +(6) +LT otal = LSupCL + LCE + λ ∗ LAdv + β ∗ LCrossCL +(7) + +CDA: Contrastive-adversarial Domain Adaptation +9 +with +λ = +� +0 +for epoch 0 ≤ e < E′ +2 +1+exp−γp − 1 +for epoch e ≥ E′ +(8) +and +β = +� +0 +for epoch e ≤ E′′ +min(1, α ∗ +� +e−E′′ +E′′ +� +) +for epoch E′′ < e ≤ E +(9) +where, E′ and E′′ (with E′′ ≥ E′) indicate the epochs when Stage-I ends and +LCrossCL is introduced in the objective function, repectively. At any given epoch, +only one type of contrastive learning is performed, i.e. for e ≥ E′′, LSupCL = 0 +(see Algorithm 1). The scaling variables λ and β (hyperparameters) control the +rate at which LAdv and LCrossCL are added to the overall objective function to +maintain the stability of the training process. The values of α and β increase +from 0 to 1. +4 +Experiments +4.1 +Datasets +We use two public benchmarks datasets to evaluate our method: +Office-31 is a common UDA benchmark that contains 4,110 images from +three distinct domains – Amazon (A with 2,817 images), DSLR (D with 498 +images) and Webcam (W with 795 images). Each domain consists of 31 object +classes. Our method is evaluated by performing UDA on each pair of domains, +which generates 6 different tasks (Table 1). +Digits-5 comprises a set of five datasets of digits 0-9 (MNIST, MNIST-M, +USPS, SVHN and Synthetic-Digits) most commonly used to evaluate domain +adaptation models. We use four of the five datasets and generate 3 different +tasks (Table 2). Both MNIST and MNIST-M contain 60,000 and 10,000 samples +for training and testing respectively. SVHN is a more complex real-world image +dataset with 73,257 samples for training and 26,032 samples for testing. The +digits in SVHN are captured from house numbers in Google Street View images. +SVHN has an additional class for the digit ’10’ which is ignored to match the +label range of other datasets. Finally, USPS is smaller dataset with 7,291 training +and 2,007 testing samples. We use all the available training samples for each task. +4.2 +Baselines +We compare the performance of CDA with the following well-known method +(a) DANN, which originally proposed the idea of adversarial learning for do- + +10 +Yadav, N. et al. +Table 1. Classification Accuracy on Office-31 Dataset +Method +A → D A → W D → A D → W W → A W → D Avg. +DANN [15] +79.5 +81.8 +65.2 +96.4 +63.2 +99.1 +80.8 +MADA [25] +87.8 +90.0 +70.3 +97.4 +66.4 +99.6 +85.2 +iCAN [39] +90.1 +92.5 +72.1 +98.8 +69.6 +100 +87.2 +CDAN [22] +91.7 +93.1 +71.3 +98.6 +69.3 +100 +87.3 +CDAN+BSP [9] +93.0 +93.3 +73.6 +98.2 +72.6 +100 +88.4 +GTA [28] +87.7 +89.5 +72.8 +97.9 +71.4 +99.8 +86.5 +GVB [13] +95.0 +94.8 +73.4 +98.7 +73.7 +100 +89.3 +CDA (ours) +93.6 +94.0 +74.7 +98.6 +78.9 +100 +89.9 +Table 2. Classification Accuracy on Digits-5 Dataset +Method +MNIST → MNIST-M MNIST → USPS SVHN → MNIST +DANN [15] +84.1 +90.8 +81.9 +ADDA [31] +- +89.4 +76.0 +CDAN [22] +- +95.6 +89.2 +CDAN+BSP [9] +- +95.0 +92.1 +MCD [27] +- +96.5 +96.2 +CDA (ours) +96.6 +97.4 +96.8 +* Best accuracy shown in bold and the second best as underlined. +main adaptation, and state-of-the-art methods that go beyond just domain- +level alignment - (b) MADA and (c) iCAN, which use multiple domain dis- +criminators to capture the multimode structures in the data distribution; (d) +CDAN and (e) CDAN+BSP, which condition the domain discriminator on +class-discriminative information obtained from the classifier; (f) GTA, which +proposes an adversarial image generation approach to directly learn the shared +feature embeddings; (g) GVB, which proposes a gradually vanishing bridge +mechanism for adversarial-based domain adaptation; (h) ADDA, which uses a +separate discriminative loss in addition to the adversarial loss to facilitate class- +level alignment; (i) MCD, which uses task-specific classifiers and maximizes the +discrepancy between them. +4.3 +Implementation Details +Network Architecture: We use a ResNet-50 model pre-trained on ImageNet +as the feature generator G. The last fully-connected (FC) layer in ResNet-50 +is replaced with a new FC layer to match the dimensions of the intermediate +feature embedding. Both the classifier C and domain discriminator D are three- +layer dense networks (512 → 256 → 128) with output dimensions of 10 (for 10 +classes) and 1 (for identifying the domain), respectively. + +CDA: Contrastive-adversarial Domain Adaptation +11 +DANN +CDA +Fig. 3. t-SNE visualizations for DANN and CDA to extract the contribution of the +proposed contrastive module in learning domain-invariant yet class-discriminative em- +beddings. The analysis is for the MNIST (source) → MNIST-M (target) experiment. +Each color represents one of the digits (0-9). (best viewed in color). +Training Details The CDA network is trained using the AdamW optimizer +with a batch size of 32 and 128 for the Office-31 and Digits-5 dataset, respectively. +The initial learning rate is set as 5e − 4; a learning rate scheduler is used with +a step decay of 0.8 every 20 epochs. We use one NVIDIA V100 GPU for the +experiments. For detailed discussion, see the supplementary material. +4.4 +Results +The results on the Office-31 and Digits-5 datasets are reported in Table 1 and +2, respectively. Our proposed method outperforms several baselines across dif- +ferent UDA tasks. Moreover, CDA achieves the best average accuracies on both +datasets. Where CDA is unable to better state-of-the-art accuracy, it reports +comparable results with the best accuracy score. A direct comparison can be +made with DANN (see section 4.5), with which it shares the same adversarial +component, to highlight the effectiveness of the contrastive module. On average, +CDA improves the accuracy on Office-31 and Digits-5 by approximately 9% and +11%, respectively, compared to DANN. Furthermore, CDA significantly outper- +forms two well-known approaches - MADA and CDAN - that also explicitly align +domains at class-level. +4.5 +Ablation Study +To tease out the individual contribution of the contrastive module in CDA, we +perform a comparative analysis between CDA and DANN as they both share +the same adversarial learning component. We plot the t-SNE embeddings corre- +sponding to the last layer in the respective classifiers (of DANN and CDA) for +the MNIST to MNIST-M task (Figure 3). It can be seen that the contrastive +module improves the adaptation performance. For DANN, although the source +and target domain align with each other, labels are not well discriminated. The +reason is that the original DANN approach does not consider class-discriminative + +12 +Yadav, N. et al. +information and only aligns at the domain level. As a result, feature embeddings +near the class boundaries are prone to be misclassified, resulting in a lower classi- +fication accuracy on the target domain, as can be seen in case of DANN in Table +2. For CDA, the contrastive module first increases the inter-class separation in +the source domain. It then aligns samples belonging to the same class across +domains close to each other - leading to well-separated decision boundaries and +improved classification accuracy. We conclude that with minimal tweaks to the +training process, the proposed contrastive module in CDA can be embedded in +existing adversarial methods for UDA for improved performance. +5 +Conclusion +This paper proposes a new method for unsupervised domain adaptation (UDA) +called Contrastive-adversarial Domain Adaptation (CDA). CDA improves upon +existing adversarial methods for UDA by using a simple two-stage contrastive +learning module that achieves well-separated class-level alignment in addition to +the domain-level alignment achieved by adversarial approaches. CDA achieves +this end-to-end in a single training regime unlike some of the existing approaches. +Furthermore, the contrastive module is proposed as a standalone component that +can be embedded with existing adversarial methods for UDA. Our proposed +method achieves better performance than several state-of-the-art methods on +two benchmark datasets, demonstrating the effectiveness of our approach. Lastly, +this work further motivates an emerging research area exploring the synergy +between contrastive learning and domain adaptation. + +CDA: Contrastive-adversarial Domain Adaptation +13 +References +1. Belkin, M., Hsu, D., Ma, S., Mandal, S.: Reconciling modern machine-learning +practice and the classical bias–variance trade-off. 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In addition +to the variables presented, we use a learning rate scheduler with a step decay +of 0.8 every 10 (20) epochs for training CDA on Office-31 (Digits-5). We also a +dropout value of 0.2-0.5 in the second to last dense layer in the classifier C and +domain discriminator D. +Table 3. Hyperparameters +Variable +Description +Value +lr +Learning Rate +5e-4 +bs +Batch Size +32, 128* +τ +Temperature scaling in contrastive loss +0.5 +E +Total no. of training epochs +90, 200* +E′ +No. epochs when stage-I of training ends +25, 40* +E′′ +No. of epochs when LCrossCL is added to +the overall objective function +35, 60* +λ +Scaling factor for adverarial loss LAdv +varies+ +β +Scaling factor for LCrossCL +varies+ +* The first values corresponds to the Office-31 dataset and second value is for +experiments involving Digits-5. ++ The values increase from 0-1 using Eq. 8 and 9 (main text) to vary E′ and +E′′. +2 +Training Process +We use the AdamW optimizer with a learning rate scheduler and a multi-step +objective function for training CDA. For the first E′ epochs, we only optimize +the supervised contrastive loss and the standard cross-entropy loss (for the clas- +sification task of interest) on the labeled source domain. The loss L till E′ is: +Le 0, r > 0, ядра Ψβ(t) вигляду (2) є узагальненими +ядрами Пуассона, тобто Ψβ(t) = Pα,r,β(t), де +Pα,r,β(t) = +∞ +� +k=1 +ψ(k) cos +� +kt − βπ +2 +� +, α > 0, r > 0, β ∈ R. +(4) +При цьому множини Cψ +β L1 та Cψ +β,1 позначатимемо вiдповiдно через Cα,r +β L1 та Cα,r +β,1 i на- +зиватемо множинами узагальнених iнтегралiв Пуассона, а вiдповiднi (ψ, β)–похiднi f ψ +β та +(ψ, β)–iнтеграли J ψ +β ϕ позначатимемо через f α,r +β +та J α,r +β +ϕ вiдповiдно. + +3 +Простiр усiх тригонометричних полiномiв tn−1 порядку не вищого за n − 1 будемо по- +значати через T2n−1. Нехай En(f)L1 — найкращi наближення в середньому тригонометри- +чними полiномами tn−1 ∈ T2n−1, тобто +En(f)L1 = +inf +tn−1∈T2n−1 ∥f − tn−1∥1. +Позначимо через ρn(f; x) вiдхилення вiд функцiї f з L1 її частинної суми Фур’є Sn−1(f; ·) +порядку n − 1 +ρn(f; x) := f(x) − Sn−1(f; x). +(5) +Норми ∥ρn(f; ·)∥C можна оцiнити зверху через найкращi рiвномiрнi наближення +En(f)C = +inf +tn−1∈T2n−1 ∥f − tn−1∥C за допомогою нерiвностi Лебега +∥ρn(f; ·)∥C ≤ (1 + Ln−1)En(f)C, n ∈ N, f ∈ C, +(6) +де величини Ln−1 — константи Лебега сум Фур’є +Ln−1 = 1 +π +π +� +−π +|Dn−1(t)|dt = 2 +π +π +2 +� +0 +| sin(2n − 1)t| +sin t +dt, +Dn−1(t) := 1 +2 + +∞ +� +k=1 +cos kt = sin(n − 1 +2)t +2 sin t +2 +. +При цьому, як встановив Фейєр [3], для констант Лебега Ln має мiсце асимптотична +рiвнiсть +Ln = 4 +π2 ln n + O(1), +n → ∞, +(7) +де O(1) — рiвномiрно обмежена по n величина. +Бiльш точнi оцiнки для рiзниць Ln − +4 +π2 ln(n + a), a > 0, при n ∈ N можна знайти в +роботах [1], [2], [4], [10], [23], [5] та iн. +З урахуванням (7), нерiвнiсть (6) можна записати у виглядi +∥ρn(f; ·)∥C ≤ +� 4 +π2 ln n + O(1) +� +En(f)C, +f ∈ C. +(8) +Незважаючи на загальнiсть, нерiвнiсть (8) на всьому просторi C є точною за порядком. +Бiльш того, вона є асимптотично непокращуваною в тому сенсi, що константа +4 +π2 у формулi +(8) зменшена бути не може. +Разом з тим, використання нерiвностей (6) i (8) для функцiй f iз функцiональних +множин Cψ +β L1 чи Cψ +β,1 може виявитись неефективним. Бiльш того, iснують послiдовностi ψ +такi, що для f ∈ Cψ +β L1 зазначенi нерiвностi є неточними навiть за порядком. Щоб у цьому +переконатись, покладемо ψ(k) = e−αk i розглянемо породженi такими послiдовностями + +4 +класи Cψ +β,1 = Cα,1 +β,1. Як показано в [12] при всiх α > 0, β ∈ R справедлива асимптотична +при n → ∞ рiвнiсть +En(Cα,1 +β,1)C = sup +f∈Cα,1 +β,1 +∥ρn(f; ·)∥C = e−αn +� +1 +π(1 − e−α) + O(1) +n +e−α +(1 − e−α)2 +� +, +(9) +в якiй O(1) — рiвномiрно обмежена по α, β, n величина. +Крiм того, (див., наприклад, [27, c. 48]) для найкращих наближень En(Cα,1 +β,1)C справе- +дливi точнi за порядком оцiнки +K(1)e−αn ≤ En(Cα,1 +β,1)C ≤ K(2)e−αn, +(10) +в яких K(1) i K(2) — деякi додатнi сталi. +Тодi для f ∈ Cα,1 +β,1 в силу (9) виконується нерiвнiсть +∥ρn(f; ·)∥C ≤ e−αn +� +1 +π(1 − e−α) + O(1) +n +e−α +n(1 − e−α)2 +� +, +(11) +в той час як використання класичної нерiвностi Лебега та оцiнки (8) дозволяє записати +бiльш грубу за порядком оцiнку +∥ρn(f; ·)∥C ≤ e−αn +�4K(2) +π2 +ln n + O(1) +� +. +У роботi [28] О.I. Степанець, розглядаючи послiдовностi ψ(k), що спадають до нуля +повiльнiше за будь-яку геометричну прогресiю, встановив аналоги нерiвностей Лебега для +множин (ψ, β)–диференцiйовних функцiй Cψ +β C ⊂ Cψ +β L1, (Cψ +β C — множина 2π–перiодичних +функцiй f(x), якi при всiх x ∈ R зображуються у виглядi (1), де ϕ ∈ C) в яких норми +вiдхилень ∥ρn(f; ·)∥C виражаються через найкращi наближення En(f ψ +β )C. Одержанi в [28] +нерiвностi виявились асимптотично точними не тiльки на всiх множинах Cψ +β C, але i на де- +яких важливих пiдмножинах iз Cψ +β C, зокрема на класах Cψ +β C0 = +� +f ∈ Cψ +β C : ∥f ψ +β ∥C ≤ 1 +� +. +Згодом дослiдження по встановленню асимптотично точних нерiвностей типу Лебега на +множинах (ψ, β)–диференцiйовних функцiй були продовженi в роботах [29], [8], [9], [14], +[30], [21], [22]. +В данiй роботi буде знайдено нерiвностi типу Лебега на множинах Cψ +β L1, в яких норми +∥ρn(f; x)∥C виражаються через En(f ψ +β )L1 i доведено їх асимптотичну непокращуванiсть у +випадку, коли +lim +n→∞ +1 +n +∞ +� +k=1 +kψ(k + n) +∞ +� +k=n +ψ(k) += 0. +(12) +Також у роботi, за умови (12) знайдено розв’язок задачi Колмогорова–Нiкольського +для сум Фур’є на класах Cψ +β,1, яка полягає у вiдшуканнi асимптотичних рiвностей величин +En(Cψ +β,1)C = sup +f∈Cψ +β,1 +∥f(·) − Sn−1(f; ·)∥C. +(13) + +5 +Проблеми, пов’язанi зi знаходженням розв’язку задачi Колмогорова–Нiкольського для +сум Фур’є на класах згорток дослiджувались в роботах [6], [11], [33], [35], [36], [24], [26], +[12], [13], [20], [34], [15], [16] та iн. +2 +Основнi результати +Має мiсце наступне твердження. +Теорема 2.1. Нехай +∞ +� +k=1 +kψ(k) < ∞, ψ(k) ≥ 0, k = 1, 2, ..., β ∈ R i n ∈ N. Тодi для +довiльної функцiї f ∈ Cψ +β L1 має мiсце нерiвнiсть +∥f(·) − Sn−1(f; ·)∥C ≤ 1 +π +∞ +� +k=n +ψ(k)En(f ψ +β )L1. +(14) +Крiм того, для довiльної функцiї f ∈ Cψ +β L1 можна знайти функцiю F(x) = F(f; n, x) +з множини Cψ +β L1 таку, що En(F ψ +β )L1 = En(f ψ +β )L1 i має мiсце рiвнiсть +∥F(·) − Sn−1(F; ·)∥C = +� +1 +π +∞ +� +k=n +ψ(k) + ξ +n +∞ +� +k=1 +kψ(k + n) +� +En(f ψ +β )L1. +(15) +В (15) величина ξ = ξ(f; n; ψ; β) є такою, що −2 ≤ ξ ≤ 0. +Доведення Теореми 2.1. Нехай f ∈ Cψ +β L1. Тодi згiдно з (1) в кожнiй точцi x ∈ R має мiсце +iнтегральне представлення +ρn(f; x) = f(x) − Sn−1(f; x) = 1 +π +π +� +−π +f ψ +β (t)Ψβ,n(x − t)dt, +(16) +де +Ψβ,n(t) := +∞ +� +k=n +ψ(k) cos +� +kt − βπ +2 +� +, β ∈ R. +(17) +Функцiя Ψβ,n(t) ортогональна до будь–якого тригонометричного полiнома tn−1 порядку +не вищого за n − 1. Тодi. для довiльного полiнома tn−1 ∈ T2n−1 отримаємо +ρn(f; x) = 1 +π +π +� +−π +δn(t)Ψβ,n(x − t)dt, +(18) +де +δn(·) = δn(ψ, β, tn−1; ·) := f ψ +β (·) − tn−1(·). +(19) +Виберемо в якостi tn−1 в формулi (18) полiном t∗ +n−1 найкращого наближення функцiї +f ψ +β в просторi L1, тобто такий що +∥f ψ +β − t∗ +n−1∥1 = En(f ψ +β )L1, + +6 +Тодi, використовуючи нерiвнiсть +���� +π +� +−π +K(t − u)ϕ(u)du +���� +C +≤ ∥K∥p′∥ϕ∥p, +(20) +ϕ ∈ Lp, +K ∈ Lp′, +1 ≤ p ≤ ∞, +1 +p + 1 +p′ = 1 +(див., наприклад, [7, c. 43]), отримуємо +∥f(·) − Sn−1(f; ·)∥C ≤ +������ +1 +π +π +� +−π +(f ψ +β (t) − t∗ +n−1(t))Ψβ,n(· − t)dt +������ +C +≤ 1 +π∥Ψβ,n∥CEn(f ψ +β )L1. +(21) +Знайдемо двостороннi оцiнки норми ∥Ψβ,n∥C. Покажемо, що при всiх n ∈ N i β ∈ R +справедлива оцiнка +∞ +� +k=n +ψ(k) − π +n +∞ +� +k=1 +kψ(k + n) ≤ ∥Ψβ,n∥C ≤ +∞ +� +k=n +ψ(k). +(22) +Оцiнка зверху для ∥Ψβ,n∥C в (22) випливає безпосередньо з (17). +Для оцiнки ∥Ψβ,n∥C знизу представимо функцiю Ψβ,n(t), яка означена формулою (17), +у виглядi +Ψβ,n(t) = gψ,n(t) cos +� +nt − βπ +2 +� ++ hψ,n(t) sin +� +nt − βπ +2 +� +, +(23) +де +gψ,n(t) := +∞ +� +k=0 +ψ(k + n) cos kt, +(24) +hψ,n(t) := − +∞ +� +k=0 +ψ(k + n) sin kt. +(25) +Оскiльки величина ∥Ψβ,n∥C перiодична з перiодом 2 за параметром β, то, не зменшуючи +загальностi, можна вважати, що β ∈ [0, 2]. +Позначимо +t0 := βπ +2n , +β ∈ [0, 2]. +(26) +В силу (23) +Ψβ,n(t0) = gψ,n(t0). +(27) +Тодi +∥Ψβ,n∥C ≥ |Ψβ,n(t0)| = |gψ,n(t0)| = |gψ,n(0) + (gψ,n(t0) − gψ,n(0))| +≥ |gψ,n(0)| − +����(gψ,n +�βπ +2n +� +− gψ,n(0)) +���� = +∞ +� +k=n +ψ(k) − +����(gψ,n +�βπ +2n +� +− gψ,n(0)) +���� . +(28) + +7 +Використовуючи теорему про середнє, маємо +����gψ,n +�βπ +2n +� +− gψ,n(0) +���� ≤ ∥g +′ +ψ,n∥C +βπ +2n ≤ π +n +∞ +� +k=1 +kψ(k + n). +(29) +Iз (28) i (29) випливає шукана оцiнка знизу для норм ∥Ψβ,n∥C в спiввiдношеннi (22) +∥Ψβ,n∥C ≥ +∞ +� +k=n +ψ(k) − π +n +∞ +� +k=1 +kψ(k + n). +(30) +Оцiнку (22) можна записати у виглядi формули +∥Ψβ,n∥C = +∞ +� +k=n +ψ(k) + Θ1π +n +∞ +� +k=1 +kψ(k + n), +(31) +де для величини Θ1 = Θ1(n, β, ψ) виконуються нерiвностi +− 1 ≤ Θ1 ≤ 0. +(32) +Отже, iз (21) i (31) випливає нерiвнiсть (14) з Теореми 2.1. +Доведемо другу частину Теореми 2.1. Для цього нам необхiдно для довiльної функцiї +ϕ ∈ L1 знайти функцiю Φ(·) = Φ(ϕ, ·) ∈ L1 таку, що En(Φ)L1 = En(ϕ)L1 i для якої викону- +ється рiвнiсть +1 +π +������ +π +� +−π +� +Φ(t) − t∗ +n−1(t) +� +Ψβ,n(0 − t)dt +������ += +� +1 +π +∞ +� +k=n +ψ(k) + ξ +n +∞ +� +k=1 +kψ(k + n) +� +En(ϕ)L1, +(33) +де t∗ +n−1 — полiном найкращого наближення порядку n − 1 функцiї Φ в просторi L1. +В цьому випадку для функцiї f ∈ Cψ +β L1 iснує функцiя Φ(·) = Φ(f ψ +β ; ·) така, що En(Φ)L1 = +En(f ψ +β )L1, i має мiсце формула (33), де в ролi ϕ виступає функцiя f ψ +β . +Розглянемо функцiю +F(·) = J ψ +β (Φ(·) − a0 +2 ), +де +a0 = a0(Φ) := 1 +π +π +� +−π +Φ(t)dt. +Функцiя F є шуканою функцiєю, оскiльки F ∈ Cψ +β L1 i +En(F ψ +β )L1 = En(Φ − a0 +2 )L1 = En(Φ)L1 = En(f ψ +β )L1, +i на пiдставi (18) i (33) має мiсце оцiнка (15). + +8 +Доведемо (33). Нехай t∗ — точка з промiжку T = +� +π(1−β) +2n +, 2π + π(1−β) +2n +� +, в якiй функцiя +|Ψ−β,n(t)| набуває свого найбiльшого значення, тобто, +|Ψ−β,n(t∗)| = ∥Ψ−β,n∥C = ∥Ψβ,n∥C. +Покладемо ∆n +k := +� +(k−1)π +n ++ π(1−β) +2n +, kπ +n + π(1−β) +2n +� +, k = 1, ..., 2n. Через k∗ позначимо номер +такий, що t∗ ∈ ∆n +k∗. Оскiльки функцiя Ψ−β,n є абсолютно неперервною, то для довiльного +ε > 0 iснує сегмент ℓ∗ = [ξ∗, ξ∗ + δ] ⊂ ∆n +k∗ такий, що для довiльного t ∈ ℓ∗ виконується +нерiвнiсть |Ψβ,n(t)| > ∥Ψβ,n∥C − ε. Ясно, що mes ℓ∗ = |ℓ∗| = δ < π +n. +Для довiльного ϕ ∈ L1 i ε > 0 розглянемо функцiю Φε(t), яка на промiжку T означена +за допомогою рiвностей +Φε(t) = + + + +En(ϕ)L1 +1−ε(2π−δ) +δ +sign cos +� +nt + βπ +2 +� +, +t ∈ ℓ∗, +En(ϕ)L1ε sign cos +� +nt + βπ +2 +� +, +t ∈ T \ ℓ∗. +(34) +Для функцiї Φε(t) при достатньо малих значеннях ε > 0 (ε ∈ (0, 1 +2π)) має мiсце наступна +рiвнiсть: +∥Φε∥1 = En(ϕ)L1 +1 − ε(2π − δ) +δ +� +ℓ∗ +���sign cos +� +nt + βπ +2 +����dt ++ En(ϕ)L1ε +� +T\ℓ∗ +���sign cos +� +nt + βπ +2 +����dt += En(ϕ)L1 +�1 − ε(2π − δ) +δ +δ + ε(2π − δ) +� += En(ϕ)L1. +(35) +Крiм того, згiдно з (34) +sign Φε(t) = sign cos +� +nt + βπ +2 +� +. +(36) +Оскiльки для довiльного тригонометричного полiнома tn−1 ∈ T2n−1 +2π +� +0 +tn−1(t)sign cos +� +nt + βπ +2 +� +dt = 0, +то, з урахуванням (36), виконується рiвнiсть +2π +� +0 +tn−1(t)sign +� +Φε(t) − 0 +� +dt = 0, +tn−1 ∈ T2n−1. +Згiдно з Теоремою 1.4.5 роботи [7, с.28], полiном t∗ +n−1 ≡ 0 є полiномом найкращого +наближення функцiї Φε в метрицi простору L1, тобто, En(Φε)L1 = ∥Φε∥1, отже з (35) +випливає En(Φε)L1 = En(ϕ)L1. + +9 +Крiм того, для функцiї Φε +1 +π +π +� +−π +(Φε(t) − t∗ +n−1(t))Ψβ,n(−t)dt = 1 +π +π +� +−π +Φε(t)Ψ−β,n(t)dt +=1 − ε(2π − δ) +πδ +En(ϕ)L1 +� +ℓ∗ +sign cos +� +nt + βπ +2 +� +Ψ−β,n(t)dt ++ ε +πEn(ϕ)L1 +� +T\ℓ∗ +sign cos +� +nt + βπ +2 +� +Ψ−β,n(t)dt. +(37) +Враховуючи, що sign Φε(t) = (−1)k, t ∈ ∆(n) +k , k = 1, ..., 2n, а також вкладення ℓ∗ ⊂ ∆(n) +k∗ , +отримуємо +������ +1 − ε(2π − δ) +πδ +En(ϕ)L1 +� +ℓ∗ +sign cos +� +nt + βπ +2 +� +Ψ−β,n(t)dt +������ += +������ +(−1)k∗ 1 − ε(2π − δ) +πδ +En(ϕ)L1 +� +ℓ∗ +Ψ−β,n(t)dt +������ +≥1 − ε(2π − δ) +π +En(ϕ)L1 (∥Ψβ,n∥C − ε) +>1 − 2πε +π +En(ϕ)L1 (∥Ψβ,n∥C − ε) += 1 +πEn(ϕ)L1 +� +∥Ψβ,n|C − 2πε∥Ψβ,n∥C − ε + 2πε2� +>En(ϕ)L1 +� 1 +π∥Ψβ,n∥C − ε +� +2∥Ψβ,n∥C + 1 +π +�� +. +(38) +Крiм того, неважко переконатись, що +������� +ε +πEn(ϕ)L1 +� +T\ℓ∗ +sign cos +� +nt + βπ +2 +� +Ψ−β,n(t)dt +������� +≤ ε +πEn(ϕ)L1∥Ψβ,n∥C(2π − δ) < 2εEn(ϕ)L1∥Ψβ,n∥C. +(39) +З формул (37)–(39) випливає наступна оцiнка: +������ +π +� +−π +1 +π(Φε(t) − t∗ +n−1(t))Ψβ,n(−t)dt +������ +>En(ϕ)L1 +� 1 +π∥Ψβ,n∥C − ε +� +4∥Ψβ,n∥C + 1 +π +�� +. +(40) +Виберемо ε настiльки малим, щоб +ε < +π +∞ +� +k=1 +kψ(k + n) +n +� +1 + 4π +∞ +� +k=n +ψ(k) +� +(41) + +10 +i для цього ε покладемо +Φ(t) = Φε(t). +(42) +Функцiя Φ(t) є шуканою функцiєю, оскiльки En(Φ)L1 = En(ϕ)L1 i згiдно з (22), (40) i +(41) +������ +1 +π +π +� +−π +(Φ(t) − t∗ +n−1(t))Ψβ,n(−t)dt +������ +≥ +� +1 +π +∞ +� +k=n +ψ(k) − 1 +n +∞ +� +k=1 +kψ(k + n) − ε +� +4 +∞ +� +k=n +ψ(k) + 1 +π +�� +En(ϕ)L1 +≥ + + + + +1 +π +∞ +� +k=n +ψ(k) − 1 +n +∞ +� +k=1 +kψ(k + n) − +π +∞ +� +k=1 +kψ(k + n) +n +� +1 + 4π +∞ +� +k=n +ψ(k) +� +� +4 +∞ +� +k=n +ψ(k) + 1 +π +� + + + + En(ϕ)L1 +≥ +� +1 +π +∞ +� +k=n +ψ(k) − 2 +n +∞ +� +k=1 +kψ(k + n) +� +En(ϕ)L1. +(43) +З формул (43), (21) i (22) випливає (33). Теорему 2.1 доведено. +Зрозумiло, що формулу (14) теореми 2.1 можна записати однотипно з формулою (15) +у наступному виглядi: +∥f(·) − Sn−1(f; ·)∥C ≤ +� +1 +π +∞ +� +k=n +ψ(k) + Θ1 +n +∞ +� +k=1 +kψ(k + n) +� +En(f ψ +β )L1, +(44) +де Θ1 = Θ1(n, β, ψ) задовольняє нерiвностi (32). +Теорема 2.2. Нехай +∞ +� +k=1 +kψ(k) < ∞, ψ(k) ≥ 0, k = 1, 2, ... i β ∈ R. Тодi при усiх n ∈ N має +мiсце формула +En(Cψ +β,1)C = 1 +π +∞ +� +k=n +ψ(k) + Θ2 +n +∞ +� +k=1 +kψ(k + n), +(45) +де для величини Θ2 = Θ2(n, β, ψ) виконуються нерiвностi −1 ≤ Θ2 ≤ 0. +Доведення. Згiдно з (13) i (16) отримуємо, що +En(Cψ +β,1)C = 1 +π sup +ϕ∈B0 +1 +���� +π +� +−π +ϕ(t)Ψβ,n(x − t)dt +���� +C +, +(46) +де Ψβ,n(·) означена рiвнiстю (17), а B0 +1 := {ϕ ∈ L1 : ||ϕ||1 ≤ 1, ϕ ⊥ 1} . +Беручи до уваги iнварiантнiсть множини B0 +1 вiдносно зсуву аргументу, з (46) отримуємо +En(Cψ +β,1)C = 1 +π sup +ϕ∈B0 +1 +π +� +−π +ϕ(t)Ψβ,n(t)dt. +(47) + +11 +На основi спiввiдношення двоїстостi (див., напр., [7]) маємо +sup +ϕ∈B0 +1 +π +� +−π +Ψβ,n(t)ϕ(t)dt = inf +λ∈R ∥Ψβ,n(t) − λ∥C, +(48) +Для знаходження двосторонньої оцiнки величини inf +λ∈R ∥Ψβ,n(t) − λ∥C нам буде корисним +наступне твердження, яке може знайти i самостiйне застосування. +Лема 2.3. Нехай ψ(k) ≥ 0, +∞ +� +k=1 +kψ(k) < ∞. Тодi при всiх β ∈ R i n ∈ N для кожної з +величин +I(1) +n += I(1) +n (ψ, β) := ∥Ψβ,n∥C, +(49) +I(2) +n += I(2) +n (ψ, β) := inf +λ∈R ∥Ψβ,n(t) − λ∥C, +(50) +I(3) +n += I(3) +n (ψ, β) := 1 +2 +���Ψβ,n +� +t + π +n +� +− Ψβ,n(t) +��� +C +(51) +виконуються формули +I(j) +n += +∞ +� +k=n +ψ(k) + Θjπ +n +∞ +� +k=1 +kψ(k + n), +j = 1, 2, 3, +(52) +в яких для будь-якої з величин Θj = Θj(n, β, ψ), j = 1, 2, 3, виконуються двостороннi +оцiнки +−1 ≤ Θj ≤ 0, +j = 1, 2, 3. +Доведення Леми 2.3. Оскiльки +inf +λ∈R ∥Ψβ,n(t) − λ∥C ≤ ∥Ψβ,n∥C +(53) +i +1 +2 +���Ψβ,n +� +t + π +n +� +− Ψβ,n(t) +��� +C ≤ inf +λ∈R ∥Ψβ,n(t) − λ∥C, +(54) +то +I(3) +n +≤ I(2) +n +≤ I(1) +n , +i, отже, необхiдна оцiнка зверху для кожної з величин I(j) +n , j = 1, 2, 3 випливає з (22). +Залишається знайти оцiнку знизу для I(3) +n . В силу (23)–(25) i (51) + +12 +I(3) +n +=1 +2 +���Ψβ,n +� +t + π +n +� +− Ψβ,n(t) +��� +C +≥1 +2 +���Ψβ,n +� +t0 + π +n +� +− Ψβ,n(t0) +��� +=1 +2 +���gψ,n +� +t0 + π +n +� +cos +� +n +� +t0 + π +n +� +− βπ +2 +� ++ hψ,n +� +t0 + π +n +� +sin +� +n +� +t0 + π +n +� +− βπ +2 +� +− +� +gψ,n(t0) cos +� +nt0 − βπ +2 +� ++ hψ,n(t0) sin +� +nt0 − βπ +2 +�� ��� +=1 +2 +��� − gψ,n +� +t0 + π +n +� +− gψ,n(t0) +��� = 1 +2 +���gψ,n +�βπ − 2π +2n +� ++ gψ,n +�βπ +2n +���� +=1 +2 +����2gψ,n(0) + +� +gψ,n +�(β − 2)π +2n +� +− gψ,n(0) +� ++ +� +gψ,n +�βπ +2n +� +− gψ,n(0) +����� +≥ |gψ,n(0)| − 1 +2 +����gψ,n +�(β − 2)π +2n +� +− gψ,n(0) +���� − 1 +2 +����gψ,n +�βπ +2n +� +− gψ,n(0) +���� , +(55) +де, як i ранiше, t0 = βπ +2n, β ∈ [0, 2]. +За теоремою про середнє значення +����gψ,n +�(β − 2)π +2n +� +− gψ,n(0) +���� ≤ ∥g +′ +ψ,n∥C +|β − 2|π +2n +≤ π +n +∞ +� +k=1 +kψ(k + n). +(56) +Аналогiчно (див. (29)) +����gψ,n +�βπ +2n +� +− gψ,n(0) +���� ≤ π +n +∞ +� +k=1 +kψ(k + n). +(57) +Об’єднуючи (55)-(57), одержуємо шукану оцiнку знизу для I(3) +n +I(3) +n +≥ +∞ +� +k=n +ψ(k) − π +n +∞ +� +k=1 +kψ(k + n). +(58) +Лему 2.3 доведено. +З формул (47), (48), (50) i (52) випливає, що +En(Cψ +β,1)C = 1 +π +∞ +� +k=n +ψ(k) + Θ2 +n +∞ +� +k=1 +kψ(k + n). +Теорему 2.2 доведено. +Зазначимо, що оцiнки (14), (15) i (45) є асимптотичними рiвностями при n → ∞, якщо +виконується граничне спiввiдношення (12), тобто коли +1 +n +∞ +� +k=1 +kψ(k + n) = o +� ∞ +� +k=n +ψ(k) +� +, +n → ∞. +(59) +Умова (59), як буде показано нижче, має мiсце у рядi важливих випадкiв, зокрема, коли +послiдовнiсть ψ(k) спадає до нуля при k → ∞ швидше за довiльну степеневу послiдовнiсть +1 +kr , r > 1. + +13 +3 +Наслiдки з Теореми 2.2 для класiв аналiтичних та цiлих фун- +кцiй +Наведемо приклади важливих функцiональних компактiв Cψ +β,1, для яких формула (45) +дозволяє записати асимптотичнi рiвностi для En(Cψ +β,1)C при n → ∞. +Розглянемо випадок, коли послiдовностi ψ(k) задовольняють умову Даламбера Dq, q ∈ +[0, 1): +lim +k→∞ +ψ(k + 1) +ψ(k) += q, +ψ(k) > 0. +(60) +Якщо ψ(k) задовольняє умову (60) при деякому q ∈ [0, 1), то будемо записувати, що +ψ ∈ Dq. Нехай спочатку q = 0. +Згiдно з Теоремою 5 роботи [32], твердження про iснування послiдовностi ψ ∈ D0 такої, +що для функцiї f вiрне включення f ∈ Cψ +β L1 при будь-якому β ∈ R, еквiвалентне твер- +дженню про включення f ∈ E, де E — множина всiх 2π–перiодичних дiйснозначних на +дiйснiй осi функцiй, якi допускають аналiтичне продовження на всю комплексну площи- +ну. Отже, класи Cψ +β,1 при ψ ∈ D0 належать до множини 2π–перiодичних дiйснозначних на +R цiлих функцiй. +Наслiдок 3.1. Нехай +∞ +� +k=n+1 +kψ(k) < ∞, ψ(k) ≥ 0, k = 1, 2, ..., n ∈ N i β ∈ R, тодi має +мiсце рiвномiрна вiдносно всiх параметрах оцiнка +En(Cψ +β,1)C = 1 +πψ(n) + O(1) +n +∞ +� +k=n+1 +kψ(k). +(61) +Якщо, крiм того, ψ ∈ D0, то оцiнка (61) є асимптотичною рiвнiстю при n → ∞. +Доведення. Користуючись формулою (45) Теореми 2.2, можна записати +En(Cψ +β,1)C = 1 +πψ(n) + O(1) +� ∞ +� +k=1 +ψ(k + n) + 1 +n +∞ +� +k=1 +kψ(k + n) +� += 1 +πψ(n) + O(1) +n +∞ +� +k=1 +(k + n)ψ(k + n) += 1 +πψ(n) + O(1) +n +∞ +� +k=n+1 +kψ(k). +Тим самим оцiнку (61) доведено. Покажемо, що при ψ ∈ D0 +1 +n +∞ +� +k=n+1 +kψ(k) = o (ψ(n)) , n → ∞. +(62) +Виберемо номери n такими, щоб +ψ(k + 1) +ψ(k) +< 1 +2, +k = n, n + 1, ... +(63) + +14 +Тодi, з урахуванням (63), маємо +1 +n +∞ +� +k=1 +kψ(k) = +� +1 + 1 +n +� +ψ(n + 1) + ψ(n + 1) +n +∞ +� +j=2 +(n + j) +j−1 +� +ℓ=1 +ψ(n + ℓ + 1) +ψ(n + ℓ) +< ψ(n + 1) +� +2 + 1 +n +∞ +� +j=2 +2j +2j−1 +� +< ψ(n + 1) +� +2 + 4 +n +∞ +� +j=1 +j +2j +� +< 10ψ(n + 1). +(64) +Оскiльки, в силу ψ ∈ D0 +ψ(n + 1) = o (ψ(n)) , +n → ∞, +(65) +то iз (64) i (65) випливає (62). +Наслiдок 3.1 доведено. +Зауважимо, що асимптотичну рiвнiсть (61) з залишковим членом, записаним в iншiй +формi, було отримано ранiше в [12] i [13]. При ψ ∈ D0 оцiнки залишкового члена в [12] i +[13] є бiльш точними, нiж у формулi (61). +Типовими представниками послiдовностей, що задовольняють умову D0 є послiдовностi +ψ(k) = e−αk−r, r > 1, α > 0. Для породжуваних такими послiдовностями класiв Cψ +β,1 = Cα,r +β,1, +одержуємо наступне твердження. +Наслiдок 3.2. Нехай r > 1, α > 0 i β ∈ R. Тодi, при n ≥ +� 3 +αr +� 1 +r − 1, n ∈ N, має мiсце +рiвномiрна по всiх розглядуваних параметрах оцiнка +En(Cα,r +β,1)C = e−αnr�1 +π + O(1)e−αrnr−1� +1 + +1 +αr(n + 1)r−2 +�� +. +(66) +Доведення. З формули (61) випливає, що +En(Cα,r +β,1)C = 1 +πe−αnr + O(1) +n +∞ +� +k=n+1 +ke−αkr. +(67) +Легко переконатись, що при номерах n таких, що (n + 1)r > +1 +αr +1 +n +∞ +� +k=n+1 +ke−αkr < 1 +n + +(n + 1)e−α(n+1)r + +∞ +� +n+1 +te−αtrdt + + . +(68) +Iнтегруючи частинами, отримуємо +∞ +� +n+1 +te−αtrdt = +∞ +� +n+1 +t2 +1 +αrtr +� +−e−αtr�′ dt ≤ +1 +αr(n + 1)r +∞ +� +n+1 +t2 � +−e−αtr�′ dt += +1 +αr(n + 1)r + +(n + 1)2e−α(n+1)r + 2 +∞ +� +n+1 +te−αtrdt + + . +(69) + +15 +З останньої нерiвностi маємо +� +1 − +2 +αr(n + 1)r +� +∞ +� +n+1 +te−αtrdt ≤ (n + 1)2e−α(n+1)r +αr(n + 1)r +, +(70) +що рiвносильно тому, що +∞ +� +n+1 +te−αtrdt ≤ +e−α(n+1)r +αr(n + 1)r−2 +αr(n + 1)r +αr(n + 1)r − 2 += +e−α(n+1)r +αr(n + 1)r−2 +� +1 + +2 +αr(n + 1)r − 2 +� +. +(71) +Зi спiввiдношень (68) i (71) випливає, що +1 +n +∞ +� +k=n+1 +ke−αkr = O(1) +� +e−α(n+1)r + +e−α(n+1)r +αr(n + 1)r−2 +� +1 + +2 +αr(n + 1)r − 2 +�� +. +(72) +Об’єднавши (67) i (72), одержуємо, що при всiх номерах n таких, що (n + 1)r > +3 +αr +En(Cα,r +β,1)C = 1 +πe−αnr + O(1) +� +e−α(n+1)r + +e−α(n+1)r +αr(n + 1)r−2 +� +1 + +2 +αr(n + 1)r − 2 +�� +=e−αnr +� +1 +π + O(1) +� +e−αrnr−1 + +e−αrnr−1 +αr(n + 1)r−2 +�� +. +Наслiдок 3.2 доведено. +Формулу (66) iз залишковим членом, записаним дещо в iншому виглядi було отримано +в [12] i [13]. При цьому оцiнки з [12] i [13] мiстять бiльш точнi оцiнки залишкового члена +нiж у (66). +Нехай далi q ∈ (0, 1). Згiдно з Теоремою 3 роботи [32], твердження про iснування по- +слiдовностi ψ ∈ Dq, q ∈ (0, 1) такої, що для функцiї f вiрне включення Cα,1 +β L1 при будь- +якому β ∈ R еквiвалентне твердженню про включення f ∈ A, де A — множина всiх +2π–перiодичних дiйснозначних на дiйснiй осi функцiй, якi допускають аналiтичне продов- +ження на деяку смугу |Imz| < c, c > 0, комплексної площини. Отже класи Cψ +β,1, ψ ∈ Dq, +0 < q < 1, складаються з перiодичних, аналiтичних у смузi |Im z| < c функцiй, при цьому +c = ln 1 +q (див., наприклад, [25, c. 32]). +Послiдовностi ψ(k) = e−αk, α > 0 належать до множини Dq при q = e−α, а вiдповiднi +класи Cψ +β,1 = Cα,1 +β,1 породжуються ядрами Пуассона +Pα,1,β(t) = +∞ +� +k=1 +e−αk cos +� +kt − βπ +2 +� +, α > 0, β ∈ R. +(73) +Iз Теореми 2.2 для класiв Cα,1 +β,1 отримуємо наступне твердження. + +16 +Наслiдок 3.3. Нехай α > 0 i β ∈ R. Тодi, при всiх n ∈ N має мiсце рiвнiсть +En(Cα,1 +β,1)C = e−αn +�1 +π +1 +1 − e−α + Θ +n +e−α +(1 − e−α)2 +� +, +(74) +де для величини Θ = Θ(n, α, β) виконуються нерiвностi −1 ≤ Θ ≤ 0. +Доведення. Покладемо q = e−α. Тодi, з Теореми 2.2 випливає, що при всiх n ∈ N +En(Cα,1 +β,1)C = 1 +π +∞ +� +k=n +qk + Θ +n +∞ +� +k=0 +kqk+n += 1 +π +qn +1 − q + Θ +n +� ∞ +� +k=n +kqk − n +∞ +� +k=n +qk +� += 1 +π +qn +1 − q + Θ +n +�nqn(1 − q) + qn+1 +(1 − q)2 +− nqn +1 − q +� += 1 +π +qn +1 − q + Θ +n +qn+1 +(1 − q)2, +(75) +де була використана наступна рiвнiсть: +∞ +� +k=n +kqk = nqn(1 − q) + qn+1 +(1 − q)2 +, +q ∈ (0, 1), +n ∈ N. +Наслiдок 3.3 доведено. +Оцiнка (74) уточнює асимптотичнi рiвностi для величин En(Cα,r +β,1)C, якi були встановленi +в [12] i [13]. Асимптотичнi рiвностi для величин En(Cψ +β,1)C при ψ ∈ Dq, q ∈ (0, 1), мiстяться +у наступному твердженнi. +Наслiдок 3.4. Нехай ψ ∈ Dq, q ∈ (0, 1), β ∈ R, n ∈ N. Тодi при всiх номерах n таких, +що +1 +n + εn < 1 − q +2 +, +(76) +де +εn := sup +k≥n +���� +ψ(k + 1) +ψ(k) +− q +���� , +(77) +має мiсце рiвномiрна вiдносно всiх розглядуваних параметрiв оцiнка +En(Cψ +β,1)C = ψ(n) +� +1 +π(1 − q) + O(1) +� +q +n(1 − q)2 + +εn +(1 − q)2 +�� +. +(78) +Доведення. З Леми 1 роботи [30] випливає, що при ψ ∈ Dq, 0 < q < 1, n ∈ N, має мiсце +рiвнiсть +∞ +� +k=n +ψ(k) = ψ(n) +� +1 +qn +∞ +� +k=n +qk + rn +� +, +(79) +де для залишку rn при всiх номерах n таких, що +εn < 1 − q +2 +(80) + +17 +виконується оцiнка +|rn| ≤ +εn +(1 − q − εn)(1 − q) ≤ +2εn +(1 − q)2. +(81) +Очевидно, що якщо ψ ∈ Dq, 0 < q < 1, то i послiдовнiсть kψ(k) також задовольняє умову +Dq, а тому знову ж таки в силу Леми 1 iз [30] +∞ +� +k=n+1 +kψ(k) = (n + 1)ψ(n + 1) +� +1 +qn+1 +∞ +� +k=n+1 +qk + r∗ +n+1 +� +, +(82) +де для залишку r∗ +n+1 при усiх номерах n таких, що +ε∗ +n+1 := sup +k≥n+1 +���� +ψ(k + 1)(k + 1) +ψ(k)k +− q +���� < 1 − q +2 +(83) +виконується оцiнка +��r∗ +n+1 +�� ≤ +2ε∗ +n+1 +(1 − q)2. +(84) +Iз означень величин εn i ε∗ +n+1 (див. (77) i (83)) маємо +ε∗ +n+1 ≤ sup +k≥n+1 +���� +ψ(k + 1) +ψ(k) +− q +���� + +1 +n + 1 = εn+1 + +1 +n + 1 < εn + 1 +n. +(85) +Iз (85) видно, що виконання нерiвностi (76) гарантує i виконання нерiвностей (80) i (83), +а отже i оцiнок (81) i (84) для залишкiв у рiвностях (79) i (82). +Тодi в силу оцiнки (45) Теореми 2.2 i рiвностей (79) i (82) випливає, що при всiх номерах +n, якi задовольняють умову (76), справджуються спiввiдношення +En(Cψ +β,1)C = 1 +π +∞ +� +k=n +ψ(k) + O(1) +n +∞ +� +k=1 +kψ(k + n) +=ψ(n) +π +� +1 +qn +∞ +� +k=n +qk + rn +� ++ O(1) +n +∞ +� +k=n+1 +(k − n)ψ(k) +=ψ(n) +π +� +1 +1 − q + O(1) +εn +(1 − q)2 +� ++ O(1) +� +1 +n +∞ +� +k=n+1 +kψ(k) − +∞ +� +k=n+1 +ψ(k) +� +=ψ(n) +� +1 +π(1 − q) + O(1) +εn +(1 − q)2 +� ++O(1)ψ(n + 1) +� +n + 1 +n +� +1 +qn+1 +∞ +� +k=n+1 +qk + +ε∗ +n+1 +(1 − q)2 +� +− +1 +qn+1 +∞ +� +k=n+1 +qk + +εn+1 +(1 − q)2 +� +=ψ(n) +� +1 +π(1 − q) + O(1) +εn +(1 − q)2 +� ++ O(1)ψ(n + 1) +� +1 +n(1 − q) + εn + 1 +n +(1 − q)2 +� +=ψ(n) +� +1 +π(1 − q) + O(1) +� +εn +(1 − q)2 + ψ(n + 1) +ψ(n) +1 +n(1 − q)2 +�� +=ψ(n) +� +1 +π(1 − q) + O(1) +� +εn +(1 − q)2 + +q +n(1 − q)2 +�� +. +(86) +Наслiдок 3.4 доведено. + +18 +Асимптотичнi рiвностi (78) вперше були встановленi в роботах [12] i [13]. +4 +Наслiдки з Теореми 2.2 для класiв нескiнченно диференцiйов- +них функцiй +В даному пiдроздiлi будемо вважати, що послiдовностi ψ(k), що породжують множини +Cψ +β L1 та Cψ +β,1, є звуженням на множину натуральних чисел деяких додатних неперервних +опуклих донизу функцiй ψ(t) неперервного аргументу t ≥ 1, що прямують до нуля при +t → ∞. Множину всiх таких функцiй ψ позначають через M: +M= +� +ψ∈C[1, ∞): ψ(t)>0, ψ(t1 − 2ψ((t1 + t2)/2) + ψ(t2) ≥ 0 ∀t1, t2 ∈ [1, ∞), lim +t→∞ ψ(t)=0 +� +. +(87) +Наслiдуючи О.I. Степанця (див., наприклад, [26, с. 160]), кожнiй функцiї ψ ∈ M поста- +вимо у вiдповiднiсть характеристики +η(t) = η(ψ; t) = ψ−1 +�1 +2ψ(t) +� +та +µ(t) = µ(ψ; t) = +t +η(t) − t, +де ψ−1 — обернена до ψ функцiя, i покладемо +M+ +∞ = {ψ ∈ M : µ(t) ↑, +t → ∞} . +Через Mα позначимо пiдмножину всiх функцiй ψ ∈ M, для яких величина +α(t) = α(ψ; t) := +ψ(t) +t|ψ′(t)|, +ψ′(t) := ψ′(t + 0), +(88) +спадає до нуля при t → ∞: +Mα = +� +ψ ∈ M : +lim +t→∞ α(ψ; t) = 0 +� +. +(89) +Згiдно з Теоремою 2 роботи [31], твердження про iснування послiдовностi ψ ∈ Mα (або +ψ ∈ M+ +∞), такої, що для функцiї f вiрне включення f ∈ Cψ +β L1 при будь-якому β ∈ R, +еквiвалентне твердженню про включення f ∈ D∞, де D∞ — множина всiх нескiнченно +диференцiйовних 2π-перiодичних дiйснозначних функцiй. А отже, класи Cψ +β,1 при ψ ∈ Mα +(або ψ ∈ M+ +∞), є класами нескiнченно диференцiйовних перiодичних функцiй. В тiй же +роботi було показано, що має мiсце включення +M+ +∞ ⊂ Mα ⊂ M∞ = +� +ψ ∈ M : ∀r > 0 +lim +t→∞ trψ(t) = 0 +� +, +(90) +яке означає, що функцiї ψ(t) iз Mα спадають до нуля швидше за довiльну степеневу +функцiю. + +19 +Для величин En(Cψ +β,1)C, ψ ∈ M+ +∞ за умови η(n) − n > 2 вiдомi точнi порядковi рiвностi +En(Cψ +β,1)C ≍ ψ(n)(η(n) − n), +(91) +якi спiвпрадають з точними порядковими рiвностями для найкращих рiвномiрних набли- +жень тригонометричними полiномами порядку n − 1 +En(Cψ +β,1)C = +inf +tn−1∈T ∥f − ttn−1∥C +а саме, (див., наприклад, [17]) +En(Cψ +β,1)C ≍ En(Cψ +β,1)C ≍ ψ(n)(η(n) − n), +(92) +(тут i надалi запис A(n) ≍ B(n) для додатних послiдовностей A(n) i B(n) означає iснува- +ння додатних констант K1 i K2 таких, що K1B(n) ≤ A(n) ≤ K2B(n), n ∈ N). +Як показано в [27, с. 166] для довiльної функцiї ψ iз M+ +∞ має мiсце порядкова рiвiнсть +η(t) − t ≍ λ(t), t ≥ 1, +(93) +де λ(t) — характеристика вигляду +λ(t) = λ(ψ; t) := ψ(t) +|ψ′(t)|. +(94) +З урахуванням (91) можна записати у виглядi +En(Cψ +β,1)C ≍ ψ(n)λ(n). +(95) +Наступне твердження мiстить сильну асимптотику величин En(Cψ +β,1)C , ψ ∈ Mα при +деяких природних обмеженнях на α(t) i λ(t). +Теорема 4.1. Нехай β ∈ R, ψ ∈ M i характеристики (88) i (94) задовольняють умови +α(t) ↓ 0, +(96) +λ(t) ↑ ∞, +t → ∞. +(97) +Тодi для всiх n ∈ N таких, що +α(n) ≤ 1 +4 +(98) +виконується оцiнка +En(Cψ +β,1)C = ψ(n)λ(n) +�1 +π + +ξ1 +λ(n) + ξ2α(n) +� +, +(99) +де −1 ≤ ξ1 ≤ 1 + 1 +π та −4 ≤ ξ2 ≤ 4 +3 +� +1 + 1 +π +� +. + +20 +Доведення Теореми 4.1. Для оцiнки величини En(Cψ +β,1)C використаємо формулу (45) iз +Теореми 2.2. При цьому нам буде необхiдно знайти оцiнки рядiв Σ1 = +∞ +� +k=n +ψ(k) та +Σ2 = +∞ +� +k=n +kψ(k). +В силу монотонного спадання функцiї ψ ∈ M бачимо, що +∞ +� +n +ψ(t)dt ≤ +∞ +� +k=n +ψ(k) ≤ ψ(n) + +∞ +� +n +ψ(t)dt, +(100) +а, отже, +∞ +� +k=n +ψ(k) = +∞ +� +n +ψ(t)dt + Θ4ψ(n), +0 ≤ Θ4 ≤ 1. +(101) +Оцiнка iнтеграла +∞� +n +ψ(t)dt випливає з наступного твердження, яке може мати i само- +стiйний iнтерес. +Лема 4.2. Нехай ψ ∈ M, λ(t) монотонно неспадає, а α(t) монотонно незростає на [1, ∞). +Тодi при всiх a ≥ 1, таких, що α(a) < 1, виконуються оцiнки +λ(a)ψ(a) ≤ +∞ +� +a +ψ(t)dt ≤ λ(a)ψ(a) +� +1 + +α(a) +1 − α(a) +� +. +(102) +Доведення Леми 4.2. Оскiльки в силу включення ψ ∈ M, функцiя ψ(t) є локально аб- +солютно неперервною на [1, ∞), то, враховуючи монотонне неспадання λ(t), одержуємо +шукану оцiнку знизу +I1 := +∞ +� +a +ψ(t)dt = +∞ +� +a +−ψ′(t)λ(t)dt ≥ λ(a) +∞ +� +a +(−ψ′(t))dt = ψ(a)λ(a), +(103) +З iншого боку, враховуючи монотонне незростання функцiї α(t), i застосовуючи метод +iнтегрування частинами, маємо +I1 = +∞ +� +a +ψ(t)dt = +∞ +� +a +α(t)(−ψ′(t)t)dt ≤ α(a) +∞ +� +a +(−ψ′(t)t) +=α(a) + +ψ(a)a + +∞ +� +a +ψ(t)dt + + = ψ(a)λ(a) + α(a)I1. +(104) +З (104) одержуємо, що +I1 ≤ λ(a)ψ(a) +1 − α(a) = λ(a)ψ(a) +� +1 + +α(a) +1 − α(a) +� +. +(105) +Iз (103) i (105) випливає (102). Лему 4.2 доведено. + +21 +Застосування Леми 4.2 при a = n, n ∈ N, за умови (98) дозволяє записати, що +I1 = +∞ +� +a +ψ(t)dt = ψ(n)λ(n) (1 + Θ5α(n)) , +0 ≤ Θ5 ≤ 4 +3. +(106) +Отже, з урахуванням (101) i (106) при α(n) ≤ 1 +4 +∞ +� +k=n +ψ(k) = ψ(n)λ(n) +� +1 + Θ4 +λ(n) + Θ5α(n) +� +, +0 ≤ Θ5 ≤ 4 +3, 0 ≤ Θ4 ≤ 1. +(107) +Далi знайдемо оцiнку для Σ2 = +∞ +� +k=n +kψ(k). В силу (98) функцiя tψ(t) спадає на [n, ∞), +а тому +∞ +� +n +tψ(t)dt ≤ +∞ +� +k=n +kψ(k) ≤ nψ(n) + +∞ +� +n +tψ(t)dt, +(108) +i, отже, +∞ +� +k=n +kψ(k) = +∞ +� +n +tψ(t)dt + Θ6nψ(n), +0 ≤ Θ6 ≤ 1. +(109) +Для оцiнки iнтеграла I2 = +∞� +n +tψ(t)dt знову використаємо метод iнтегрування частинами +i врахуємо (90) та умову незростання α(n) +I2 = +∞ +� +n +tψ(t)dt = +∞ +� +n +t2 +ψ(t) +−tψ′(t)(−ψ′(t))dt ≤ α(n) +∞ +� +n +t2(−ψ′(t))dt +=α(n) +� +n2ψ(n) + 2I2 +� +, +(110) +З останнiх спiввiдношень i умови (98) маємо +I2 (1 − 2α(n)) ≤ α(n)n2ψ(n) +i, отже, з урахуванням умови (98) маємо +I2 ≤ψ(n)n2α(n) +1 +1 − 2α(n) = ψ(n)n2α(n) +� +1 + +2α(n) +1 − 2α(n) +� +≤ψ(n)n2α(n) (1 + 4α(n)) = ψ(n)nλ(n)(1 + 4α(n)). +(111) +З iншого боку, з урахуванням умови (106), +I2 = +∞ +� +n +tψ(t)dt ≥ n +∞ +� +n +ψ(t)dt ≥ ψ(n)nλ(n). +(112) + +22 +Об’єднання (111) i (112) дозволяє записати, що при α(n) ≤ 1 +4 +∞ +� +n +tψ(t)dt = ψ(n)nλ(n) (1 + Θ7α(n)) , +0 ≤ Θ7 ≤ 4. +(113) +Iз формул (109) i (113) випливає, що за умов (94) i (98) +∞ +� +k=n +kψ(k) = ψ(n)nλ(n) +� +1 + Θ7α(n) + Θ6 +λ(n) +� +, +0 ≤ Θ7 ≤ 4, +0 ≤ Θ6 ≤ 1. +(114) +Користуючись оцiнками (108) i (107), одержуємо +1 +n +∞ +� +k=1 +kψ(k + n) = 1 +n +∞ +� +k=0 +kψ(k + n) +=1 +n +� ∞ +� +k=n +kψ(k) − n +∞ +� +k=n +ψ(k) +� +=1 +n +∞ +� +k=n +kψ(k) − +∞ +� +k=n +ψ(k) +=ψ(n)λ(n) +� +1 + Θ7α(n) + Θ6 +λ(n) +� +− ψ(n)nλ(n) +� +1 + Θ5α(n) + Θ4 +λ(n) +� +=ψ(n)λ(n) +� +(Θ7 − Θ5)α(n) + Θ6 − Θ4 +λ(n) +� +. +(115) +На пiдставi формули (45) iз Теореми 2.2 та оцiнок (107) та (115), отримуємо, що для +всiх номерiв n таких, що виконується нерiвнiсть (98) +En(Cψ +β,1)C = 1 +π +∞ +� +k=n +ψ(k) + Θ2 +1 +n +∞ +� +k=1 +kψ(k + n) += 1 +πψ(n)λ(n) +� +1 + Θ4 +λ(n) + Θ5α(n) +� ++Θ2ψ(n)λ(n) +� +1 + Θ6 − Θ4 +λ(n) ++ (Θ7 − Θ5)α(n) +� +=ψ(n)λ(n) +�1 +π + Θ4/π + Θ2(Θ6 − Θ4) +λ(n) ++ +�Θ5 +π + Θ2(Θ7 − Θ5) +� +α(n) +� +. +(116) +Оскiльки для величини ξ1 = Θ4 +π + Θ2(Θ6 − Θ4) виконується оцiнка +−1 ≤ ξ1 ≤ 1 + 1 +π, +а для ξ2 = Θ5 +π + Θ2(Θ7 − Θ5) — оцiнка +−4 ≤ ξ2 ≤ 4 +3 +� +1 + 1 +π +� +, +то iз (116) випливає (99). Теорему 4.1 доведено. + +23 +Наведемо наслiдок з Теореми 4.1 у випадку, коли ψ(t) = e−αt−r, α > 0, 0 < r ≤ 1, тобто +коли класи Cψ +β,1 є класами узагальнених iнтегралiв Пуассона Cα,r +β,1. Легко переконатись, що +для вказаних ψ(t) при всiх t ≥ 1, +λ(t) = t1−r +αr , +α(t) = +1 +αrtr . +(117) +Iз (117) видно, що умови (96) i (97) Теореми 4.1 виконуються. При цьому виконання +нерiвностi +1 +αrnr ≤ +1 +4 рiвносильне виконанню умови (98). Отже, з Теореми 4.1 випливає +наступне твердження. +Наслiдок 4.3. Нехай 0 < r < 1, α > 0, β ∈ R, n ∈ N. Тодi при всiх n ≥ +� 4 +αr +� 1 +r справедлива +рiвномiрно обмежена по всiх розглядуваних параметрах оцiнка +En(Cα,r +β,1)C = e−αnrn1−r� 1 +παr + O(1) +� +1 +(αr)2 +1 +nr + +1 +n1−r +�� +. +(118) +Зазначимо, що оцiнка вигляду (118) при дещо жорсткiших обмеженнях на n була зна- +йдена у роботах [18]– [20]. У зазаначених роботах мiстяться двостороннi оцiнки величини +O(1) через абсолютнi сталi. +Наведемо ще декiлька прикладiв застосування Теореми 4.1 для рiзних функцiй ψ iз M, +якi задовольняють умовам (96) i (97). +Будемо розглядати ψ(t) вигляду +ψ(t) = (t + 2)− ln ln(t+2), +t ≥ 1, +(119) +ψ(t) = e− ln2(t+1), +t ≥ 1, +(120) +ψ(t) = e− +t+2 +ln(t+2), +t ≥ 1. +(121) +Для зазначених функцiй ψ(t) знайдемо характеристики λ(t) i α(t). Результати обчи- +слень вiдображено в наступнiй таблицi: +№ +Функцiя ψ(t) +α(t) +λ(t) +1. +(t + 2)− ln ln(t+2) +t+2 +t +1 +1+ln ln(t+2) +t+2 +1+ln ln(t+2) +2. +e− ln2(t+1) +t+1 +t +1 +2 ln(t+1) +t+1 +2 ln(t+1) +3. +e− +t+2 +ln(t+2) +ln2(t+2) +t(ln(t+2)−1) +ln2(t+2) +ln(t+2)−1 +Iз Теореми 4.1 i наведених в таблицi значень α(t) i λ(t) отримуємо асимптотичнi при +n → ∞ рiвностi для величин En(Cψ +β,1)C у випадку, коли ψ мають вигляд (119)–(121). +Наслiдок 4.4. Нехай ψ(k) = (k + 2)− ln ln(k+2), k = 1, 2, ..., β ∈ R i n ∈ N. Тодi при n → ∞ +виконується асимптотична рiвнiсть +En(Cψ +β,1)C = 1 +πψ(n) +n +ln ln(n + 2) + O(1)ψ(n) +n +(ln ln(n + 2))2. +(122) + +24 +Наслiдок 4.5. Нехай ψ(k) = e− ln2(k+1), k = 1, 2, ..., β ∈ R i n ∈ N.Тодi при n → ∞ має +мiсце асимптотична рiвнiсть +En(Cψ +β,1)C = 1 +2π +ψ(n)n +ln(n + 1) + O(1)ψ(n) +n +ln2(n + 1). +(123) +Наслiдок 4.6. Нехай ψ(k) = e− +k+2 +ln(k+2), k = 1, 2, ..., β ∈ R i n ∈ N Тодi при n → ∞ має +мiсце асимптотична рiвнiсть +En(Cψ +β,1)C = 1 +πψ(n) ln(n + 2) + O(1)ψ(n). +(124) +Зауважимо, що у випадку, коли ψ ∈ M i при t → ∞ α(t) → 0 i λ(t) → ∞ за додаткової +умови, що функцiя ψ(t) є диференцiйовною скрiзь на [1, ∞), граничне спiввiдношення +(12), яке гарантує той факт, що оцiнки (15) i (45) є асимптотичними рiвностями, завжди +виконується. +Дiйсно, застосувавши правило Лопiталя, маємо +lim +n→∞ +∞� +n +ψ(t)dt +ψ(n) += lim +n→∞ +ψ(n) +|ψ′(n)| = lim +n→∞ λ(n) = ∞, +(125) +lim +n→∞ +∞� +n +tψ(t)dt +nψ(n) += lim +n→∞ +−nψ(n) +ψ(n) + nψ′(n) = lim +n→∞ +λ(n) +1 − α(n) = ∞. +(126) +Тодi, з урахуванням (101) i (109) мають мiсце асимптотичнi рiвностi +∞ +� +k=n +ψ(k) = +∞ +� +n +ψ(t)dt + O(1)ψ(n), +(127) +∞ +� +k=n +kψ(k) = +∞ +� +n +tψ(t)dt + O(1)nψ(n). +(128) +Використовуючи формули (125)–(109) i застосувавши правило Лопiталя, отримуємо +lim +n→∞ +1 +n +∞ +� +k=1 +kψ(k + n) +∞ +� +k=n +ψ(k) += lim +n→∞ +1 +n +∞ +� +k=1 +kψ(k) − +∞ +� +k=n +ψ(k) +∞ +� +k=n +ψ(k) += lim +n→∞ +1 +n +∞ +� +k=1 +kψ(k) +∞ +� +k=n +ψ(k) +− 1 = lim +n→∞ +1 +n +∞� +n +tψ(t)dt +∞� +n +ψ(t)dt +− 1 + +25 += lim +n→∞ +−nψ(n) +∞� +n +ψ(t)dt − nψ(n) +− 1 = lim +n→∞ +− +∞� +n +ψ(t)dt +∞� +n +ψ(t)dt − nψ(n) += lim +n→∞ +ψ(n) +−2ψ(n) − nψ′(n) = lim +n→∞ +ψ(n) +−nψ′(n) +1 − +2ψ(n) +−nψ′(n) += lim +n→∞ +α(n) +1 − α(n) = 0. +(129) +Тим самим рiвнiсть (12) доведено. +5 +Коментарi щодо нерiвностей Лебега +У пiдроздiлах 3 i 4 були наведенi наслiдки з Теореми 2.2 для швидко спадних послiдов- +ностей ψ(k), для яких формула (45) є асимптотичною рiвнiстю, або, що те саме, коли +справджується (12). Зрозумiло, що у всiх розглянутих у пiдроздiлах 3 i 4 частинних ви- +падках для ψ(·) легко одержати i асимптотично непокращуванi нерiвностi типу Лебега +вигляду (44). Ми обмежидись лише формулюванням лише деяких тверджень, якi випли- +вають iз Теореми 2.1. Спочатку сформулюємо вiдповiднi твердження для ψ(t) = e−αtr, +α > 0 i r > 0. Випадки r > 1, r = 1 i r ∈ (0, 1) видiляються окремо. +Наслiдок 5.1. Нехай r > 1, α > 1 i β ∈ R. Тодi при n ≥ +� 3 +αr +� 1 +r − 1 для довiльної функцiї +f ∈ Cα,r +β L1 має мiсце нерiвнiсть +∥f(·) − Sn−1(f; ·)∥C ≤ e−αnr � 1 +π + O(1)e−αnr−1 � +1 + +1 +αr(n + 1)r−2 +�� +En(f α,r +β )L1. +(130) +Крiм того, для довiльної функцiї f ∈ Cα,r +β L1 можна знайти функцiю F(x) = F(f; n, x) +з множини Cα,r +β L1 таку, що En(F +α,r +β )L1 = En(f α,r +β )L1 i має мiсце рiвнiсть +∥F(·) − Sn−1(F(·); ·)∥C = e−αnr � 1 +π + O(1)e−αnr−1 � +1 + +1 +αr(n + 1)r−2 +�� +En(f α,r +β )L1. +(131) +У (130) i (131) O(1) — рiвномiрно обмежена по всiх параметрах величина. +Аналоги нерiвностi (130) i формули (131), яка доводить асимптотичну непокращува- +нiсть зазначеної нерiвностi, в яких залишковi члени записанi в дещо iншiй формi, отри- +мано в [9]. +Наслiдок 5.2. Нехай α > 0, β ∈ R i n ∈ N. Тодi для довiльної функцiї f ∈ Cα,1 +β L1 має +мiсце нерiвнiсть +∥f(·) − Sn−1(f; ·)∥C ≤ 1 +π +e−αn +1 − e−αEn(f α,1 +β )L1. +(132) +Крiм того, для довiльної функцiї f ∈ Cα,1 +β L1 можна знайти функцiю F(x) = F(f; n, x) +з множини Cα,1 +β L1 таку, що En(F +α,1 +β )L1 = En(f α,1 +β )L1 i має мiсце рiвнiсть +∥F(·) − Sn−1(F(·); ·)∥C = e−αn +�1 +π +1 +1 − e−α + ξ +n +1 +(1 − e−α)2 +� +En(f α,1 +β )L1, +(133) + +26 +де для величини ξ = ξ(f; n; α; β) виконується нерiвнiсть −2 ≤ ξ ≤ 0. +Оцiнки (132) i (133) уточнюють оцiнки рiвномiрних вiдхилень сум Фур’є на множинах +iнтегралiв Пуассона Cα,1 +β L1, що були одержанi в роботах [14] i [8]. +Наслiдок 5.3. Нехай 0 < r < 1, α > 0, β ∈ R i n ∈ N. Тодi при всiх n ≥ +� 4 +αr +� 1 +r для +довiльної функцiї f ∈ Cα,r +β L1 має мiсце нерiвнiсть +∥f(·) − Sn−1(f; ·)∥C ≤ e−αnrn1−r +� 1 +παr + O(1) +� +1 +(αr)2 +1 +nr + +1 +n1−r +�� +En(f α,r +β )L1. +(134) +Крiм того, для довiльної функцiї f ∈ Cα,r +β L1 можна знайти функцiю F(x) = F(f; n, x) +з множини Cα,r +β L1 таку, що En(F +α,r +β )L1 = En(f α,r +β )L1 i має мiсце рiвнiсть +∥F(·) − Sn−1(F(·); ·)∥C = e−αnrn1−r +� 1 +παr + O(1) +� +1 +(αr)2 +1 +nr + +1 +n1−r +�� +En(f α,r +β )L1. +(135) +У (134) i (135) O(1) — величини, що рiвномiрно обмеженi по всiх параметрах. +При дещо жорсткiших обмеженнях на n формули вигляду (134) i (135) були встановленi +ранiше в [21]. +Наслiдок 5.4. Нехай ψ ∈ Dq, q ∈ (0, 1), β ∈ R i n ∈ N. Тодi при всiх n таких, що +виконується нерiвнiсть (77) для довiльної функцiї f ∈ Cψ +β L1 має мiсце нерiвнiсть +∥f(·) − Sn−1(f; ·)∥C ≤ ψ(n) +� +1 +π(1 − q) + O(1) +� +q +n(1 − q)2 + +εn +(1 − q)2 +�� +En(f ψ +β )L1. +(136) +Крiм того, для довiльної функцiї f ∈ Cψ +β L1 можна знайти функцiю F(x) = F(f; n, x) +з множини Cψ +β L1 таку, що En(F +ψ +β )L1 = En(f ψ +β )L1 i таку, що при виконаннi (76) для неї +має мiсце рiвнiсть +∥F(·) − Sn−1(F(·); ·)∥C = ψ(n) +� +1 +π(1 − q) + O(1) +� +q +n(1 − q)2 + +εn +(1 − q)2 +�� +En(f ψ +β )L1. (137) +У (136) i (137) величина εn означена рiвнiстю (77), а O(1) — величини, що рiвномiрно +обмеженi по всiх параметрах. +Теорема 5.5. Нехай β ∈ R, ψ ∈ M i характеристики α(t) i λ(t) вигляду (88) i (94) +задовольняють умови (96) i (97). Тодi для всiх n ∈ N таких, що α(n) < 1 +4 для будь-якої +функцiї f ∈ Cψ +β L1 виконується нерiвнiсть +∥f(·) − Sn−1(f; ·)∥C ≤ ψ(n)λ(n) +�1 +π + +ξ3 +λ(n) + ξ4α(n) +� +En(f ψ +β )L1, +(138) +де 0 ≤ ξ3 ≤ +4 +3π, 0 ≤ ξ4 ≤ 1 +π. +Крiм того, для довiльної функцiї f ∈ Cψ +β L1 можна знайти функцiю F(x) = F(f; n, x) +з множини Cψ +β L1 таку, що En(F +ψ +β )L1 = En(f ψ +β )L1 i при n ∈ N таких, що α(n) < 1 +4 має + +27 +мiсце рiвнiсть +∥F(·) − Sn−1(F(·); ·)∥C = ψ(n)λ(n) +�1 +π + +ξ5 +λ(n) + ξ6α(n) +� +En(f ψ +β )L1, +(139) +де −2 ≤ ξ5 ≤ 2 + 1 +π, −8 ≤ ξ6 ≤ 4 +3 +� +2 + 1 +π +� +. +Нерiвнiсть (138) є наслiдком формул (14) та (107), а рiвнiсть (139) випливає iз (15), +(107) та (115). +1. Н.И. Ахиезер, Лекции по теории аппроксимации, Мир, Москва (1965). +2. В.К. Дзядык, Введение в теорию равномерного приближения функций полиномами, +Наука, Москва (1977). +3. L. Fejer, Lebesguesche konstanten und divergente Fourierreihen, J. Reine Angew Math. +138, 22–53 (1910). +4. П.В. Галкин, Оценки для констант Лебега, Тр. МИАН СССР, 109, 3–5 (1971). +5. В.В. Жук, Г.И.Натансон, Тригонометрические ряды и элементы теории аппрокси- +мации, Изд-во Ленинг. ун-та (1983). +6. A. Kolmogoroff, Zur Gr¨ossennordnung des Restgliedes Fourierschen Reihen differenzi- +erbarer Funktionen, (in German) Ann. Math.(2), 36, №2, 521–526 (1935). +7. Н.П. Корнейчук, Точные константы в теории приближения, Наука, Москва, (1987). +8. А.П. Мусiєнко, А.С. Сердюк, Нерiвностi типу Лебега для сум Валле Пуссена на +множинах аналiтичних функцiй, Укр. мат. журн., 65, № 4, 522-537 (2013). +9. А.П. Мусiєнко, А.С. Сердюк, Нерiвностi типу Лебега для сум Валле Пуссена на +множинах цiлих функцiй, Укр. мат. журн., 65, № 5, 642–653 (2013). +10. Г.И. Натансон, Об оценке констант Лебега сумм Валле–Пуссена, Геометрические +вопросы теории функций и множеств, Калинин (1986). +11. С. М. Никольский, Приближение функций тригонометрическими полиномами в сре- +днем, Изв. АН СССР. Сер. матем., 10, №3, 207–256 (1946). +12. А.С. Сердюк, Наближення класiв аналiтичних функцiй сумами Фур’є в рiвномiрнiй +метрицi, Укр. мат. журн., 57, № 8. 1079–1096 (2005). +13. А.С. Сердюк, Наближення класiв аналiтичних функцiй сумами Фур’є в метрицi +простору Lp, Укр. мат. журн., 57, № 10, 1395–1408 (2005). + +28 +14. А.С. Сердюк, А.П. Мусiєнко, Нерiвностi типу Лебега для сум Валле Пуссена при +наближеннi iнтегралiв Пуассона, Збiрник праць Iнституту математики НАН Укра- +їни, 7, № 1: Теорiя наближення функцiй та сумiжнi питання, Київ: Iн-т математики +НАН України, 298–316 (2010). +15. A.S. Serdyuk, I.V. Sokolenko,Approximation by Fourier sums in classes of differenti- +able functions with high exponents of smoothness, Methods of Functional Analysis and +Topology, 25, № 4, 381–387 (2019). +16. А.С. Сердюк, I. В. Соколенко, Наближення сумами Фур’є на класах диференцiйовних +у сенсi Вейля – Надя функцiй iз високим показником гладкостi, Укр. мат. журн., 74, +№ 5, 685 –700 (2022). +17. А.С. Сердюк , Оцiнки найкращих наближень класiв нескiнченно диференцiйовних +функцiй в рiвномiрнiй та iнтегральнiй метриках , Укр. мат. журн., 66, №9, 1244– +1256 (2014). +18. А.С. Сердюк, Т.А. Степанюк, Рiвномiрнi наближення сумами Фур’є на класах згор- +ток з iнтегралами Пуассона, Допов. НАН України, № 11, 10–16 (2016). +19. А.С. Сердюк, Т.А. Степанюк, Наближення класiв узагальнених iнтегралiв Пуассона +сумами Фур’є в метриках просторiв Ls, Укр. мат. журн., 69, № 5, 695-704 (2017). +20. A.S. Serdyuk, T.A. Stepanyuk, Uniform approximations by Fourier sums on classes of +generalized Poisson integrals, Analysis Mathematica, 45, №1, 201–236 (2019). +21. A.S. Serdyuk, T.A. Stepanyuk, Asymptotically best possible Lebesque-type inequalities for +the Fourier sums on sets of generalized Poisson integrals, FILOMAT, 34, №14, 4697–4707 +(2020). +22. A.S. Serdyuk, T.A. Stepanyuk, About Lebesgue inequalities on the classes of generalized +Poisson integrals, Jaen J. Approx. 12, 25–40 (2021). +23. И.А. Шакиров, О двусторонней оценке нормы оператора Фурье, Уфимск. матем. +журн., 10, №1, 96–117 (2018). +24. А.И. Степанец, Классификация периодических функций и скорость сходимости их +рядов Фурье, Изв. АН СССР. Сер. мат., 50, №1, 101–136 (1986). +25. А.И. Степанец, +Классификация и приближение периодических функций, Наукова +Думка, Киев (1987). +26. А.И. Степанец, Методы теории приближений: В 2 ч., Пр. Iн-ту математики НАН +України, Ин-т математики НАН Украины, Київ, 40, Ч. I (2002). + +29 +27. А.И.Степанец. Методы теории приближений: В 2 ч., Пр. Iн-ту математики НАН +України, Ин-т математики НАН Украины, Київ, 40, Ч. I (2002). +28. A.I. Stepanets, On the Lebesgue inequality on classes of (ψ, β)-differentiable functions, +Ukr. Math. J., 41, №4, 435–443 (1989). +29. А.И. Степанец, А.C. Сердюк, Неравенства Лебега для интегралов Пуассона, Укр. +мат. журн., 52, № 6, 798-808 (2000). +30. А.И. Степанец, А.С. Сердюк Приближение суммами Фурье и наилучшие приближе- +ния на классах аналитических функций, Укр. мат. журн., 52, №3, .375–395 (2000). +31. О.I. Степанець, А.С. Сердюк, А.Л. Шидлiч Про деякi новi критерiї нескiнченної ди- +ференцiйовностi перiодичних функцiй, Укр. мат. журн., 59, №10, 1399–1409 (2007) +32. А.И. Степанец, А.С. Сердюк, А.Л. Шидлич, Классификация бесконечно дифференци- +руемых функций Укр. мат. журн., 60, №12, 1686–1708 (2008). +33. С. Б. Стечкин Оценка остатка ряда Фурье для дифференцируемых функций. При- +ближение функций полиномами и сплайнами, Сборник статей, Тр. МИАН СССР, +145, 126–151 (1980). +34. С.А. Теляковский, О нормах тригонометрических полиномов и приближении диф- +ференцируемых функций линейными средними их рядов Фурье. I, Тр. Мат. ин-та АН +СССР, 62, 61–97 (1961). +35. С.А. Теляковский, Приближение дифференцируемых функций частными суммами +их рядов Фурье, Матем. заметки, 4, № 3, 291–300 (1968). +36. С.А. Теляковский, О приближении суммами Фурье функций высокой гладкости, +Укр. мат. журн., 41, № 4, 510–518 (1989). + diff --git a/T9A0T4oBgHgl3EQfEf8T/content/tmp_files/load_file.txt b/T9A0T4oBgHgl3EQfEf8T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8b76ef9beccf80ada0dc2f016cc7622f2a57f0f --- /dev/null +++ b/T9A0T4oBgHgl3EQfEf8T/content/tmp_files/load_file.txt @@ -0,0 +1,837 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf,len=836 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='02017v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='CA] 5 Jan 2023 УДК 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='5 А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк (Iнститут математики НАН України, Київ) Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанюк (Iнститут математики НАН України, Київ) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Serdyuk (Institute of Mathematics NAS of Ukraine, Kyiv) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Stepaniuk (Institute of Mathematics NAS of Ukraine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Kyiv) Рiвномiрнi наближення сумами Фур’є на множинах згорток перiодичних функцiй високої гладкостi Uniform approximations by Fourier sums on the sets of convolutions of periodic functions of high smoothness На множинах 2π–перiодичних функцiй f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' котрi задаються (ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' β)–iнтегралами вiд фун- кцiй ϕ iз L1 встановлено нерiвностi типу Лебега,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' в яких рiвномiрнi норми вiдхилень сум Фур’є виражаються через найкращi наближення в середньому тригонометричними полiно- мами функцiй ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Доведено асимптотичну непокращуванiсть одержаних оцiнок за умови, коли послiдовностi ψ(k) спадають до нуля швидше за довiльну степеневу функцiю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В рядi важливих випадкiв встановлено асимптотичнi рiвностi для точних верхнiх меж рiвномiрних наближень сумами Фур’є на класах (ψ, β)–iнтегралiв вiд функцiй ϕ, що на- лежать одиничнiй кулi з простору L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' On the sets of 2π–periodic functions f, which are defined with a help of (ψ, β)–integrals of the functions ϕ from L1, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' We prove that obtained estimates are best possible, in the case when the sequences ψ(k) decrease to zero faster than any power function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' In some important cases we establish the asymptotic equalities for the exact upper boundari- es of uniform approximations by Fourier sums on the classes of (ψ, β)–integrals of the functions ϕ, which belong to the unit ball of the space L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 2 1 Вступ Нехай L1 — простiр 2π–перiодичних сумовних на [0, 2π) функцiй f в якому норма задається формулою ∥f∥1 = 2π� 0 |f(t)|dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' L∞ — простiр вимiрних i суттєво обмежених 2π–перiодичних функцiй f з нормою ∥f∥∞ = ess sup t |f(t)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' C — простiр неперервних 2π–перiодичних фун- кцiй f, в якому норма означається рiвнiстю ∥f∥C = max t |f(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ(k) — довiльна фiксована послiдовнiсть дiйсних невiд’ємних чисел, i нехай β — фiксоване дiйсне число.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Позначимо через Cψ β L1 множину 2π–перiодичних функцiй, якi при всiх x ∈ R зображуються у виглядi згортки f(x) = a0 2 + 1 π π � −π Ψβ(x − t)ϕ(t)dt, a0 ∈ R, ϕ ∈ L1, ϕ ⊥ 1 (1) з твiрним ядром Ψβ вигляду Ψβ(t) = ∞ � k=1 ψ(k) cos � kt − βπ 2 � , ψ(k) ≥ 0, β ∈ R, (2) таким, що ∞ � k=1 ψ(k) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (3) Якщо функцiї f i ϕ пов’язанi рiвнiстю (1), то функцiю f в цьому спiввiдношеннi на- зивають (ψ, β)–похiдною функцiї f i позначають через f ψ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' З iншого боку функцiю f у рiвностi (1) називають (ψ, β)–iнтегралом функцiї ϕ i позначають через J ψ β ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Поняття (ψ, β)–похiдної ((ψ, β)–iнтеграла) введенi О.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанцем [24], [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Пiдмножину функцiй f з Cψ β L1 таких, що f ψ β ∈ B1, де B1 — одинична куля в просторi L1, тобто B1 := {ϕ : ||ϕ||1 ≤ 1} , будемо позначати через c Cψ β,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Зрозумiло, що умова (3) гарантує неперервнiсть твiрного ядра Ψβ(t) вигляду (2), а отже i iстиннiсть вкладення Cψ β L1 ⊂ C (Cψ β,1 ⊂ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' У випадку, коли ψ(k) = e−αkr, α > 0, r > 0, ядра Ψβ(t) вигляду (2) є узагальненими ядрами Пуассона, тобто Ψβ(t) = Pα,r,β(t), де Pα,r,β(t) = ∞ � k=1 ψ(k) cos � kt − βπ 2 � , α > 0, r > 0, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (4) При цьому множини Cψ β L1 та Cψ β,1 позначатимемо вiдповiдно через Cα,r β L1 та Cα,r β,1 i на- зиватемо множинами узагальнених iнтегралiв Пуассона, а вiдповiднi (ψ, β)–похiднi f ψ β та (ψ, β)–iнтеграли J ψ β ϕ позначатимемо через f α,r β та J α,r β ϕ вiдповiдно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 3 Простiр усiх тригонометричних полiномiв tn−1 порядку не вищого за n − 1 будемо по- значати через T2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай En(f)L1 — найкращi наближення в середньому тригонометри- чними полiномами tn−1 ∈ T2n−1, тобто En(f)L1 = inf tn−1∈T2n−1 ∥f − tn−1∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Позначимо через ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' x) вiдхилення вiд функцiї f з L1 її частинної суми Фур’є Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·) порядку n − 1 ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' x) := f(x) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (5) Норми ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C можна оцiнити зверху через найкращi рiвномiрнi наближення En(f)C = inf tn−1∈T2n−1 ∥f − tn−1∥C за допомогою нерiвностi Лебега ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ (1 + Ln−1)En(f)C, n ∈ N, f ∈ C, (6) де величини Ln−1 — константи Лебега сум Фур’є Ln−1 = 1 π π � −π |Dn−1(t)|dt = 2 π π 2 � 0 | sin(2n − 1)t| sin t dt, Dn−1(t) := 1 2 + ∞ � k=1 cos kt = sin(n − 1 2)t 2 sin t 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' При цьому, як встановив Фейєр [3], для констант Лебега Ln має мiсце асимптотична рiвнiсть Ln = 4 π2 ln n + O(1), n → ∞, (7) де O(1) — рiвномiрно обмежена по n величина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Бiльш точнi оцiнки для рiзниць Ln − 4 π2 ln(n + a), a > 0, при n ∈ N можна знайти в роботах [1], [2], [4], [10], [23], [5] та iн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' З урахуванням (7), нерiвнiсть (6) можна записати у виглядi ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ � 4 π2 ln n + O(1) � En(f)C, f ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (8) Незважаючи на загальнiсть, нерiвнiсть (8) на всьому просторi C є точною за порядком.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Бiльш того, вона є асимптотично непокращуваною в тому сенсi, що константа 4 π2 у формулi (8) зменшена бути не може.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Разом з тим, використання нерiвностей (6) i (8) для функцiй f iз функцiональних множин Cψ β L1 чи Cψ β,1 може виявитись неефективним.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Бiльш того, iснують послiдовностi ψ такi, що для f ∈ Cψ β L1 зазначенi нерiвностi є неточними навiть за порядком.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Щоб у цьому переконатись, покладемо ψ(k) = e−αk i розглянемо породженi такими послiдовностями 4 класи Cψ β,1 = Cα,1 β,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Як показано в [12] при всiх α > 0, β ∈ R справедлива асимптотична при n → ∞ рiвнiсть En(Cα,1 β,1)C = sup f∈Cα,1 β,1 ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C = e−αn � 1 π(1 − e−α) + O(1) n e−α (1 − e−α)2 � , (9) в якiй O(1) — рiвномiрно обмежена по α, β, n величина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Крiм того, (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', наприклад, [27, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 48]) для найкращих наближень En(Cα,1 β,1)C справе- дливi точнi за порядком оцiнки K(1)e−αn ≤ En(Cα,1 β,1)C ≤ K(2)e−αn, (10) в яких K(1) i K(2) — деякi додатнi сталi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi для f ∈ Cα,1 β,1 в силу (9) виконується нерiвнiсть ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ e−αn � 1 π(1 − e−α) + O(1) n e−α n(1 − e−α)2 � , (11) в той час як використання класичної нерiвностi Лебега та оцiнки (8) дозволяє записати бiльш грубу за порядком оцiнку ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ e−αn �4K(2) π2 ln n + O(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' У роботi [28] О.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанець, розглядаючи послiдовностi ψ(k), що спадають до нуля повiльнiше за будь-яку геометричну прогресiю, встановив аналоги нерiвностей Лебега для множин (ψ, β)–диференцiйовних функцiй Cψ β C ⊂ Cψ β L1, (Cψ β C — множина 2π–перiодичних функцiй f(x), якi при всiх x ∈ R зображуються у виглядi (1), де ϕ ∈ C) в яких норми вiдхилень ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C виражаються через найкращi наближення En(f ψ β )C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Одержанi в [28] нерiвностi виявились асимптотично точними не тiльки на всiх множинах Cψ β C, але i на де- яких важливих пiдмножинах iз Cψ β C, зокрема на класах Cψ β C0 = � f ∈ Cψ β C : ∥f ψ β ∥C ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Згодом дослiдження по встановленню асимптотично точних нерiвностей типу Лебега на множинах (ψ, β)–диференцiйовних функцiй були продовженi в роботах [29], [8], [9], [14], [30], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В данiй роботi буде знайдено нерiвностi типу Лебега на множинах Cψ β L1, в яких норми ∥ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' x)∥C виражаються через En(f ψ β )L1 i доведено їх асимптотичну непокращуванiсть у випадку, коли lim n→∞ 1 n ∞ � k=1 kψ(k + n) ∞ � k=n ψ(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (12) Також у роботi, за умови (12) знайдено розв’язок задачi Колмогорова–Нiкольського для сум Фур’є на класах Cψ β,1, яка полягає у вiдшуканнi асимптотичних рiвностей величин En(Cψ β,1)C = sup f∈Cψ β,1 ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (13) 5 Проблеми, пов’язанi зi знаходженням розв’язку задачi Колмогорова–Нiкольського для сум Фур’є на класах згорток дослiджувались в роботах [6], [11], [33], [35], [36], [24], [26], [12], [13], [20], [34], [15], [16] та iн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 2 Основнi результати Має мiсце наступне твердження.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Теорема 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ∞ � k=1 kψ(k) < ∞, ψ(k) ≥ 0, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', β ∈ R i n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi для довiльної функцiї f ∈ Cψ β L1 має мiсце нерiвнiсть ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ 1 π ∞ � k=n ψ(k)En(f ψ β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (14) Крiм того, для довiльної функцiї f ∈ Cψ β L1 можна знайти функцiю F(x) = F(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n, x) з множини Cψ β L1 таку, що En(F ψ β )L1 = En(f ψ β )L1 i має мiсце рiвнiсть ∥F(·) − Sn−1(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C = � 1 π ∞ � k=n ψ(k) + ξ n ∞ � k=1 kψ(k + n) � En(f ψ β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (15) В (15) величина ξ = ξ(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' β) є такою, що −2 ≤ ξ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Доведення Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай f ∈ Cψ β L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi згiдно з (1) в кожнiй точцi x ∈ R має мiсце iнтегральне представлення ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' x) = f(x) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' x) = 1 π π � −π f ψ β (t)Ψβ,n(x − t)dt, (16) де Ψβ,n(t) := ∞ � k=n ψ(k) cos � kt − βπ 2 � , β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (17) Функцiя Ψβ,n(t) ортогональна до будь–якого тригонометричного полiнома tn−1 порядку не вищого за n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' для довiльного полiнома tn−1 ∈ T2n−1 отримаємо ρn(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' x) = 1 π π � −π δn(t)Ψβ,n(x − t)dt, (18) де δn(·) = δn(ψ, β, tn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·) := f ψ β (·) − tn−1(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (19) Виберемо в якостi tn−1 в формулi (18) полiном t∗ n−1 найкращого наближення функцiї f ψ β в просторi L1, тобто такий що ∥f ψ β − t∗ n−1∥1 = En(f ψ β )L1, 6 Тодi, використовуючи нерiвнiсть ���� π � −π K(t − u)ϕ(u)du ���� C ≤ ∥K∥p′∥ϕ∥p, (20) ϕ ∈ Lp, K ∈ Lp′, 1 ≤ p ≤ ∞, 1 p + 1 p′ = 1 (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', наприклад, [7, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 43]), отримуємо ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ ������ 1 π π � −π (f ψ β (t) − t∗ n−1(t))Ψβ,n(· − t)dt ������ C ≤ 1 π∥Ψβ,n∥CEn(f ψ β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (21) Знайдемо двостороннi оцiнки норми ∥Ψβ,n∥C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Покажемо, що при всiх n ∈ N i β ∈ R справедлива оцiнка ∞ � k=n ψ(k) − π n ∞ � k=1 kψ(k + n) ≤ ∥Ψβ,n∥C ≤ ∞ � k=n ψ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (22) Оцiнка зверху для ∥Ψβ,n∥C в (22) випливає безпосередньо з (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Для оцiнки ∥Ψβ,n∥C знизу представимо функцiю Ψβ,n(t), яка означена формулою (17), у виглядi Ψβ,n(t) = gψ,n(t) cos � nt − βπ 2 � + hψ,n(t) sin � nt − βπ 2 � , (23) де gψ,n(t) := ∞ � k=0 ψ(k + n) cos kt, (24) hψ,n(t) := − ∞ � k=0 ψ(k + n) sin kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (25) Оскiльки величина ∥Ψβ,n∥C перiодична з перiодом 2 за параметром β, то, не зменшуючи загальностi, можна вважати, що β ∈ [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Позначимо t0 := βπ 2n , β ∈ [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (26) В силу (23) Ψβ,n(t0) = gψ,n(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (27) Тодi ∥Ψβ,n∥C ≥ |Ψβ,n(t0)| = |gψ,n(t0)| = |gψ,n(0) + (gψ,n(t0) − gψ,n(0))| ≥ |gψ,n(0)| − ����(gψ,n �βπ 2n � − gψ,n(0)) ���� = ∞ � k=n ψ(k) − ����(gψ,n �βπ 2n � − gψ,n(0)) ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (28) 7 Використовуючи теорему про середнє, маємо ����gψ,n �βπ 2n � − gψ,n(0) ���� ≤ ∥g ′ ψ,n∥C βπ 2n ≤ π n ∞ � k=1 kψ(k + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (29) Iз (28) i (29) випливає шукана оцiнка знизу для норм ∥Ψβ,n∥C в спiввiдношеннi (22) ∥Ψβ,n∥C ≥ ∞ � k=n ψ(k) − π n ∞ � k=1 kψ(k + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (30) Оцiнку (22) можна записати у виглядi формули ∥Ψβ,n∥C = ∞ � k=n ψ(k) + Θ1π n ∞ � k=1 kψ(k + n), (31) де для величини Θ1 = Θ1(n, β, ψ) виконуються нерiвностi − 1 ≤ Θ1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (32) Отже, iз (21) i (31) випливає нерiвнiсть (14) з Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Доведемо другу частину Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Для цього нам необхiдно для довiльної функцiї ϕ ∈ L1 знайти функцiю Φ(·) = Φ(ϕ, ·) ∈ L1 таку, що En(Φ)L1 = En(ϕ)L1 i для якої викону- ється рiвнiсть 1 π ������ π � −π � Φ(t) − t∗ n−1(t) � Ψβ,n(0 − t)dt ������ = � 1 π ∞ � k=n ψ(k) + ξ n ∞ � k=1 kψ(k + n) � En(ϕ)L1, (33) де t∗ n−1 — полiном найкращого наближення порядку n − 1 функцiї Φ в просторi L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В цьому випадку для функцiї f ∈ Cψ β L1 iснує функцiя Φ(·) = Φ(f ψ β ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·) така, що En(Φ)L1 = En(f ψ β )L1, i має мiсце формула (33), де в ролi ϕ виступає функцiя f ψ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Розглянемо функцiю F(·) = J ψ β (Φ(·) − a0 2 ), де a0 = a0(Φ) := 1 π π � −π Φ(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Функцiя F є шуканою функцiєю, оскiльки F ∈ Cψ β L1 i En(F ψ β )L1 = En(Φ − a0 2 )L1 = En(Φ)L1 = En(f ψ β )L1, i на пiдставi (18) i (33) має мiсце оцiнка (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 8 Доведемо (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай t∗ — точка з промiжку T = � π(1−β) 2n , 2π + π(1−β) 2n � , в якiй функцiя |Ψ−β,n(t)| набуває свого найбiльшого значення, тобто, |Ψ−β,n(t∗)| = ∥Ψ−β,n∥C = ∥Ψβ,n∥C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Покладемо ∆n k := � (k−1)π n + π(1−β) 2n , kπ n + π(1−β) 2n � , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Через k∗ позначимо номер такий, що t∗ ∈ ∆n k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Оскiльки функцiя Ψ−β,n є абсолютно неперервною, то для довiльного ε > 0 iснує сегмент ℓ∗ = [ξ∗, ξ∗ + δ] ⊂ ∆n k∗ такий, що для довiльного t ∈ ℓ∗ виконується нерiвнiсть |Ψβ,n(t)| > ∥Ψβ,n∥C − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Ясно, що mes ℓ∗ = |ℓ∗| = δ < π n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Для довiльного ϕ ∈ L1 i ε > 0 розглянемо функцiю Φε(t), яка на промiжку T означена за допомогою рiвностей Φε(t) = \uf8f1 \uf8f2 \uf8f3 En(ϕ)L1 1−ε(2π−δ) δ sign cos � nt + βπ 2 � , t ∈ ℓ∗, En(ϕ)L1ε sign cos � nt + βπ 2 � , t ∈ T \\ ℓ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (34) Для функцiї Φε(t) при достатньо малих значеннях ε > 0 (ε ∈ (0, 1 2π)) має мiсце наступна рiвнiсть: ∥Φε∥1 = En(ϕ)L1 1 − ε(2π − δ) δ � ℓ∗ ���sign cos � nt + βπ 2 ����dt + En(ϕ)L1ε � T\\ℓ∗ ���sign cos � nt + βπ 2 ����dt = En(ϕ)L1 �1 − ε(2π − δ) δ δ + ε(2π − δ) � = En(ϕ)L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (35) Крiм того, згiдно з (34) sign Φε(t) = sign cos � nt + βπ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (36) Оскiльки для довiльного тригонометричного полiнома tn−1 ∈ T2n−1 2π � 0 tn−1(t)sign cos � nt + βπ 2 � dt = 0, то, з урахуванням (36), виконується рiвнiсть 2π � 0 tn−1(t)sign � Φε(t) − 0 � dt = 0, tn−1 ∈ T2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Згiдно з Теоремою 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='5 роботи [7, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='28], полiном t∗ n−1 ≡ 0 є полiномом найкращого наближення функцiї Φε в метрицi простору L1, тобто, En(Φε)L1 = ∥Φε∥1, отже з (35) випливає En(Φε)L1 = En(ϕ)L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 9 Крiм того, для функцiї Φε 1 π π � −π (Φε(t) − t∗ n−1(t))Ψβ,n(−t)dt = 1 π π � −π Φε(t)Ψ−β,n(t)dt =1 − ε(2π − δ) πδ En(ϕ)L1 � ℓ∗ sign cos � nt + βπ 2 � Ψ−β,n(t)dt + ε πEn(ϕ)L1 � T\\ℓ∗ sign cos � nt + βπ 2 � Ψ−β,n(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (37) Враховуючи, що sign Φε(t) = (−1)k, t ∈ ∆(n) k , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 2n, а також вкладення ℓ∗ ⊂ ∆(n) k∗ , отримуємо ������ 1 − ε(2π − δ) πδ En(ϕ)L1 � ℓ∗ sign cos � nt + βπ 2 � Ψ−β,n(t)dt ������ = ������ (−1)k∗ 1 − ε(2π − δ) πδ En(ϕ)L1 � ℓ∗ Ψ−β,n(t)dt ������ ≥1 − ε(2π − δ) π En(ϕ)L1 (∥Ψβ,n∥C − ε) >1 − 2πε π En(ϕ)L1 (∥Ψβ,n∥C − ε) = 1 πEn(ϕ)L1 � ∥Ψβ,n|C − 2πε∥Ψβ,n∥C − ε + 2πε2� >En(ϕ)L1 � 1 π∥Ψβ,n∥C − ε � 2∥Ψβ,n∥C + 1 π �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (38) Крiм того, неважко переконатись, що ������� ε πEn(ϕ)L1 � T\\ℓ∗ sign cos � nt + βπ 2 � Ψ−β,n(t)dt ������� ≤ ε πEn(ϕ)L1∥Ψβ,n∥C(2π − δ) < 2εEn(ϕ)L1∥Ψβ,n∥C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (39) З формул (37)–(39) випливає наступна оцiнка: ������ π � −π 1 π(Φε(t) − t∗ n−1(t))Ψβ,n(−t)dt ������ >En(ϕ)L1 � 1 π∥Ψβ,n∥C − ε � 4∥Ψβ,n∥C + 1 π �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (40) Виберемо ε настiльки малим, щоб ε < π ∞ � k=1 kψ(k + n) n � 1 + 4π ∞ � k=n ψ(k) � (41) 10 i для цього ε покладемо Φ(t) = Φε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (42) Функцiя Φ(t) є шуканою функцiєю,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' оскiльки En(Φ)L1 = En(ϕ)L1 i згiдно з (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (40) i (41) ������ 1 π π � −π (Φ(t) − t∗ n−1(t))Ψβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(−t)dt ������ ≥ � 1 π ∞ � k=n ψ(k) − 1 n ∞ � k=1 kψ(k + n) − ε � 4 ∞ � k=n ψ(k) + 1 π �� En(ϕ)L1 ≥ \uf8eb \uf8ec \uf8ec \uf8ed 1 π ∞ � k=n ψ(k) − 1 n ∞ � k=1 kψ(k + n) − π ∞ � k=1 kψ(k + n) n � 1 + 4π ∞ � k=n ψ(k) � � 4 ∞ � k=n ψ(k) + 1 π � \uf8f6 \uf8f7 \uf8f7 \uf8f8 En(ϕ)L1 ≥ � 1 π ∞ � k=n ψ(k) − 2 n ∞ � k=1 kψ(k + n) � En(ϕ)L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (43) З формул (43), (21) i (22) випливає (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Теорему 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Зрозумiло, що формулу (14) теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 можна записати однотипно з формулою (15) у наступному виглядi: ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ � 1 π ∞ � k=n ψ(k) + Θ1 n ∞ � k=1 kψ(k + n) � En(f ψ β )L1, (44) де Θ1 = Θ1(n, β, ψ) задовольняє нерiвностi (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Теорема 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ∞ � k=1 kψ(k) < ∞, ψ(k) ≥ 0, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' i β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при усiх n ∈ N має мiсце формула En(Cψ β,1)C = 1 π ∞ � k=n ψ(k) + Θ2 n ∞ � k=1 kψ(k + n), (45) де для величини Θ2 = Θ2(n, β, ψ) виконуються нерiвностi −1 ≤ Θ2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Доведення.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Згiдно з (13) i (16) отримуємо, що En(Cψ β,1)C = 1 π sup ϕ∈B0 1 ���� π � −π ϕ(t)Ψβ,n(x − t)dt ���� C , (46) де Ψβ,n(·) означена рiвнiстю (17), а B0 1 := {ϕ ∈ L1 : ||ϕ||1 ≤ 1, ϕ ⊥ 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Беручи до уваги iнварiантнiсть множини B0 1 вiдносно зсуву аргументу, з (46) отримуємо En(Cψ β,1)C = 1 π sup ϕ∈B0 1 π � −π ϕ(t)Ψβ,n(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (47) 11 На основi спiввiдношення двоїстостi (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', напр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', [7]) маємо sup ϕ∈B0 1 π � −π Ψβ,n(t)ϕ(t)dt = inf λ∈R ∥Ψβ,n(t) − λ∥C, (48) Для знаходження двосторонньої оцiнки величини inf λ∈R ∥Ψβ,n(t) − λ∥C нам буде корисним наступне твердження, яке може знайти i самостiйне застосування.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Лема 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ(k) ≥ 0, ∞ � k=1 kψ(k) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при всiх β ∈ R i n ∈ N для кожної з величин I(1) n = I(1) n (ψ, β) := ∥Ψβ,n∥C, (49) I(2) n = I(2) n (ψ, β) := inf λ∈R ∥Ψβ,n(t) − λ∥C, (50) I(3) n = I(3) n (ψ, β) := 1 2 ���Ψβ,n � t + π n � − Ψβ,n(t) ��� C (51) виконуються формули I(j) n = ∞ � k=n ψ(k) + Θjπ n ∞ � k=1 kψ(k + n), j = 1, 2, 3, (52) в яких для будь-якої з величин Θj = Θj(n, β, ψ), j = 1, 2, 3, виконуються двостороннi оцiнки −1 ≤ Θj ≤ 0, j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Доведення Леми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Оскiльки inf λ∈R ∥Ψβ,n(t) − λ∥C ≤ ∥Ψβ,n∥C (53) i 1 2 ���Ψβ,n � t + π n � − Ψβ,n(t) ��� C ≤ inf λ∈R ∥Ψβ,n(t) − λ∥C, (54) то I(3) n ≤ I(2) n ≤ I(1) n , i, отже, необхiдна оцiнка зверху для кожної з величин I(j) n , j = 1, 2, 3 випливає з (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Залишається знайти оцiнку знизу для I(3) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В силу (23)–(25) i (51) 12 I(3) n =1 2 ���Ψβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n � t + π n � − Ψβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(t) ��� C ≥1 2 ���Ψβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n � t0 + π n � − Ψβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(t0) ��� =1 2 ���gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n � t0 + π n � cos � n � t0 + π n � − βπ 2 � + hψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n � t0 + π n � sin � n � t0 + π n � − βπ 2 � − � gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(t0) cos � nt0 − βπ 2 � + hψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(t0) sin � nt0 − βπ 2 �� ��� =1 2 ��� − gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n � t0 + π n � − gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(t0) ��� = 1 2 ���gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n �βπ − 2π 2n � + gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n �βπ 2n ���� =1 2 ����2gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(0) + � gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n �(β − 2)π 2n � − gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(0) � + � gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n �βπ 2n � − gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(0) ����� ≥ |gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(0)| − 1 2 ����gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n �(β − 2)π 2n � − gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(0) ���� − 1 2 ����gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n �βπ 2n � − gψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(0) ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (55) де,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' як i ранiше,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' t0 = βπ 2n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' β ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' За теоремою про середнє значення ����gψ,n �(β − 2)π 2n � − gψ,n(0) ���� ≤ ∥g ′ ψ,n∥C |β − 2|π 2n ≤ π n ∞ � k=1 kψ(k + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (56) Аналогiчно (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (29)) ����gψ,n �βπ 2n � − gψ,n(0) ���� ≤ π n ∞ � k=1 kψ(k + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (57) Об’єднуючи (55)-(57), одержуємо шукану оцiнку знизу для I(3) n I(3) n ≥ ∞ � k=n ψ(k) − π n ∞ � k=1 kψ(k + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (58) Лему 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='3 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' З формул (47), (48), (50) i (52) випливає, що En(Cψ β,1)C = 1 π ∞ � k=n ψ(k) + Θ2 n ∞ � k=1 kψ(k + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Теорему 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Зазначимо, що оцiнки (14), (15) i (45) є асимптотичними рiвностями при n → ∞, якщо виконується граничне спiввiдношення (12), тобто коли 1 n ∞ � k=1 kψ(k + n) = o � ∞ � k=n ψ(k) � , n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (59) Умова (59), як буде показано нижче, має мiсце у рядi важливих випадкiв, зокрема, коли послiдовнiсть ψ(k) спадає до нуля при k → ∞ швидше за довiльну степеневу послiдовнiсть 1 kr , r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 13 3 Наслiдки з Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 для класiв аналiтичних та цiлих фун- кцiй Наведемо приклади важливих функцiональних компактiв Cψ β,1, для яких формула (45) дозволяє записати асимптотичнi рiвностi для En(Cψ β,1)C при n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Розглянемо випадок, коли послiдовностi ψ(k) задовольняють умову Даламбера Dq, q ∈ [0, 1): lim k→∞ ψ(k + 1) ψ(k) = q, ψ(k) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (60) Якщо ψ(k) задовольняє умову (60) при деякому q ∈ [0, 1), то будемо записувати, що ψ ∈ Dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай спочатку q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Згiдно з Теоремою 5 роботи [32], твердження про iснування послiдовностi ψ ∈ D0 такої, що для функцiї f вiрне включення f ∈ Cψ β L1 при будь-якому β ∈ R, еквiвалентне твер- дженню про включення f ∈ E, де E — множина всiх 2π–перiодичних дiйснозначних на дiйснiй осi функцiй, якi допускають аналiтичне продовження на всю комплексну площи- ну.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Отже, класи Cψ β,1 при ψ ∈ D0 належать до множини 2π–перiодичних дiйснозначних на R цiлих функцiй.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ∞ � k=n+1 kψ(k) < ∞, ψ(k) ≥ 0, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', n ∈ N i β ∈ R, тодi має мiсце рiвномiрна вiдносно всiх параметрах оцiнка En(Cψ β,1)C = 1 πψ(n) + O(1) n ∞ � k=n+1 kψ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (61) Якщо, крiм того, ψ ∈ D0, то оцiнка (61) є асимптотичною рiвнiстю при n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Доведення.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Користуючись формулою (45) Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2, можна записати En(Cψ β,1)C = 1 πψ(n) + O(1) � ∞ � k=1 ψ(k + n) + 1 n ∞ � k=1 kψ(k + n) � = 1 πψ(n) + O(1) n ∞ � k=1 (k + n)ψ(k + n) = 1 πψ(n) + O(1) n ∞ � k=n+1 kψ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тим самим оцiнку (61) доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Покажемо, що при ψ ∈ D0 1 n ∞ � k=n+1 kψ(k) = o (ψ(n)) , n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (62) Виберемо номери n такими, щоб ψ(k + 1) ψ(k) < 1 2, k = n, n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (63) 14 Тодi, з урахуванням (63), маємо 1 n ∞ � k=1 kψ(k) = � 1 + 1 n � ψ(n + 1) + ψ(n + 1) n ∞ � j=2 (n + j) j−1 � ℓ=1 ψ(n + ℓ + 1) ψ(n + ℓ) < ψ(n + 1) � 2 + 1 n ∞ � j=2 2j 2j−1 � < ψ(n + 1) � 2 + 4 n ∞ � j=1 j 2j � < 10ψ(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (64) Оскiльки, в силу ψ ∈ D0 ψ(n + 1) = o (ψ(n)) , n → ∞, (65) то iз (64) i (65) випливає (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Зауважимо, що асимптотичну рiвнiсть (61) з залишковим членом, записаним в iншiй формi, було отримано ранiше в [12] i [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' При ψ ∈ D0 оцiнки залишкового члена в [12] i [13] є бiльш точними, нiж у формулi (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Типовими представниками послiдовностей, що задовольняють умову D0 є послiдовностi ψ(k) = e−αk−r, r > 1, α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Для породжуваних такими послiдовностями класiв Cψ β,1 = Cα,r β,1, одержуємо наступне твердження.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай r > 1, α > 0 i β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi, при n ≥ � 3 αr � 1 r − 1, n ∈ N, має мiсце рiвномiрна по всiх розглядуваних параметрах оцiнка En(Cα,r β,1)C = e−αnr�1 π + O(1)e−αrnr−1� 1 + 1 αr(n + 1)r−2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (66) Доведення.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' З формули (61) випливає, що En(Cα,r β,1)C = 1 πe−αnr + O(1) n ∞ � k=n+1 ke−αkr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (67) Легко переконатись, що при номерах n таких, що (n + 1)r > 1 αr 1 n ∞ � k=n+1 ke−αkr < 1 n \uf8eb \uf8ed(n + 1)e−α(n+1)r + ∞ � n+1 te−αtrdt \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (68) Iнтегруючи частинами, отримуємо ∞ � n+1 te−αtrdt = ∞ � n+1 t2 1 αrtr � −e−αtr�′ dt ≤ 1 αr(n + 1)r ∞ � n+1 t2 � −e−αtr�′ dt = 1 αr(n + 1)r \uf8eb \uf8ed(n + 1)2e−α(n+1)r + 2 ∞ � n+1 te−αtrdt \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (69) 15 З останньої нерiвностi маємо � 1 − 2 αr(n + 1)r � ∞ � n+1 te−αtrdt ≤ (n + 1)2e−α(n+1)r αr(n + 1)r , (70) що рiвносильно тому, що ∞ � n+1 te−αtrdt ≤ e−α(n+1)r αr(n + 1)r−2 αr(n + 1)r αr(n + 1)r − 2 = e−α(n+1)r αr(n + 1)r−2 � 1 + 2 αr(n + 1)r − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (71) Зi спiввiдношень (68) i (71) випливає, що 1 n ∞ � k=n+1 ke−αkr = O(1) � e−α(n+1)r + e−α(n+1)r αr(n + 1)r−2 � 1 + 2 αr(n + 1)r − 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (72) Об’єднавши (67) i (72), одержуємо, що при всiх номерах n таких, що (n + 1)r > 3 αr En(Cα,r β,1)C = 1 πe−αnr + O(1) � e−α(n+1)r + e−α(n+1)r αr(n + 1)r−2 � 1 + 2 αr(n + 1)r − 2 �� =e−αnr � 1 π + O(1) � e−αrnr−1 + e−αrnr−1 αr(n + 1)r−2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Формулу (66) iз залишковим членом, записаним дещо в iншому виглядi було отримано в [12] i [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' При цьому оцiнки з [12] i [13] мiстять бiльш точнi оцiнки залишкового члена нiж у (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай далi q ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Згiдно з Теоремою 3 роботи [32], твердження про iснування по- слiдовностi ψ ∈ Dq, q ∈ (0, 1) такої, що для функцiї f вiрне включення Cα,1 β L1 при будь- якому β ∈ R еквiвалентне твердженню про включення f ∈ A, де A — множина всiх 2π–перiодичних дiйснозначних на дiйснiй осi функцiй, якi допускають аналiтичне продов- ження на деяку смугу |Imz| < c, c > 0, комплексної площини.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Отже класи Cψ β,1, ψ ∈ Dq, 0 < q < 1, складаються з перiодичних, аналiтичних у смузi |Im z| < c функцiй, при цьому c = ln 1 q (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', наприклад, [25, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Послiдовностi ψ(k) = e−αk, α > 0 належать до множини Dq при q = e−α, а вiдповiднi класи Cψ β,1 = Cα,1 β,1 породжуються ядрами Пуассона Pα,1,β(t) = ∞ � k=1 e−αk cos � kt − βπ 2 � , α > 0, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (73) Iз Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 для класiв Cα,1 β,1 отримуємо наступне твердження.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 16 Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай α > 0 i β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi, при всiх n ∈ N має мiсце рiвнiсть En(Cα,1 β,1)C = e−αn �1 π 1 1 − e−α + Θ n e−α (1 − e−α)2 � , (74) де для величини Θ = Θ(n, α, β) виконуються нерiвностi −1 ≤ Θ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Доведення.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Покладемо q = e−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi, з Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 випливає, що при всiх n ∈ N En(Cα,1 β,1)C = 1 π ∞ � k=n qk + Θ n ∞ � k=0 kqk+n = 1 π qn 1 − q + Θ n � ∞ � k=n kqk − n ∞ � k=n qk � = 1 π qn 1 − q + Θ n �nqn(1 − q) + qn+1 (1 − q)2 − nqn 1 − q � = 1 π qn 1 − q + Θ n qn+1 (1 − q)2, (75) де була використана наступна рiвнiсть: ∞ � k=n kqk = nqn(1 − q) + qn+1 (1 − q)2 , q ∈ (0, 1), n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='3 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Оцiнка (74) уточнює асимптотичнi рiвностi для величин En(Cα,r β,1)C, якi були встановленi в [12] i [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Асимптотичнi рiвностi для величин En(Cψ β,1)C при ψ ∈ Dq, q ∈ (0, 1), мiстяться у наступному твердженнi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ ∈ Dq, q ∈ (0, 1), β ∈ R, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при всiх номерах n таких, що 1 n + εn < 1 − q 2 , (76) де εn := sup k≥n ���� ψ(k + 1) ψ(k) − q ���� , (77) має мiсце рiвномiрна вiдносно всiх розглядуваних параметрiв оцiнка En(Cψ β,1)C = ψ(n) � 1 π(1 − q) + O(1) � q n(1 − q)2 + εn (1 − q)2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (78) Доведення.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' З Леми 1 роботи [30] випливає, що при ψ ∈ Dq, 0 < q < 1, n ∈ N, має мiсце рiвнiсть ∞ � k=n ψ(k) = ψ(n) � 1 qn ∞ � k=n qk + rn � , (79) де для залишку rn при всiх номерах n таких, що εn < 1 − q 2 (80) 17 виконується оцiнка |rn| ≤ εn (1 − q − εn)(1 − q) ≤ 2εn (1 − q)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (81) Очевидно, що якщо ψ ∈ Dq, 0 < q < 1, то i послiдовнiсть kψ(k) також задовольняє умову Dq, а тому знову ж таки в силу Леми 1 iз [30] ∞ � k=n+1 kψ(k) = (n + 1)ψ(n + 1) � 1 qn+1 ∞ � k=n+1 qk + r∗ n+1 � , (82) де для залишку r∗ n+1 при усiх номерах n таких, що ε∗ n+1 := sup k≥n+1 ���� ψ(k + 1)(k + 1) ψ(k)k − q ���� < 1 − q 2 (83) виконується оцiнка ��r∗ n+1 �� ≤ 2ε∗ n+1 (1 − q)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (84) Iз означень величин εn i ε∗ n+1 (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (77) i (83)) маємо ε∗ n+1 ≤ sup k≥n+1 ���� ψ(k + 1) ψ(k) − q ���� + 1 n + 1 = εn+1 + 1 n + 1 < εn + 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (85) Iз (85) видно, що виконання нерiвностi (76) гарантує i виконання нерiвностей (80) i (83), а отже i оцiнок (81) i (84) для залишкiв у рiвностях (79) i (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi в силу оцiнки (45) Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 i рiвностей (79) i (82) випливає,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' що при всiх номерах n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' якi задовольняють умову (76),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' справджуються спiввiдношення En(Cψ β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1)C = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='ψ(k) + O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='kψ(k + n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='=ψ(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='qn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='qk + rn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='+ O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(k − n)ψ(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='=ψ(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 − q + O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='εn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='+ O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='kψ(k) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='ψ(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='=ψ(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='π(1 − q) + O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='εn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='+O(1)ψ(n + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='qn+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='qk + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='ε∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='qn+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='k=n+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='qk + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='εn+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='=ψ(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='π(1 − q) + O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='εn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='+ O(1)ψ(n + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(1 − q) + εn + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='=ψ(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='π(1 − q) + O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='εn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 + ψ(n + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='ψ(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='=ψ(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='π(1 − q) + O(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='εn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='(1 − q)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='n(1 − q)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (86) Наслiдок 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='4 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 18 Асимптотичнi рiвностi (78) вперше були встановленi в роботах [12] i [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 4 Наслiдки з Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 для класiв нескiнченно диференцiйов- них функцiй В даному пiдроздiлi будемо вважати, що послiдовностi ψ(k), що породжують множини Cψ β L1 та Cψ β,1, є звуженням на множину натуральних чисел деяких додатних неперервних опуклих донизу функцiй ψ(t) неперервного аргументу t ≥ 1, що прямують до нуля при t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Множину всiх таких функцiй ψ позначають через M: M= � ψ∈C[1, ∞): ψ(t)>0, ψ(t1 − 2ψ((t1 + t2)/2) + ψ(t2) ≥ 0 ∀t1, t2 ∈ [1, ∞), lim t→∞ ψ(t)=0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (87) Наслiдуючи О.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанця (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', наприклад, [26, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 160]), кожнiй функцiї ψ ∈ M поста- вимо у вiдповiднiсть характеристики η(t) = η(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' t) = ψ−1 �1 2ψ(t) � та µ(t) = µ(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' t) = t η(t) − t, де ψ−1 — обернена до ψ функцiя, i покладемо M+ ∞ = {ψ ∈ M : µ(t) ↑, t → ∞} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Через Mα позначимо пiдмножину всiх функцiй ψ ∈ M, для яких величина α(t) = α(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' t) := ψ(t) t|ψ′(t)|, ψ′(t) := ψ′(t + 0), (88) спадає до нуля при t → ∞: Mα = � ψ ∈ M : lim t→∞ α(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' t) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (89) Згiдно з Теоремою 2 роботи [31], твердження про iснування послiдовностi ψ ∈ Mα (або ψ ∈ M+ ∞), такої, що для функцiї f вiрне включення f ∈ Cψ β L1 при будь-якому β ∈ R, еквiвалентне твердженню про включення f ∈ D∞, де D∞ — множина всiх нескiнченно диференцiйовних 2π-перiодичних дiйснозначних функцiй.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А отже, класи Cψ β,1 при ψ ∈ Mα (або ψ ∈ M+ ∞), є класами нескiнченно диференцiйовних перiодичних функцiй.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В тiй же роботi було показано, що має мiсце включення M+ ∞ ⊂ Mα ⊂ M∞ = � ψ ∈ M : ∀r > 0 lim t→∞ trψ(t) = 0 � , (90) яке означає, що функцiї ψ(t) iз Mα спадають до нуля швидше за довiльну степеневу функцiю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 19 Для величин En(Cψ β,1)C, ψ ∈ M+ ∞ за умови η(n) − n > 2 вiдомi точнi порядковi рiвностi En(Cψ β,1)C ≍ ψ(n)(η(n) − n), (91) якi спiвпрадають з точними порядковими рiвностями для найкращих рiвномiрних набли- жень тригонометричними полiномами порядку n − 1 En(Cψ β,1)C = inf tn−1∈T ∥f − ttn−1∥C а саме, (див.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', наприклад, [17]) En(Cψ β,1)C ≍ En(Cψ β,1)C ≍ ψ(n)(η(n) − n), (92) (тут i надалi запис A(n) ≍ B(n) для додатних послiдовностей A(n) i B(n) означає iснува- ння додатних констант K1 i K2 таких, що K1B(n) ≤ A(n) ≤ K2B(n), n ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Як показано в [27, с.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 166] для довiльної функцiї ψ iз M+ ∞ має мiсце порядкова рiвiнсть η(t) − t ≍ λ(t), t ≥ 1, (93) де λ(t) — характеристика вигляду λ(t) = λ(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' t) := ψ(t) |ψ′(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (94) З урахуванням (91) можна записати у виглядi En(Cψ β,1)C ≍ ψ(n)λ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (95) Наступне твердження мiстить сильну асимптотику величин En(Cψ β,1)C , ψ ∈ Mα при деяких природних обмеженнях на α(t) i λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Теорема 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай β ∈ R, ψ ∈ M i характеристики (88) i (94) задовольняють умови α(t) ↓ 0, (96) λ(t) ↑ ∞, t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (97) Тодi для всiх n ∈ N таких, що α(n) ≤ 1 4 (98) виконується оцiнка En(Cψ β,1)C = ψ(n)λ(n) �1 π + ξ1 λ(n) + ξ2α(n) � , (99) де −1 ≤ ξ1 ≤ 1 + 1 π та −4 ≤ ξ2 ≤ 4 3 � 1 + 1 π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 20 Доведення Теореми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Для оцiнки величини En(Cψ β,1)C використаємо формулу (45) iз Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' При цьому нам буде необхiдно знайти оцiнки рядiв Σ1 = ∞ � k=n ψ(k) та Σ2 = ∞ � k=n kψ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В силу монотонного спадання функцiї ψ ∈ M бачимо, що ∞ � n ψ(t)dt ≤ ∞ � k=n ψ(k) ≤ ψ(n) + ∞ � n ψ(t)dt, (100) а, отже, ∞ � k=n ψ(k) = ∞ � n ψ(t)dt + Θ4ψ(n), 0 ≤ Θ4 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (101) Оцiнка iнтеграла ∞� n ψ(t)dt випливає з наступного твердження, яке може мати i само- стiйний iнтерес.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Лема 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ ∈ M, λ(t) монотонно неспадає, а α(t) монотонно незростає на [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при всiх a ≥ 1, таких, що α(a) < 1, виконуються оцiнки λ(a)ψ(a) ≤ ∞ � a ψ(t)dt ≤ λ(a)ψ(a) � 1 + α(a) 1 − α(a) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (102) Доведення Леми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Оскiльки в силу включення ψ ∈ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' функцiя ψ(t) є локально аб- солютно неперервною на [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' то,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' враховуючи монотонне неспадання λ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' одержуємо шукану оцiнку знизу I1 := ∞ � a ψ(t)dt = ∞ � a −ψ′(t)λ(t)dt ≥ λ(a) ∞ � a (−ψ′(t))dt = ψ(a)λ(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (103) З iншого боку,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' враховуючи монотонне незростання функцiї α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' i застосовуючи метод iнтегрування частинами,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' маємо I1 = ∞ � a ψ(t)dt = ∞ � a α(t)(−ψ′(t)t)dt ≤ α(a) ∞ � a (−ψ′(t)t) =α(a) \uf8eb \uf8edψ(a)a + ∞ � a ψ(t)dt \uf8f6 \uf8f8 = ψ(a)λ(a) + α(a)I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (104) З (104) одержуємо, що I1 ≤ λ(a)ψ(a) 1 − α(a) = λ(a)ψ(a) � 1 + α(a) 1 − α(a) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (105) Iз (103) i (105) випливає (102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Лему 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 21 Застосування Леми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 при a = n, n ∈ N, за умови (98) дозволяє записати, що I1 = ∞ � a ψ(t)dt = ψ(n)λ(n) (1 + Θ5α(n)) , 0 ≤ Θ5 ≤ 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (106) Отже, з урахуванням (101) i (106) при α(n) ≤ 1 4 ∞ � k=n ψ(k) = ψ(n)λ(n) � 1 + Θ4 λ(n) + Θ5α(n) � , 0 ≤ Θ5 ≤ 4 3, 0 ≤ Θ4 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (107) Далi знайдемо оцiнку для Σ2 = ∞ � k=n kψ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В силу (98) функцiя tψ(t) спадає на [n, ∞), а тому ∞ � n tψ(t)dt ≤ ∞ � k=n kψ(k) ≤ nψ(n) + ∞ � n tψ(t)dt, (108) i, отже, ∞ � k=n kψ(k) = ∞ � n tψ(t)dt + Θ6nψ(n), 0 ≤ Θ6 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (109) Для оцiнки iнтеграла I2 = ∞� n tψ(t)dt знову використаємо метод iнтегрування частинами i врахуємо (90) та умову незростання α(n) I2 = ∞ � n tψ(t)dt = ∞ � n t2 ψ(t) −tψ′(t)(−ψ′(t))dt ≤ α(n) ∞ � n t2(−ψ′(t))dt =α(n) � n2ψ(n) + 2I2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (110) З останнiх спiввiдношень i умови (98) маємо I2 (1 − 2α(n)) ≤ α(n)n2ψ(n) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' отже,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' з урахуванням умови (98) маємо I2 ≤ψ(n)n2α(n) 1 1 − 2α(n) = ψ(n)n2α(n) � 1 + 2α(n) 1 − 2α(n) � ≤ψ(n)n2α(n) (1 + 4α(n)) = ψ(n)nλ(n)(1 + 4α(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (111) З iншого боку, з урахуванням умови (106), I2 = ∞ � n tψ(t)dt ≥ n ∞ � n ψ(t)dt ≥ ψ(n)nλ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (112) 22 Об’єднання (111) i (112) дозволяє записати, що при α(n) ≤ 1 4 ∞ � n tψ(t)dt = ψ(n)nλ(n) (1 + Θ7α(n)) , 0 ≤ Θ7 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (113) Iз формул (109) i (113) випливає, що за умов (94) i (98) ∞ � k=n kψ(k) = ψ(n)nλ(n) � 1 + Θ7α(n) + Θ6 λ(n) � , 0 ≤ Θ7 ≤ 4, 0 ≤ Θ6 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (114) Користуючись оцiнками (108) i (107), одержуємо 1 n ∞ � k=1 kψ(k + n) = 1 n ∞ � k=0 kψ(k + n) =1 n � ∞ � k=n kψ(k) − n ∞ � k=n ψ(k) � =1 n ∞ � k=n kψ(k) − ∞ � k=n ψ(k) =ψ(n)λ(n) � 1 + Θ7α(n) + Θ6 λ(n) � − ψ(n)nλ(n) � 1 + Θ5α(n) + Θ4 λ(n) � =ψ(n)λ(n) � (Θ7 − Θ5)α(n) + Θ6 − Θ4 λ(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (115) На пiдставi формули (45) iз Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 та оцiнок (107) та (115), отримуємо, що для всiх номерiв n таких, що виконується нерiвнiсть (98) En(Cψ β,1)C = 1 π ∞ � k=n ψ(k) + Θ2 1 n ∞ � k=1 kψ(k + n) = 1 πψ(n)λ(n) � 1 + Θ4 λ(n) + Θ5α(n) � +Θ2ψ(n)λ(n) � 1 + Θ6 − Θ4 λ(n) + (Θ7 − Θ5)α(n) � =ψ(n)λ(n) �1 π + Θ4/π + Θ2(Θ6 − Θ4) λ(n) + �Θ5 π + Θ2(Θ7 − Θ5) � α(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (116) Оскiльки для величини ξ1 = Θ4 π + Θ2(Θ6 − Θ4) виконується оцiнка −1 ≤ ξ1 ≤ 1 + 1 π, а для ξ2 = Θ5 π + Θ2(Θ7 − Θ5) — оцiнка −4 ≤ ξ2 ≤ 4 3 � 1 + 1 π � , то iз (116) випливає (99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Теорему 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 23 Наведемо наслiдок з Теореми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 у випадку, коли ψ(t) = e−αt−r, α > 0, 0 < r ≤ 1, тобто коли класи Cψ β,1 є класами узагальнених iнтегралiв Пуассона Cα,r β,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Легко переконатись, що для вказаних ψ(t) при всiх t ≥ 1, λ(t) = t1−r αr , α(t) = 1 αrtr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (117) Iз (117) видно, що умови (96) i (97) Теореми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 виконуються.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' При цьому виконання нерiвностi 1 αrnr ≤ 1 4 рiвносильне виконанню умови (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Отже, з Теореми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 випливає наступне твердження.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай 0 < r < 1, α > 0, β ∈ R, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при всiх n ≥ � 4 αr � 1 r справедлива рiвномiрно обмежена по всiх розглядуваних параметрах оцiнка En(Cα,r β,1)C = e−αnrn1−r� 1 παr + O(1) � 1 (αr)2 1 nr + 1 n1−r �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (118) Зазначимо, що оцiнка вигляду (118) при дещо жорсткiших обмеженнях на n була зна- йдена у роботах [18]– [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' У зазаначених роботах мiстяться двостороннi оцiнки величини O(1) через абсолютнi сталi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наведемо ще декiлька прикладiв застосування Теореми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 для рiзних функцiй ψ iз M, якi задовольняють умовам (96) i (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Будемо розглядати ψ(t) вигляду ψ(t) = (t + 2)− ln ln(t+2), t ≥ 1, (119) ψ(t) = e− ln2(t+1), t ≥ 1, (120) ψ(t) = e− t+2 ln(t+2), t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (121) Для зазначених функцiй ψ(t) знайдемо характеристики λ(t) i α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Результати обчи- слень вiдображено в наступнiй таблицi: № Функцiя ψ(t) α(t) λ(t) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (t + 2)− ln ln(t+2) t+2 t 1 1+ln ln(t+2) t+2 1+ln ln(t+2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' e− ln2(t+1) t+1 t 1 2 ln(t+1) t+1 2 ln(t+1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' e− t+2 ln(t+2) ln2(t+2) t(ln(t+2)−1) ln2(t+2) ln(t+2)−1 Iз Теореми 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1 i наведених в таблицi значень α(t) i λ(t) отримуємо асимптотичнi при n → ∞ рiвностi для величин En(Cψ β,1)C у випадку, коли ψ мають вигляд (119)–(121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ(k) = (k + 2)− ln ln(k+2), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', β ∈ R i n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при n → ∞ виконується асимптотична рiвнiсть En(Cψ β,1)C = 1 πψ(n) n ln ln(n + 2) + O(1)ψ(n) n (ln ln(n + 2))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (122) 24 Наслiдок 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ(k) = e− ln2(k+1), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', β ∈ R i n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='Тодi при n → ∞ має мiсце асимптотична рiвнiсть En(Cψ β,1)C = 1 2π ψ(n)n ln(n + 1) + O(1)ψ(n) n ln2(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (123) Наслiдок 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ(k) = e− k+2 ln(k+2), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', β ∈ R i n ∈ N Тодi при n → ∞ має мiсце асимптотична рiвнiсть En(Cψ β,1)C = 1 πψ(n) ln(n + 2) + O(1)ψ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (124) Зауважимо, що у випадку, коли ψ ∈ M i при t → ∞ α(t) → 0 i λ(t) → ∞ за додаткової умови, що функцiя ψ(t) є диференцiйовною скрiзь на [1, ∞), граничне спiввiдношення (12), яке гарантує той факт, що оцiнки (15) i (45) є асимптотичними рiвностями, завжди виконується.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Дiйсно, застосувавши правило Лопiталя, маємо lim n→∞ ∞� n ψ(t)dt ψ(n) = lim n→∞ ψ(n) |ψ′(n)| = lim n→∞ λ(n) = ∞, (125) lim n→∞ ∞� n tψ(t)dt nψ(n) = lim n→∞ −nψ(n) ψ(n) + nψ′(n) = lim n→∞ λ(n) 1 − α(n) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (126) Тодi, з урахуванням (101) i (109) мають мiсце асимптотичнi рiвностi ∞ � k=n ψ(k) = ∞ � n ψ(t)dt + O(1)ψ(n), (127) ∞ � k=n kψ(k) = ∞ � n tψ(t)dt + O(1)nψ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (128) Використовуючи формули (125)–(109) i застосувавши правило Лопiталя,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' отримуємо lim n→∞ 1 n ∞ � k=1 kψ(k + n) ∞ � k=n ψ(k) = lim n→∞ 1 n ∞ � k=1 kψ(k) − ∞ � k=n ψ(k) ∞ � k=n ψ(k) = lim n→∞ 1 n ∞ � k=1 kψ(k) ∞ � k=n ψ(k) − 1 = lim n→∞ 1 n ∞� n tψ(t)dt ∞� n ψ(t)dt − 1 25 = lim n→∞ −nψ(n) ∞� n ψ(t)dt − nψ(n) − 1 = lim n→∞ − ∞� n ψ(t)dt ∞� n ψ(t)dt − nψ(n) = lim n→∞ ψ(n) −2ψ(n) − nψ′(n) = lim n→∞ ψ(n) −nψ′(n) 1 − 2ψ(n) −nψ′(n) = lim n→∞ α(n) 1 − α(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (129) Тим самим рiвнiсть (12) доведено.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 5 Коментарi щодо нерiвностей Лебега У пiдроздiлах 3 i 4 були наведенi наслiдки з Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2 для швидко спадних послiдов- ностей ψ(k), для яких формула (45) є асимптотичною рiвнiстю, або, що те саме, коли справджується (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Зрозумiло, що у всiх розглянутих у пiдроздiлах 3 i 4 частинних ви- падках для ψ(·) легко одержати i асимптотично непокращуванi нерiвностi типу Лебега вигляду (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Ми обмежидись лише формулюванням лише деяких тверджень, якi випли- вають iз Теореми 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Спочатку сформулюємо вiдповiднi твердження для ψ(t) = e−αtr, α > 0 i r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Випадки r > 1, r = 1 i r ∈ (0, 1) видiляються окремо.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай r > 1, α > 1 i β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при n ≥ � 3 αr � 1 r − 1 для довiльної функцiї f ∈ Cα,r β L1 має мiсце нерiвнiсть ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ e−αnr � 1 π + O(1)e−αnr−1 � 1 + 1 αr(n + 1)r−2 �� En(f α,r β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (130) Крiм того, для довiльної функцiї f ∈ Cα,r β L1 можна знайти функцiю F(x) = F(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n, x) з множини Cα,r β L1 таку, що En(F α,r β )L1 = En(f α,r β )L1 i має мiсце рiвнiсть ∥F(·) − Sn−1(F(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C = e−αnr � 1 π + O(1)e−αnr−1 � 1 + 1 αr(n + 1)r−2 �� En(f α,r β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (131) У (130) i (131) O(1) — рiвномiрно обмежена по всiх параметрах величина.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Аналоги нерiвностi (130) i формули (131), яка доводить асимптотичну непокращува- нiсть зазначеної нерiвностi, в яких залишковi члени записанi в дещо iншiй формi, отри- мано в [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай α > 0, β ∈ R i n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi для довiльної функцiї f ∈ Cα,1 β L1 має мiсце нерiвнiсть ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ 1 π e−αn 1 − e−αEn(f α,1 β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (132) Крiм того, для довiльної функцiї f ∈ Cα,1 β L1 можна знайти функцiю F(x) = F(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n, x) з множини Cα,1 β L1 таку, що En(F α,1 β )L1 = En(f α,1 β )L1 i має мiсце рiвнiсть ∥F(·) − Sn−1(F(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C = e−αn �1 π 1 1 − e−α + ξ n 1 (1 − e−α)2 � En(f α,1 β )L1, (133) 26 де для величини ξ = ξ(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' β) виконується нерiвнiсть −2 ≤ ξ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Оцiнки (132) i (133) уточнюють оцiнки рiвномiрних вiдхилень сум Фур’є на множинах iнтегралiв Пуассона Cα,1 β L1, що були одержанi в роботах [14] i [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай 0 < r < 1, α > 0, β ∈ R i n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при всiх n ≥ � 4 αr � 1 r для довiльної функцiї f ∈ Cα,r β L1 має мiсце нерiвнiсть ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ e−αnrn1−r � 1 παr + O(1) � 1 (αr)2 1 nr + 1 n1−r �� En(f α,r β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (134) Крiм того, для довiльної функцiї f ∈ Cα,r β L1 можна знайти функцiю F(x) = F(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n, x) з множини Cα,r β L1 таку, що En(F α,r β )L1 = En(f α,r β )L1 i має мiсце рiвнiсть ∥F(·) − Sn−1(F(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C = e−αnrn1−r � 1 παr + O(1) � 1 (αr)2 1 nr + 1 n1−r �� En(f α,r β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (135) У (134) i (135) O(1) — величини, що рiвномiрно обмеженi по всiх параметрах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' При дещо жорсткiших обмеженнях на n формули вигляду (134) i (135) були встановленi ранiше в [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Наслiдок 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай ψ ∈ Dq, q ∈ (0, 1), β ∈ R i n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi при всiх n таких, що виконується нерiвнiсть (77) для довiльної функцiї f ∈ Cψ β L1 має мiсце нерiвнiсть ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ ψ(n) � 1 π(1 − q) + O(1) � q n(1 − q)2 + εn (1 − q)2 �� En(f ψ β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (136) Крiм того, для довiльної функцiї f ∈ Cψ β L1 можна знайти функцiю F(x) = F(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n, x) з множини Cψ β L1 таку, що En(F ψ β )L1 = En(f ψ β )L1 i таку, що при виконаннi (76) для неї має мiсце рiвнiсть ∥F(·) − Sn−1(F(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C = ψ(n) � 1 π(1 − q) + O(1) � q n(1 − q)2 + εn (1 − q)2 �� En(f ψ β )L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' (137) У (136) i (137) величина εn означена рiвнiстю (77), а O(1) — величини, що рiвномiрно обмеженi по всiх параметрах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Теорема 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нехай β ∈ R, ψ ∈ M i характеристики α(t) i λ(t) вигляду (88) i (94) задовольняють умови (96) i (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Тодi для всiх n ∈ N таких, що α(n) < 1 4 для будь-якої функцiї f ∈ Cψ β L1 виконується нерiвнiсть ∥f(·) − Sn−1(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C ≤ ψ(n)λ(n) �1 π + ξ3 λ(n) + ξ4α(n) � En(f ψ β )L1, (138) де 0 ≤ ξ3 ≤ 4 3π, 0 ≤ ξ4 ≤ 1 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Крiм того, для довiльної функцiї f ∈ Cψ β L1 можна знайти функцiю F(x) = F(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' n, x) з множини Cψ β L1 таку, що En(F ψ β )L1 = En(f ψ β )L1 i при n ∈ N таких, що α(n) < 1 4 має 27 мiсце рiвнiсть ∥F(·) − Sn−1(F(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ·)∥C = ψ(n)λ(n) �1 π + ξ5 λ(n) + ξ6α(n) � En(f ψ β )L1, (139) де −2 ≤ ξ5 ≤ 2 + 1 π, −8 ≤ ξ6 ≤ 4 3 � 2 + 1 π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Нерiвнiсть (138) є наслiдком формул (14) та (107), а рiвнiсть (139) випливає iз (15), (107) та (115).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Н.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Ахиезер, Лекции по теории аппроксимации, Мир, Москва (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='К.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Жук, Г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='Натансон, Тригонометрические ряды и элементы теории аппрокси- мации, Изд-во Ленинг.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' ун-та (1983).' metadata={'source': 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+page_content=' Мусiєнко, А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, Нерiвностi типу Лебега для сум Валле Пуссена на множинах аналiтичних функцiй, Укр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 65, № 4, 522-537 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Мусiєнко, А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, Нерiвностi типу Лебега для сум Валле Пуссена на множинах цiлих функцiй, Укр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 65, № 5, 642–653 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Г.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Натансон, Об оценке констант Лебега сумм Валле–Пуссена, Геометрические вопросы теории функций и множеств, Калинин (1986).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 10, №3, 207–256 (1946).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, Наближення класiв аналiтичних функцiй сумами Фур’є в рiвномiрнiй метрицi, Укр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 57, № 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 1079–1096 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, Наближення класiв аналiтичних функцiй сумами Фур’є в метрицi простору Lp, Укр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 57, № 10, 1395–1408 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 28 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Мусiєнко, Нерiвностi типу Лебега для сум Валле Пуссена при наближеннi iнтегралiв Пуассона, Збiрник праць Iнституту математики НАН Укра- їни, 7, № 1: Теорiя наближення функцiй та сумiжнi питання, Київ: Iн-т математики НАН України, 298–316 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Serdyuk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Sokolenko,Approximation by Fourier sums in classes of differenti- able functions with high exponents of smoothness, Methods of Functional Analysis and Topology, 25, № 4, 381–387 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Соколенко, Наближення сумами Фур’є на класах диференцiйовних у сенсi Вейля – Надя функцiй iз високим показником гладкостi, Укр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 74, № 5, 685 –700 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк , Оцiнки найкращих наближень класiв нескiнченно диференцiйовних функцiй в рiвномiрнiй та iнтегральнiй метриках , Укр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 66, №9, 1244– 1256 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанюк, Рiвномiрнi наближення сумами Фур’є на класах згор- ток з iнтегралами Пуассона, Допов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' НАН України, № 11, 10–16 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк, Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанюк, Наближення класiв узагальнених iнтегралiв Пуассона сумами Фур’є в метриках просторiв Ls, Укр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 69, № 5, 695-704 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Serdyuk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Stepanyuk, Uniform approximations by Fourier sums on classes of generalized Poisson integrals, Analysis Mathematica, 45, №1, 201–236 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Serdyuk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Stepanyuk, Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals, FILOMAT, 34, №14, 4697–4707 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Serdyuk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Stepanyuk, About Lebesgue inequalities on the classes of generalized Poisson integrals, Jaen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 12, 25–40 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Шакиров, О двусторонней оценке нормы оператора Фурье, Уфимск.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 10, №1, 96–117 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанец, Классификация периодических функций и скорость сходимости их рядов Фурье, Изв.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' АН СССР.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' мат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 50, №1, 101–136 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанец, Классификация и приближение периодических функций, Наукова Думка, Киев (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанец, Методы теории приближений: В 2 ч.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', Пр.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Iн-ту математики НАН України, Ин-т математики НАН Украины, Київ, 40, Ч.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' I (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 29 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='Степанец.' metadata={'source': 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журн.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=', 52, № 6, 798-808 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='И.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Степанец, А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content='С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} +page_content=' Сердюк 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+page_content=', 41, № 4, 510–518 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9A0T4oBgHgl3EQfEf8T/content/2301.02017v1.pdf'} diff --git a/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf b/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..019f072f4db2f9e6295def87324d98e1c53377e6 --- /dev/null +++ b/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:73ee24fe7892a2a226de376a005f42c22b9ea454c5aee92ae9e99dd0e1851c32 +size 11688649 diff --git a/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf b/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5d68c408ada635ff5dc3b22d98f9d41677eac723 --- /dev/null +++ b/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Gerard,1, 2 Daren Dillon,2 Sylvain Cetre,3, 4 and Rebecca Jensen-Clem2 +1Lawrence Livermore National Laboratory +2University of California Santa Cruz +3Durham University +4Wakea Consulting +(Received Dec. 20, 2022; Revised Jan. 23, 2023; Accepted Jan. 25, 2025) +Submitted to PASP +ABSTRACT +Focal plane wavefront sensing and control is a critical approach to reducing non- +common path errors between the a conventional astronomical adaptive optics (AO) +wavefront sensor (WFS) detector and science camera. However, in addition to mitigat- +ing non-common path errors, recent focal plane wavefront sensing techniques have been +developed to operate at speeds fast enough to enable “multi-WFS” AO, where residual +atmospheric errors are further corrected by a focal plane WFS. Although a number +of such techniques have been recently developed for coronagraphic imaging, here we +present one designed for non-coronagraphic imaging. Utilizing conventional AO system +components, this concept additionaly requires (1) a detector imaging the focal plane of +the WFS light source and (2) a pupil plane optical chopper device that is non-common +path to the first WFS and is synchronized to the focal plane imager readout. These min- +imal hardware requirements enable the temporal amplitude modulation to resolve the +sine ambiguity of even wavefront modes for both low, mid, and high wavefront spatial +frequencies. Similar capabilities have been demonstrated with classical phase diversity +by defocusing the detector, but such techniques are incompatible with simultaneous +science observations. This optical chopping techniqe, however, enables science imaging +at up to a 50% duty cycle. We present both simulations and laboratory validation of +this concept on SEAL, the Santa Cruz Extreme AO Laboratory testbed. +Keywords: adaptive optics +1. INTRODUCTION +Astronomical adaptive optics (AO) has enabled ground-breaking discoveries over the past three +decades, from the Nobel prize-winning imaging and dynamical analysis of the Galactic center (Ghez +et al. 2005) to the first images of planets around other stars on Solar System scales (Marois et al. +2008). However, advancements over the years has illuminated a new limitation to further improving +Corresponding author: Benjamin L. Gerard +gerard3@llnl.gov +arXiv:2301.11282v1 [astro-ph.IM] 26 Jan 2023 + +2 +performance of AO systems: non-common path aberrations (NCPAs) between an AO wavefront +sensor (WFS) and AO-fed science detector. These NCPAs are fundamentally not measurable with +a standard AO WFS and therefore not correctable (spatially or temporally) by an AO deformable +mirror (DM) without additional wavefront sensing methods. +NCPAs are among the dominant terms in error budgets of current AO systems (e.g., Poyneer et al. +2016), and as such there has been significant recent development of NCPA correction techniques +for AO systems (e.g., Bottom et al. 2016; Potier et al. 2022). Considering the amplitude, A, and +phase, φ, of the electromagnetic field from a star at infinity placed on the DM of an AO system (i.e., +conjugated to the telescope pupil), a downstream focal plane point spread function (PSF) image is +described by +PSF = |F{A ei φ}|2, +where F is a Fourier transform operator, | |2 represents square modulus of the focal plane electric +field and i is √−1. A PSF at perfect focus is inherently degenerate to certain modes of φ, including +the sine of symmetric Zernike modes (e.g., focus, spherical aberration, etc.) and the relative phase of +Fourier modes (e.g., a 3 cycle/pupil sine vs. cosine), and therefore additional techniques are needed +to sense the wavefront from a science image and then update DM commands and/or AO WFS offsets +to accordingly compensate for NCPAs. +NCPA measurement and correction techniques, also known as focal plane wavefront sensing and +control, generally fall into one of two categories to resolve the wavefront measurement degeneracy +with a PSF: temporal or spatial modulation. Temporal modulation involves recording a series of two +or more images with the science camera where a known probe is applied between images, resolving +the wavefront ambiguity with “temporal diversity” (e.g., Mugnier et al. 2008; Bord´e & Traub 2006). +Conversely, techniques with spatial modulation can in principle use a single focal image for wavefront +reconstruction, leveraging custom hardware and/or software to resolve the wavefront measurement +degeneracy, e.g., with fringes (Baudoz et al. 2006) or calibrated symmetry (Miller et al. 2017) in the +focal plane image. +In general, temporal modulation-based focal plane wavefront sensing techniques use iterative algo- +rithms designed for stable, space telescope-like environments. In this paper, we introduce a temporal +modulation-based focal plane WFS designed for correction of residual AO turbulence down to mil- +lisecond timescales using an optical chopper device. +We note that this technique was originally +introduced in Gerard et al. (2022b), but is expanded and presented more verbosely here. In §2 we +present the concept of pupil chopping for focal plane wavefront sensing and present simulations. We +then present validating laboratory results using the Santa Cruz Extreme AO Laboratory (SEAL) +testbed in §3, provide further discussion in §4, and then conclude in §5. Unless otherwise noted, +simulations in this paper assume an operating wavelength of λ = 1.6µm, a 256×256 pixel image size, +and beam ratio (number of pixels per λ/D) of 3. +2. CONCEPT AND SIMULATIONS +2.1. Concept +Figure 1 illustrates the concept of focal plane wavefront sensing with a pupil plane optical chopper. +Two focal plane images are required to enable a WFS measurement: one with the chopper blade +partially blocking the pupil, and one with the pupil unblocked; the latter can be used for science +while the former is limited to wavefront sensing. A “reference” WFS frame is saved before on-sky + +3 +DM +Optics + +first stage +WFS +Light from +telescope +Real-time +control +science +camera + +second +stage +WFS +Optics + +pupil +chopper +(dichroic/ +beam-splitter) +1. Chopped pupil +2. Airy disk PSF +3. Chopped pupil PSF +4. 3-2 (WFS frame) +5. Tip response +(double difference) +6. Tilt response +7. Focus response +8. Negative focus +response +9. Fourier mode response +(x,y=5,0 c/p sin) +10. x,y=5,0 c/p cos +11. x,y=0,5 c/p sin +12. x,y=0,5 c/p cos +(a) +(b) +Figure 1. Illustration of the pupil chopping concept, both from a geometric (a) and Fourier optics (b) +perspective. In (b), all focal plane images are shown with the same 10×10 λ/D field of view. Panels 2-3 +are shown on a log scale; all other panels are shown on a linear scale. Zernike and Fourier modes in panels +5-12 all use 1 nm amplitudes. Panels 7-12 indicate that pupil chopping for focal plane wavefront sensing +can resolve the sign ambiguity of focus and the relative phase of Fourier modes, which is not possible with +a single PSF image. +operations and then used analogously to the concept of on-sky reference slopes for other pupil plane +WFSs. Such reference slopes enable a real-time WFS frame to measure the wavefront down to the +flatness level of the pre-calibrated reference image. In equation form, the on-sky WFS frame, w, at +an instance in time is given by +w = chop{φon-sky} − chop{φref}, with +chop{φ} ≡ +� +|F{Achopeiφ}|2 − |F{Anormeiφ}|2� +/Σ{|F{Anormeiφ}|2}, +where Anorm is an unobscured pupil wavefront amplitude (i.e., an open slot of the chopper wheel), +Achop represents a chopped pupil wavefront amplitude, φon-sky is the on-sky pupil wavefront phase, +φref is the reference pupil wavefront present during the above mentioned daytime calibration, and +Σ{} represents an operator for summing pixel values within a given image. To be clear, φref cannot +be generated or reconstructed with this technique, and must be obtained by some other non-linear +iterative wavefront sensing and/or phase retrieval technique to flatten the absolute wavefront at the +science detector plane (e.g., with phase diversity from Lamb et al. 2017 and/or the asymmetric pupil +mask from Martinache 2013; see §4). However, the advantage of this double difference approach is +that it enables a linear wavefront reconstruction from w space to DM command space (i.e., optimal +for real-time wavefront control of AO residuals), which we will describe and illustrate in the next +section. +Due to the implementation of a continuously spinning blade, the duty cycle for a chopped or un- +chopped image is limited to a maximum of 1 +2(1 − dbeam +dblade), where dbeam is the beam diameter and dblade +is the chopper blade width (assuming equal spacing between obscuring and unobscuring slots of the +blade; so if dblade = dbeam, there is no time to keep the beam un-chopped and the duty cycle is 0, +whereas the maximum duty cycle with a very tiny pupil and/or large blade, dbeam +dblade ≈ 0, is 0.5). To +clarify, for this technique dblade ≥ dbeam (i.e., a blade size smaller than the beam size would prevent + +4 +either frame from seeing a unobscured pupil). A phase delay is also needed to acquire chopped and +un-chopped pupils in consecutive focal plane images, without which in every other frame the pupil +would be either completely blocked and then un-blocked or chopped by a fraction, f, on one side and +then symmetrically chopped by 1-f on the other side in the next frame. +By being a focal plane wavefront sensor, this method benefits from natural wavefront spatial fil- +tering properties (i.e., with the focal plane image representing the Fourier plane of the pupil plane +wavefront). This configuration enables binary masks to be placed on the focal plane image WFS +that act as a natural anti-aliasing filter and minimize non-linear cross talk between modes. Using the +focal plane image and such binary masks for wavefront sensing with this technique also distinguishes +this technique from the differential optical transfer function method (Codona 2013). This method is +also complementary to pupil amplitude diversity-based absolute phase retrieval methods, such as the +asymmetric pupil Fourier WFS (Martinache 2013). Our pupil chopping technique enables a linear- +least squares reconstruction (as we will show next in §2.2), optimal for high speed wavefront control +of AO residuals but not able to flatten the absolute phase below a pre-calibrated best flat (e.g., due to +evolving on-sky quasi-static aberrations), whereas phase retrieval methods can use the chopped pupil +PSF image to recover the absolute phase to track such quasi-static effects, but typically requiring +non-linear iterative algorithms that cannot be run at high speed. See §4 for a further discussion +about combining these two complementary approaches. +2.2. Low Order Wavefront Reconstruction and Control +With the setup described in §2.1, we enable a linear matrix vector multiply (MVM) reconstructor +for real-time wavefront control as described in Appendix A. Figure 2 shows the result of open loop +MVM-based modal coefficients for 13 Zernike modes. Fig. 2 clearly shows the potential of this focal +Figure 2. Simulated open-loop reconstruction of modal coefficients for different Zernike modes and ampli- +tudes (mode 0 in this case refers to tip). The peak-to-valley IM amplitude is 1 nm for all modes, mapping +modal coefficients to reconstructed wavefront in nm. +plane wavefront sensing technique as a linear WFS for input wavefront phases with wavefront errors +less than 1 radian rms (i.e., the standard regime in which a Taylor expansion of the PSF is linearly +related to the pupil wavefront phase). + +10.0 +input: mode0,10nm +input: mode3,8nm +input: mode0,-10nm +input: mode3,-8nm +7.5 +input: mode1,-6nm +input: mode10,5nm +input: mode1,6nm +input: mode10,-5nm +5.0 +2.5 +(wu) indno +0.0 +-2.5 +-5.0 +-7.5 +-10.0 +0 +2 +4 +6 +8 +10 +12 +Zernike mode number5 +2.3. Limitations to High Order Wavefront Reconstruction +Using this pupil chopping technique, we carried out a detailed analysis of achievable residual wave- +front error on input extreme AO residual turbulence (i.e., a static phase screen normalized to 100 nm +rms from 0 to 10 c/p with a -2 power law, similar to Poyneer et al. 2016, with results medianed over +100 different random realizations) as a function of DM actuator count, illustrated in Figure 3. The +15 +20 +25 +30 +Nact = # of DM actuators across pupil +10 +1 +102 +105 +108 +1011 +maximum achievable WFE gain +0.0 +0.2 +0.4 +0.6 +chopped pupil fraction +Nact=14 +Nact=16 +Figure 3. Grid search simulations, optimizing all adjustable parameters in the wavefront reconstruction +code as a function of DM actuator count (left) and chopped pupil fraction for two different actuator counts +(right). The y-axis for both panels shows the ratio of input (100 nm rms) to convergent output wavefront +errors due to non-linearities (i.e. without aliasing or photon noise) after 12 iterations, medianed over 100 +random realizations for each data point shown. Interaction matrix amplitude, SVD cutoff, and the radius +for a binary mask to isolate sine spots from Fourier modes are optimized for all data points in both panels, +with chopped pupil fraction additionally optimized for the left panel. +left panel of Fig. 3 shows that pupil wavefront spatial frequencies greater than around 9 c/p do not +converge to a deeper WFE than the input AO residual level, while spatial frequencies less than this +limit in principle can gain in convergent vs. input WFE by many orders of magnitude. Intuitively, +this 9 c/p wavefront measurement limit is because light compared with an un-modulated Airy disk +is only modulated in a chopped frame out to about 9 λ/D with the current chopper blade concept. +We will propose possible solutions to this limit and discuss this further in §4. The right panel of Fig. +3 further illustrates performance dependence on what fraction of the pupil is chopped, showing that +∼30-50% enables sufficient gains, which will justify our later laboratory implementation in §3. +3. LABORATORY TESTING +3.1. Setup +Our laboratory setup, mimicking Fig. 1a, uses SEAL—the Santa Cruz Extreme AO Laboratory +(Gerard et al. 2022a, Jensen-Clem et al. 2021). We use a Thorlabs KLS635 laser (set at 0.15 mW +unless otherwise noted), ALPAO 97 actuator DM (which also serves as the system pupil stop, Andor +Zyla 5.5 sCMOS detector, Thorlabs WFS-20 SHWFS (for this paper used only to flatten the ALPAO +DM), a Thorlabs MC2000B optical chopper with blade MC1F10, and a Stanford Instruments DG535 + +6 +controller and an in-house electrical doubler so that the Andor detector readout is synchronized +with the optical chopper (with the chopper as the leader and the Andor camera as the follower1) to +produce a chopped and un-chopped pupil every other frame, respectively. As in Gerard et al. (2022a), +ALPAO DM units are converted to WFE units via a single scalar multiplication, calibrated via open +tip/tilt measurements and corresponding PSF images, but for which higher order conversions become +increasingly wrong due to the decreasing stroke limits of the DM as a function of mode order. +Unless otherwise noted, our standard laboratory configuration for chopper WFS image acquisition +is with the chopper blade running at 100 Hz, using the doubler so the Andor camera follower runs +at 200 Hz with a 0.1 ms exposure per frame (i.e., a 2 % duty cycle), and also implementing a 2.5ms +phase delay for the doubled chopper signal with the Stanford controller. As in Gerard et al. (2022a), +we also implement a serial 20 ms pause in between acquiring a chopper pair of images and sending +any DM commands (both for calibration and real-time control software) due to additional latency +from our Python interface to hardware components. Two random chopper imager pair acquisitions +are not necessarily in the same order (i.e., one could have the chopped frame first while the other +could have the un-chopped frame first), but for consistency we order every chopper sequence with +the chopped frame first, as in Equation 1, determined as the frame with less cumulative flux. A serial +timing diagram outlining the procedure described in this section, which is used for real-time control, +is shown in Table 1. Due to the continuous sequence of images (camera integrations starting at 0, +time (ms) +0 +0.1 +5 +7.5 +7.5 +7.6 +12.5 +32.5 +loop +component +|F{Achopeiφon-sky}|2 +phase delay +|F{Anormeiφon-sky}|2 +20 ms pause +loop +timing +start +camera +integration +stop +camera +integration +finish +camera +duty +cycle +end +delay +start +camera +integration +stop +camera +integration +finish +camera +duty +cycle +apply +DM +commands +Table 1. Chopper timing diagram used for real-time control in §3.4. +7.5, 10, 17.5, 20, ... etc ms) being a non-integer multiple of the 32.5 ms total loop time shown in +Table 1, a continuous real-time buffer is constantly acquiring chopped and un-chopped images and +the real-time loop just grabs the most recent images from this buffer, meaning that effectively the 20 +ms pause dominates the lag budget in our setup, is why subsequent real-time control results in §3.4 +assume a ∼50 Hz loop rate. +Since the light source remains at a constant intensity, we also just use a single normalization +value for all data rather than generate a new value for each chopper pair as equation 1 suggests, +but for changing light source intensities a moving average cumulative intensity should be considered +to minimize noise propagation into this normalization term. The 0.15 mW setting for our testbed +light source is set to place the PSF core at ∼2000 analog to digital units (ADUs, ∼half the Andor +Zyla’s 4196 ADU dynamic range), which from photon scaling relations (Bessell et al. 1998) for real- +time purposes translates to a ∼90% Strehl ratio from an extreme AO residual wavefront for 1 ms +exposure, mH=7 star, 3% spectral bandpass centered at 1.6 µm (reasonable for a narrow band focal +1 Although the AO field has historically adopted “master and slave” terminology from the computer science field to +refer to the chopper and camera in this configuration, respectively, it is now clear that such discriminatory terminology +should no longer be used. Following the computer science field, instead we will refer to such configuration as “leader +and follower” throughout this paper, encouraging other members of the AO field to do the same. + +7 +plane WFS setup to avoid blurring effects from PSF magnification with wavelength), and 50% total +sky-to-photoelectron throughput. Given that we can only control the 97 actuator ALPAO DM up +to ∼5 cycles per pupil, only pixel values within a pre-defined 5λ/D radius of the star are used for +subsequent low order signal processing. The real-time differential image architecture of this technique +lends itself to self-calibration, so dark subtraction or other “cosmetic” calibrations are not needed +here. We found better performance when applying Zernike modes were non-illuminated actuators +were set to the commands of their nearest neighbor rather than set to zero, and so we use the former +hereafter in this paper unless noted otherwise. +3.2. Low Order Linearity +After following the procedure outlined in §2.2, in Fig. 4 we generate reconstructed modal coefficients +for low order Zernike modes, probing a range of input amplitudes for each mode in open loop and +recording corresponding reconstructed output modal coefficients. In general, the cross terms in Fig. +5 are negligible2 when the given input mode is reconstructed within it’s linear regime (≲200 nm +PV for each modal group, for tip/tilt corresponding to 5 mas for a 8m telescope at λ = 1.6µm), +which as expected is only feasible for AO residual WFEs. This also confirms that, as expected, +a diffraction-limited AO residual wavefront is needed an an input in order for this pupil chopping +technique to enable closing the loop. Figure 4 also shows that linearities are computed separately +in groups for (1) tip, tilt, and focus, astigmatism and coma, and (2) Zernike modes with n=4→5. +We found that the increased flexibility to use different SVD cutoffs for each group helped improve +linearity and lower cross talk, motivating our choice to use this modal grouping. We also found that +averaging 100 chopper frames per target image used for wavefront reconstruction to measure linearity +gave significantly better stability than shorter sequences. The cause of such instability is described +in the next section. +3.3. Limitations to High Order Wavefront Reconstruction +Unfortunately, we were not able to measure and control of Zernike modes with n > 5 due to chopper +blade phase jitter causing a time-varying chopped pupil fraction at the 1% rms level, illustrated in +Fig. 5. This phase jitter is due to (1) chopper motor control variations and (2) imperfections of in the +uniformity of the chopper blades. Our measurements in Fig. 5 are consistent with the phase jitter +specifications from Thorlabs, with no other available models able to reach smaller phase jitter levels. +We found that ≳100 nm amplitudes were needed for a given Fourier mode to detect sine spots at +spatial frequencies ≳3 c/p, which if used in the IM produces a quadratic linearity curve (i.e., creating +a reconstructor that is unable to tell positive from negative input amplitudes). From simulations, ≲2 +nm amplitudes for such Fourier modes are needed to instead produce a good linearity curve, meaning +a ≳50x reduction in phase jitter would be needed to enable higher wavefront order control. See §4 +for further discussion on future solutions to improve this phase jitter limitation. +3.4. Closed-Loop Operations +Building on our measured linear modal basis in §3.2, in this section we will present closed-loop +results on-air (§3.4.1) and with DM-based input AO residual turbulence (§3.4.2). +2 Ignoring n,m=5,±1 (which we use Fig. 4c to motivate not controlling and accordingly note in the subfigure caption), +in this context we use “negligible” to mean that for a given probed Zernike mode (1) it shows close-to-linear behavior +for both positive and negative input amplitudes, and (2) cross terms do not reach amplitudes beyond the probed +dominant mode over the linear range of (1). That being said, we acknowledge that decreased performance for both +dominant mode and cross term non-linearities will result in decreased achievable closed-loop WFE and may require a +modified calibration and/or control scheme, which are also discussed further in §3.4 and §4. + +8 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +reconstructed output +( m WFE, PV) +n,m=1,-1 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=1,1 +0.25 0.00 0.25 +input ( m WFE, PV) +0.25 +0.00 +0.25 +n,m=2,0 +y=x +n,m=1,-1 +n,m=1,1 +n,m=2,0 +(a) Linearity for tip, tilt, and focus. +0.25 0.00 0.25 +0.25 +0.00 +0.25 +reconstructed output +( m WFE, PV) +n,m=2,-2 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=2,2 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=3,-3 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=3,-1 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=3,1 +0.25 0.00 0.25 +input ( m WFE, PV) +0.25 +0.00 +0.25 +n,m=3,3 +y=x +n,m=2,-2 +n,m=2,2 +n,m=3,-3 +n,m=3,-1 +n,m=3,1 +n,m=3,3 +(b) Linearity for n = 2 →3. +0.25 0.00 0.25 +0.25 +0.00 +0.25 +reconstructed output +( m WFE, PV) +n,m=4,-4 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=4,-2 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=4,0 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=4,2 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=4,4 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=5,-5 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=5,-3 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=5,-1 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=5,1 +0.25 0.00 0.25 +0.25 +0.00 +0.25 +n,m=5,3 +0.25 0.00 0.25 +input ( m WFE, PV) +0.25 +0.00 +0.25 +n,m=5,5 +y=x +n,m=4,-4 +n,m=4,-2 +n,m=4,0 +n,m=4,2 +n,m=4,4 +n,m=5,-5 +n,m=5,-3 +n,m=5,-1 +n,m=5,1 +n,m=5,3 +n,m=5,5 +(c) Linearity for n = 4 →5, informing our decision to not control the highly non-linear modes n,m=5,±1. +Figure 4. Linearity curves for Zernike modes n = 1 → 5, separated into different modal groups (a-c), which +are separately optimized by interaction matrix amplitudes and different command matrix SVD cutoffs. + +9 +Figure 5. +Chopper phase jitter analysis, showing the cumulative chopped pupil frame intensity as a +fraction relative to the sequence-averaged cumulative un-chopped pupil frame intensity. The light source is +saturating all illuminated pixels during this dataset, ensuring that variations shown here are due to chopper +phase jitter, which are shown here to be present at the ±1% level, ultimately preventing high order wavefront +reconstruction for spatial frequencies ≳ 2.5 c/p. +3.4.1. On-air Stabilization +On-air closed-loop results are shown in Fig. 6. Although the stabilized environment from our +0 +200 +400 +600 +800 +1000 +iteration number +10 +3 +10 +2 +low order WFE ( m rms) +TTF +n=1 +5 +loop closed +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +log10[low order WFE ( m rms)] +0.0 +0.5 +1.0 +1.5 +2.0 +normalized kernel density +TTF +n=1 +5 +open loop +closed loop +(a) +(b) +Figure 6. Real-time correction of on-air wavefront error (measured by the chopper WFS at a frame rate +of about 50 Hz), showing both tip/tilt/focus (TTF) and all controlled modes (n = 1 → 5), both in the +time domain (a) and by WFE distribution (computed using the kdeplot function of the seaborn package; +Waskom 2021) comparing the open and closed loop sequences (b; i.e., respectively to the left and right of +the dotted line in panel a). +granite optical table allows only a moderate gain, we still demonstrate a closed-loop WFE reduction + +mean +/-std=0.805+/-0.01 +0.83 +100 +choppedpupililluminationfraction +0.82 +10-2 +0.81 +10-4 +PSD +0.80 +10-6 +0.79 +10-8 +0.78 +0 +1000 +2000 +3000 +4000 +5000 +100 +101 +choppedpupilframenumber(at11oHz) +temporalfrequency(Hz)10 +of 1.2x for all controlled modes relative to ambient air (i.e., the median of the solid vs. +dotted +orange distributions in Fig. 6b is 1.2), illustrating the potential of this technique to stabilize quasi- +static temporal disturbances on-sky (e.g., thermal variations, flexure, and additional sources of slow +beam wander). Closed-loop control was manually tuned for optimal performance using a integrator +controller with gains of 0.01, 0.003, and 0.008 for the TTF, low order Zernike, and high order Zernike +modal groups, respectively; such low gains were necessary for loop stability due to SEAL’s highly +stabilized ∼nm-level on-air WFE (Gerard et al. 2022a), which as shown is mostly below the noise floor +of individual frames of Fig. 6. Note these results help address concerns that the spinning chopper +wheel would be generating additional turbulence, since telemetry from closing the loop on-air clearly +shows improvement compared to open-loop. +It is important to note here that we obtained the results in Fig. 6 with a high-speed referencing setup +designed to minimize chopper blade phase jitter limitations, as discussed in §3.3, and/or integrated +effects of on-air turbulence that the chopper may be generating. Specifically, both chop{φref} from +equation 1 and all command matrices are obtained just before the real-time open and closed loop +telemetry sequence begins. This strategy ensures that chop{φref} and the command matrices do +not suffer from the loss of information due to wavefront “blurring” effects, such as chopper blade +phase jitter, on-air turbulence evolving, and/or quasi-static evolution of optics, since we found that +a reference flat frame and 18 Zernike mode probes (each of which are taken relative to a reference +flat before the next mode is probed) acquired at ∼50 Hz sufficiently freezes the on-air turbulence and +chopper blade phase jitter effects. Other strategies we tried to obtain chop{φref} did not work as well, +such as longer exposure acquisition in attempt to “average out” these turbulent on-air effects, which, +corroborated by Fig. 5, did not work as well. The disadvantage of this fast referencing technique, +however, is that single frames can be noisier than other approaches, preventing the deepest convergent +WFE this approach can provide; we discuss this further in §4. +3.4.2. Real-time AO Residual Turbulence +Fig. 7 summarizes our results for closing the loop on DM-induced AO residual turbulence, clearly +demonstrating a performance improvement in both WFS telemetry and observed Strehl ratio. Input +turbulence is normalized to a 100 nm rms, -2 power law phase screen and translated assuming a +single 10 m/s frozen flow ground layer for a 10m telescope, with chopper pair images acquired at 50 +Hz. We also use the same high-speed referencing and calibration technique for simulated AO residual +turbulence here as described in §3.4.1 for on-air real-time control. Closed-loop control was manually +tuned for optimal performance using a leaky integrator controller with gains of 0.7, 0.5, and 0.3 for +the TTF, low order Zernike, and high order Zernike modal groups, respectively, and with a leak of +0.95 for all modal groups. +Fig. +7a and b shows a total WFE reduction of ∼2.2x, roughly consistent with the measured +Strehl ratio enhancement from 73 to 92% (in Fig. 7c) via the Mar´ecehal approximation. A clear +reduction of all controlled modes empirically demonstrates sufficiently minimal cross-talk between +different modal groups that use different IM amplitudes and SVD cutoffs, as initially motivated in +§3.2. Although it could still be the case that non-linearities dominate the closed-loop error budget in +Fig. 7, such an error budget analysis is beyond the scope of this first introductory paper of the pupil +chopping concept, and regardless our results show that these non-linearities are at least ∼2.2x below +an extreme AO residual WFE, which is an important result. Although the closed loop vs. open loop +WFE histogram mean values clearly decrease, their distribution widths remain unchanged, suggesting + +11 +0 +500 +1000 +1500 +2000 +iteration number +10 +2 +10 +1 +low order WFE ( m rms) +TTF +n=1 +5 +loop closed +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +log10[low order WFE ( m rms)] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +normalized kernel density +TTF +n=1 +5 +open loop +closed loop +(a) +(b) +open loop +closed loop +0 +2 +4 +radial separation ( /D) +10 +2 +10 +1 +100 +median normalized intensity +OL +CL +flat +(c) +Figure 7. Analogous to Figure 6, (a) and (b) show WFE vs. time and open and closed loop distributions for +AO residual turbulence applied on the DM. Note that open loop WFE values for the n = 1 → 5 group may be +under- or over-estimated as these levels approach the linear ranges of some modes in Fig. 4. However, panel +(c) shows the stack of un-chopped images over the open and closed loop sequences and the corresponding +intensity profile for both sequences compared to a static best flat image with no turbulence applied, showing +a ∼20% Strehl improvement by closing the loop on AO residuals. +either our chosen control parameters are not yet fully optimized for robust closed-loop stability and/or +similar sources of noise (e.g., chopper phase jitter) account for the distribution width in both cases. +Although these results assume use of a second stage “cascade” AO system (Cerpa-Urra et al. +2022; i.e., a separate DM and WFS have produced the residual AO phases used as input into these +experiments, this is an important first step in demonstrating the potential of this technique. The +clear benefit in Fig. 7 of closed loop control in the stack of un-chopped images (i.e., science frames) +and corresponding radial profiles already illustrates how our existing lab setup could be deployed as +a second stage AO system to enhance performance, albeit with AO control limitations as outlined +in Cerpa-Urra et al. (2022). It is also possible to develop an RTC for two common path WFSs + +12 +(e.g, a SHWFS and chopper focal plane WFS) to control one or more common path DM(s), e.g., +using the framework developed in Gerard et al. (2021). This multi-WFS architecture is particularly +interesting to explore further for our chopper-based focal plane WFS proposed here, as temporal +modulation from the chopper wheel is non-common path to a first-stage WFS (e.g., unlike DM-based +phase diversity approaches that would use a common path DM, which further decrease science duty +cycle). +4. DISCUSSION AND FUTURE WORK +Expanding further on the chopper phase jitter limitations presented in §3.3, the first option to +improving chopper performance would be to reduce the phase jitter by at least 10x. Our inquiries +to vendors such as Thorlabs and Scitec Instruments indicate that off-the-self optical choppers with +such capabilities are not immediately available. More custom options have nonetheless demonstrated +the ability to reach such limits (Johnson et al. 2022) and should be explored further. However, the +DM-based chopping approach presented in Gerard et al. (2022b) does not suffer from a similar phase +jitter problem. Of particular importance to highlight after these demonstrations is the potential +for non-coronagraphic focal plane WFS applications, including single conjugate AO, laser guide star +(LGS) AO (compatible with off-axis sensing), wavefront sensing in crowded fields, and tomographic +reconstruction. All of these applications are enabled simply by recording consecutive chopped and +un-chopped focal plane images, with the latter used for science. LGS and off-axis sensing applications +would additionally require the detector to image the off-axis guide star (and for LGSs placed to the +appropriate position to image the focal plane), Crowded field sensing would require point-source +detection algorithms to accommodate a different position distributions of stars for a given on-sky +pointing and/or for LGS tomography pre-defined image positions. Multiple sources in crowded fields +could also potentially limit the number of controllable modes, as in principle the same pixels on a +detector cannot be used for wavefront reconstruction from two different incoherent sources without +a multi-star wavefront control strategy (Sirbu et al. 2017). Tomographic reconstruction with such +multiple sources could then enable wide-field AO correction, either for ground-layer AO for a single +pupil-conjugated DM or for multi-congugate AO. Relatedly, guiding on resolved astrophysical objects +should be further explored. +There are many areas to further explore and develop such that this pupil chopping technique +can be operational on-sky. +Perhaps the most pressing is overcoming the ∼10 c/p sensing limit +presented in §2.3, currently limiting higher order wavefront control to DMs with less than 20 × 20 +actuators. For the DM-based chopper technique, it is possible that more complex chopper “ridge- +line”3 geometries could enable improved high order measurement sensitivity. Although this would +be a challenging endavor for an optical chopper blade, it is a simple application for a DM, and +particularly for a segmented DM. Expanding on the discussion in §3.4.1, generating chop{φref} from +simulation and/or using a fully synthetic interaction matrix would be interesting to explore further. +Like the fast referencing approach previously discussed, this approach does not suffer from wavefront +blurring effects, but furthermore it is not photon and/or detector noise-limited, potentially allowing +deeper achievable closed-loop convergent WFEs but with the added risk of additional model-based +errors. Such model errors could be less problematic for this method as presented in this manuscript +3 By “ridge-line,” we mean the shape of the line defining the transition between the modulated and un-modulated pupil +fractions. In this paper we have just considered this to be a straight line. + +13 +compared to coronagraphic focal plane wavefront sensing techniques that have additional degrees of +freedom to model (Potier et al. 2020). Coronagraphic applications of this technique should also be +explored; in principle the chopper temporal modulation concept can remain identical to what we +presented in this paper, but additional coronagraph-specific questions should be investigated (e.g., +chopping in an apodizer vs. +Lyot stop plane, and WFS linearity and sensitivity dependence on +different coronagraph designs). Higher duty cycle chopper operations could also be explored further; +the 2% duty used in laboratory testing presented in this paper (see §3.1) is effectively generating +a PSF with an obscured aperture at a single chop fraction, but higher duty cycles would cover a +range of fractions over a single camera integration due to the continuous motion of the chopper +blade. Although these effects could be simulated and tested, again the DM-based chopping approach +introduced in Gerard et al. (2022b) does not have this problem and can in principle reach higher than +50 % duty cycles with a custom camera read out scheme. Another interesting area to further explore +is absolute phase retrieval; we have only presented this technique so far as differential relative to a +pre-calibrated best flat enabled by some other method, but this method could in principle enable +non-linear (e.g., Gerchberg–Saxton) absolute wavefront reconstruction (either with a single chopper +pair for the chopper blade “amplitude diversity” approach and/or a single chopper image for the +DM-based chopper “phase diversity” approach). Performance over a large spectral bandpass should +be explored; although PSF images inherently suffer from chromatic magnification without a Wynne +corrector, in this technique the wavefront diversity is applied in the pupil plane, potentially enabling +robustness to higher spectral bandwidth operations with this technique. +5. CONCLUSION +We have presented a new focal plane wavefront sensing technique, which uses an optical chopper +designed to partially block and then unblock the pupil in consecutive frames while recording syn- +chronized focal plane images (§2.1). This pupil chopping provides sufficient amplitude diversity to +reconstruct wavefront modes ≲10 c/p (§2.3). It is also optimal for high speed wavefront control, +requiring only two consecutive images (one of which is the science image) and a MVM to generate +DM commands for residual AO correction (§2.2). In addition to simulations of this new concept, we +presented laboratory results using SEAL (§3), summarized as follows: +1. In §3.2 we measured good linearity for reconstructed low order Zernike modes (n < 6), but +higher order Zernikes were not measurable due to chopper phase jitter (§3.3; but see §4). +2. We first closed the loop on air at ∼50 Hz (i.e., consecutive image acquisition at 100 Hz) in +§3.4.1, clearly improving the WFE and demonstrating the potential for real-time correction of +quasi-static errors. +3. We next closed the loop in §3.4.2 on DM-induced AO residual turbulence, also at 50 Hz, showing +a clear performance gain, including by a measured 20% Strehl ratio increase in closed vs. open +loop cases. +More topics need to be addressed before this technique can be deployed at observatories on-sky +(§4), but thus far we foresee no showstoppers towards enabling this. The power of this technique— +particularly in the DM-based chopping approach presented first in Gerard et al. (2022b) and in a +forthcoming more-detailed paper (Soto, Gerard et al., in prep)—is its simplicity. Hardware simplicity + +14 +(only requiring a focal plane imager in addition to a conventional AO system), software simplicity (op- +erating with a linear MVM, in comparison to other iterative, non-linear focal plane WFS techniques), +and broad compatibility (with applications non-coronagraphic and/or coronagraphic systems, LGS +and/or natural guide star systems, and correction of high-speed atmospheric residuals and/or quasi- +static WFEs) illusrate a promising and powerful potential for this new technique. +ACKNOWLEDGEMENTS +We gratefully acknowledge research support of the University of California Observatories for funding +this research. Author B. Gerard thanks the 2018 SCExAO team for hosting discussions that led to +the pupil chopping concept. This work performed under the auspices of the U.S. Department of +Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The +document number is LLNL-JRNL-843401. +The authors thank the anonymous reviewer for their +detailed consideration and feedback of this manuscript. +APPENDIX +A. LINEAR LEAST-SQUARES RECONSTRUCTOR SUMMARY +Below we summarize the widely-used calibration routine to convert WFS measurememts in real- +time into closed-loop DM commands as applied to this paper. +1. For k DM modes (e.g., Zernike modes), w from equation 1 is recorded, for low order modes +selecting pixel values within a given control radius of the detector focal plane optical axis (e.g., +if controlling up to 5 radial orders of Zernike modes, only using w pixel values within a 5 λ/D +radius from the optical axis). Thus, for DM each mode we generate a vector wk of dimensions +1 × j, where j, is the number of w pixel values used for a given image. +2. an Interaction Matrix (IM) concatenates wk for all k modes into a k × j matrix. A reference +vector of DM actuator commands for all modes, R, is also saved during this modal calibration, +generating a k × p matrix in units of DM command space (e.g., volts), where p is the number +of actuators within the two-dimensional DM pupil footprint. +3. The command matrix (CM), used to convert w space to DM command space assuming linearity +between the two, is then +CM = IM† · −R, +(A1) +where · and † represent a matrix dot product and a matrix pseudo inverse process such as +Singular Value Decomposition (SVD), respectively. +4. Open loop DM commands are then reconstructed by an on-sky input w via +DMCopen-loop = won-sky · CM +(A2) +where won-sky is a 1×j vector, representing a real-time w value, and DMCon-sky is a 1×p vector +of real-time DM commands (DMCs). +5. 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L. 2021, Journal of Open Source +Software, 6, 3021, doi: 10.21105/joss.03021 + diff --git a/UtFIT4oBgHgl3EQfgCvc/content/tmp_files/load_file.txt b/UtFIT4oBgHgl3EQfgCvc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2b54cb939411a742934c24c8b6e2e3f514a5d6f --- /dev/null +++ b/UtFIT4oBgHgl3EQfgCvc/content/tmp_files/load_file.txt @@ -0,0 +1,786 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf,len=785 +page_content='High Speed Focal Plane Wavefront Sensing with an Optical Chopper Benjamin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Gerard,1, 2 Daren Dillon,2 Sylvain Cetre,3, 4 and Rebecca Jensen-Clem2 1Lawrence Livermore National Laboratory 2University of California Santa Cruz 3Durham University 4Wakea Consulting (Received Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 20, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Revised Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Accepted Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 25, 2025) Submitted to PASP ABSTRACT Focal plane wavefront sensing and control is a critical approach to reducing non- common path errors between the a conventional astronomical adaptive optics (AO) wavefront sensor (WFS) detector and science camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' However, in addition to mitigat- ing non-common path errors, recent focal plane wavefront sensing techniques have been developed to operate at speeds fast enough to enable “multi-WFS” AO, where residual atmospheric errors are further corrected by a focal plane WFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Although a number of such techniques have been recently developed for coronagraphic imaging, here we present one designed for non-coronagraphic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Utilizing conventional AO system components, this concept additionaly requires (1) a detector imaging the focal plane of the WFS light source and (2) a pupil plane optical chopper device that is non-common path to the first WFS and is synchronized to the focal plane imager readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' These min- imal hardware requirements enable the temporal amplitude modulation to resolve the sine ambiguity of even wavefront modes for both low, mid, and high wavefront spatial frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Similar capabilities have been demonstrated with classical phase diversity by defocusing the detector, but such techniques are incompatible with simultaneous science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This optical chopping techniqe, however, enables science imaging at up to a 50% duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We present both simulations and laboratory validation of this concept on SEAL, the Santa Cruz Extreme AO Laboratory testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Keywords: adaptive optics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' INTRODUCTION Astronomical adaptive optics (AO) has enabled ground-breaking discoveries over the past three decades, from the Nobel prize-winning imaging and dynamical analysis of the Galactic center (Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2005) to the first images of planets around other stars on Solar System scales (Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' However, advancements over the years has illuminated a new limitation to further improving Corresponding author: Benjamin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Gerard gerard3@llnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='gov arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='11282v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='IM] 26 Jan 2023 2 performance of AO systems: non-common path aberrations (NCPAs) between an AO wavefront sensor (WFS) and AO-fed science detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' These NCPAs are fundamentally not measurable with a standard AO WFS and therefore not correctable (spatially or temporally) by an AO deformable mirror (DM) without additional wavefront sensing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' NCPAs are among the dominant terms in error budgets of current AO systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', Poyneer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2016), and as such there has been significant recent development of NCPA correction techniques for AO systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', Bottom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Potier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Considering the amplitude, A, and phase, φ, of the electromagnetic field from a star at infinity placed on the DM of an AO system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', conjugated to the telescope pupil), a downstream focal plane point spread function (PSF) image is described by PSF = |F{A ei φ}|2, where F is a Fourier transform operator, | |2 represents square modulus of the focal plane electric field and i is √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' A PSF at perfect focus is inherently degenerate to certain modes of φ, including the sine of symmetric Zernike modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', focus, spherical aberration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=') and the relative phase of Fourier modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', a 3 cycle/pupil sine vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' cosine), and therefore additional techniques are needed to sense the wavefront from a science image and then update DM commands and/or AO WFS offsets to accordingly compensate for NCPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' NCPA measurement and correction techniques, also known as focal plane wavefront sensing and control, generally fall into one of two categories to resolve the wavefront measurement degeneracy with a PSF: temporal or spatial modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Temporal modulation involves recording a series of two or more images with the science camera where a known probe is applied between images, resolving the wavefront ambiguity with “temporal diversity” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', Mugnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Bord´e & Traub 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Conversely, techniques with spatial modulation can in principle use a single focal image for wavefront reconstruction, leveraging custom hardware and/or software to resolve the wavefront measurement degeneracy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', with fringes (Baudoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2006) or calibrated symmetry (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2017) in the focal plane image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In general, temporal modulation-based focal plane wavefront sensing techniques use iterative algo- rithms designed for stable, space telescope-like environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In this paper, we introduce a temporal modulation-based focal plane WFS designed for correction of residual AO turbulence down to mil- lisecond timescales using an optical chopper device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We note that this technique was originally introduced in Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2022b), but is expanded and presented more verbosely here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In §2 we present the concept of pupil chopping for focal plane wavefront sensing and present simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We then present validating laboratory results using the Santa Cruz Extreme AO Laboratory (SEAL) testbed in §3, provide further discussion in §4, and then conclude in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Unless otherwise noted, simulations in this paper assume an operating wavelength of λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='6µm, a 256×256 pixel image size, and beam ratio (number of pixels per λ/D) of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' CONCEPT AND SIMULATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Concept Figure 1 illustrates the concept of focal plane wavefront sensing with a pupil plane optical chopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Two focal plane images are required to enable a WFS measurement: one with the chopper blade partially blocking the pupil, and one with the pupil unblocked;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' the latter can be used for science while the former is limited to wavefront sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' A “reference” WFS frame is saved before on-sky 3 DM Optics + first stage WFS Light from telescope Real-time control science camera + second stage WFS Optics + pupil chopper (dichroic/ beam-splitter) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Chopped pupil 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Airy disk PSF 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Chopped pupil PSF 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3-2 (WFS frame) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Tip response (double difference) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Tilt response 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Focus response 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Negative focus response 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Fourier mode response (x,y=5,0 c/p sin) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' x,y=5,0 c/p cos 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' x,y=0,5 c/p sin 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' x,y=0,5 c/p cos (a) (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Illustration of the pupil chopping concept, both from a geometric (a) and Fourier optics (b) perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In (b), all focal plane images are shown with the same 10×10 λ/D field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Panels 2-3 are shown on a log scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' all other panels are shown on a linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Zernike and Fourier modes in panels 5-12 all use 1 nm amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Panels 7-12 indicate that pupil chopping for focal plane wavefront sensing can resolve the sign ambiguity of focus and the relative phase of Fourier modes, which is not possible with a single PSF image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' operations and then used analogously to the concept of on-sky reference slopes for other pupil plane WFSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Such reference slopes enable a real-time WFS frame to measure the wavefront down to the flatness level of the pre-calibrated reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In equation form, the on-sky WFS frame, w, at an instance in time is given by w = chop{φon-sky} − chop{φref}, with chop{φ} ≡ � |F{Achopeiφ}|2 − |F{Anormeiφ}|2� /Σ{|F{Anormeiφ}|2}, where Anorm is an unobscured pupil wavefront amplitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', an open slot of the chopper wheel), Achop represents a chopped pupil wavefront amplitude, φon-sky is the on-sky pupil wavefront phase, φref is the reference pupil wavefront present during the above mentioned daytime calibration, and Σ{} represents an operator for summing pixel values within a given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' To be clear, φref cannot be generated or reconstructed with this technique, and must be obtained by some other non-linear iterative wavefront sensing and/or phase retrieval technique to flatten the absolute wavefront at the science detector plane (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', with phase diversity from Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2017 and/or the asymmetric pupil mask from Martinache 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' see §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' However, the advantage of this double difference approach is that it enables a linear wavefront reconstruction from w space to DM command space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', optimal for real-time wavefront control of AO residuals), which we will describe and illustrate in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Due to the implementation of a continuously spinning blade, the duty cycle for a chopped or un- chopped image is limited to a maximum of 1 2(1 − dbeam dblade), where dbeam is the beam diameter and dblade is the chopper blade width (assuming equal spacing between obscuring and unobscuring slots of the blade;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' so if dblade = dbeam, there is no time to keep the beam un-chopped and the duty cycle is 0, whereas the maximum duty cycle with a very tiny pupil and/or large blade, dbeam dblade ≈ 0, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' To clarify, for this technique dblade ≥ dbeam (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', a blade size smaller than the beam size would prevent 4 either frame from seeing a unobscured pupil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' A phase delay is also needed to acquire chopped and un-chopped pupils in consecutive focal plane images, without which in every other frame the pupil would be either completely blocked and then un-blocked or chopped by a fraction, f, on one side and then symmetrically chopped by 1-f on the other side in the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' By being a focal plane wavefront sensor, this method benefits from natural wavefront spatial fil- tering properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', with the focal plane image representing the Fourier plane of the pupil plane wavefront).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This configuration enables binary masks to be placed on the focal plane image WFS that act as a natural anti-aliasing filter and minimize non-linear cross talk between modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Using the focal plane image and such binary masks for wavefront sensing with this technique also distinguishes this technique from the differential optical transfer function method (Codona 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This method is also complementary to pupil amplitude diversity-based absolute phase retrieval methods, such as the asymmetric pupil Fourier WFS (Martinache 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Our pupil chopping technique enables a linear- least squares reconstruction (as we will show next in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2), optimal for high speed wavefront control of AO residuals but not able to flatten the absolute phase below a pre-calibrated best flat (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', due to evolving on-sky quasi-static aberrations), whereas phase retrieval methods can use the chopped pupil PSF image to recover the absolute phase to track such quasi-static effects, but typically requiring non-linear iterative algorithms that cannot be run at high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' See §4 for a further discussion about combining these two complementary approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Low Order Wavefront Reconstruction and Control With the setup described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1, we enable a linear matrix vector multiply (MVM) reconstructor for real-time wavefront control as described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Figure 2 shows the result of open loop MVM-based modal coefficients for 13 Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2 clearly shows the potential of this focal Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Simulated open-loop reconstruction of modal coefficients for different Zernike modes and ampli- tudes (mode 0 in this case refers to tip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The peak-to-valley IM amplitude is 1 nm for all modes, mapping modal coefficients to reconstructed wavefront in nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' plane wavefront sensing technique as a linear WFS for input wavefront phases with wavefront errors less than 1 radian rms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', the standard regime in which a Taylor expansion of the PSF is linearly related to the pupil wavefront phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 input: mode0,10nm input: mode3,8nm input: mode0,-10nm input: mode3,-8nm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 input: mode1,-6nm input: mode10,5nm input: mode1,6nm input: mode10,-5nm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 (wu) indno 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 0 2 4 6 8 10 12 Zernike mode number5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Limitations to High Order Wavefront Reconstruction Using this pupil chopping technique, we carried out a detailed analysis of achievable residual wave- front error on input extreme AO residual turbulence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', a static phase screen normalized to 100 nm rms from 0 to 10 c/p with a -2 power law, similar to Poyneer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2016, with results medianed over 100 different random realizations) as a function of DM actuator count, illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The 15 20 25 30 Nact = # of DM actuators across pupil 10 1 102 105 108 1011 maximum achievable WFE gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='6 chopped pupil fraction Nact=14 Nact=16 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Grid search simulations, optimizing all adjustable parameters in the wavefront reconstruction code as a function of DM actuator count (left) and chopped pupil fraction for two different actuator counts (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The y-axis for both panels shows the ratio of input (100 nm rms) to convergent output wavefront errors due to non-linearities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' without aliasing or photon noise) after 12 iterations, medianed over 100 random realizations for each data point shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Interaction matrix amplitude, SVD cutoff, and the radius for a binary mask to isolate sine spots from Fourier modes are optimized for all data points in both panels, with chopped pupil fraction additionally optimized for the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3 shows that pupil wavefront spatial frequencies greater than around 9 c/p do not converge to a deeper WFE than the input AO residual level, while spatial frequencies less than this limit in principle can gain in convergent vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' input WFE by many orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Intuitively, this 9 c/p wavefront measurement limit is because light compared with an un-modulated Airy disk is only modulated in a chopped frame out to about 9 λ/D with the current chopper blade concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We will propose possible solutions to this limit and discuss this further in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3 further illustrates performance dependence on what fraction of the pupil is chopped, showing that ∼30-50% enables sufficient gains, which will justify our later laboratory implementation in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' LABORATORY TESTING 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Setup Our laboratory setup, mimicking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 1a, uses SEAL—the Santa Cruz Extreme AO Laboratory (Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2022a, Jensen-Clem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We use a Thorlabs KLS635 laser (set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='15 mW unless otherwise noted), ALPAO 97 actuator DM (which also serves as the system pupil stop, Andor Zyla 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 sCMOS detector, Thorlabs WFS-20 SHWFS (for this paper used only to flatten the ALPAO DM), a Thorlabs MC2000B optical chopper with blade MC1F10, and a Stanford Instruments DG535 6 controller and an in-house electrical doubler so that the Andor detector readout is synchronized with the optical chopper (with the chopper as the leader and the Andor camera as the follower1) to produce a chopped and un-chopped pupil every other frame, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' As in Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2022a), ALPAO DM units are converted to WFE units via a single scalar multiplication, calibrated via open tip/tilt measurements and corresponding PSF images, but for which higher order conversions become increasingly wrong due to the decreasing stroke limits of the DM as a function of mode order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Unless otherwise noted, our standard laboratory configuration for chopper WFS image acquisition is with the chopper blade running at 100 Hz, using the doubler so the Andor camera follower runs at 200 Hz with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1 ms exposure per frame (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', a 2 % duty cycle), and also implementing a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5ms phase delay for the doubled chopper signal with the Stanford controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' As in Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2022a), we also implement a serial 20 ms pause in between acquiring a chopper pair of images and sending any DM commands (both for calibration and real-time control software) due to additional latency from our Python interface to hardware components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Two random chopper imager pair acquisitions are not necessarily in the same order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', one could have the chopped frame first while the other could have the un-chopped frame first), but for consistency we order every chopper sequence with the chopped frame first, as in Equation 1, determined as the frame with less cumulative flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' A serial timing diagram outlining the procedure described in this section, which is used for real-time control, is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Due to the continuous sequence of images (camera integrations starting at 0, time (ms) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 loop component |F{Achopeiφon-sky}|2 phase delay |F{Anormeiφon-sky}|2 20 ms pause loop timing start camera integration stop camera integration finish camera duty cycle end delay start camera integration stop camera integration finish camera duty cycle apply DM commands Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Chopper timing diagram used for real-time control in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5, 10, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' etc ms) being a non-integer multiple of the 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 ms total loop time shown in Table 1, a continuous real-time buffer is constantly acquiring chopped and un-chopped images and the real-time loop just grabs the most recent images from this buffer, meaning that effectively the 20 ms pause dominates the lag budget in our setup, is why subsequent real-time control results in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4 assume a ∼50 Hz loop rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Since the light source remains at a constant intensity, we also just use a single normalization value for all data rather than generate a new value for each chopper pair as equation 1 suggests, but for changing light source intensities a moving average cumulative intensity should be considered to minimize noise propagation into this normalization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='15 mW setting for our testbed light source is set to place the PSF core at ∼2000 analog to digital units (ADUs, ∼half the Andor Zyla’s 4196 ADU dynamic range), which from photon scaling relations (Bessell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 1998) for real- time purposes translates to a ∼90% Strehl ratio from an extreme AO residual wavefront for 1 ms exposure, mH=7 star, 3% spectral bandpass centered at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='6 µm (reasonable for a narrow band focal 1 Although the AO field has historically adopted “master and slave” terminology from the computer science field to refer to the chopper and camera in this configuration, respectively, it is now clear that such discriminatory terminology should no longer be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Following the computer science field, instead we will refer to such configuration as “leader and follower” throughout this paper, encouraging other members of the AO field to do the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 7 plane WFS setup to avoid blurring effects from PSF magnification with wavelength), and 50% total sky-to-photoelectron throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Given that we can only control the 97 actuator ALPAO DM up to ∼5 cycles per pupil, only pixel values within a pre-defined 5λ/D radius of the star are used for subsequent low order signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The real-time differential image architecture of this technique lends itself to self-calibration, so dark subtraction or other “cosmetic” calibrations are not needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We found better performance when applying Zernike modes were non-illuminated actuators were set to the commands of their nearest neighbor rather than set to zero, and so we use the former hereafter in this paper unless noted otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Low Order Linearity After following the procedure outlined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 4 we generate reconstructed modal coefficients for low order Zernike modes, probing a range of input amplitudes for each mode in open loop and recording corresponding reconstructed output modal coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In general, the cross terms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 5 are negligible2 when the given input mode is reconstructed within it’s linear regime (≲200 nm PV for each modal group, for tip/tilt corresponding to 5 mas for a 8m telescope at λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='6µm), which as expected is only feasible for AO residual WFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This also confirms that, as expected, a diffraction-limited AO residual wavefront is needed an an input in order for this pupil chopping technique to enable closing the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Figure 4 also shows that linearities are computed separately in groups for (1) tip, tilt, and focus, astigmatism and coma, and (2) Zernike modes with n=4→5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We found that the increased flexibility to use different SVD cutoffs for each group helped improve linearity and lower cross talk, motivating our choice to use this modal grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We also found that averaging 100 chopper frames per target image used for wavefront reconstruction to measure linearity gave significantly better stability than shorter sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The cause of such instability is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Limitations to High Order Wavefront Reconstruction Unfortunately, we were not able to measure and control of Zernike modes with n > 5 due to chopper blade phase jitter causing a time-varying chopped pupil fraction at the 1% rms level, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This phase jitter is due to (1) chopper motor control variations and (2) imperfections of in the uniformity of the chopper blades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Our measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 5 are consistent with the phase jitter specifications from Thorlabs, with no other available models able to reach smaller phase jitter levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We found that ≳100 nm amplitudes were needed for a given Fourier mode to detect sine spots at spatial frequencies ≳3 c/p, which if used in the IM produces a quadratic linearity curve (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', creating a reconstructor that is unable to tell positive from negative input amplitudes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' From simulations, ≲2 nm amplitudes for such Fourier modes are needed to instead produce a good linearity curve, meaning a ≳50x reduction in phase jitter would be needed to enable higher wavefront order control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' See §4 for further discussion on future solutions to improve this phase jitter limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Closed-Loop Operations Building on our measured linear modal basis in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2, in this section we will present closed-loop results on-air (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1) and with DM-based input AO residual turbulence (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2 Ignoring n,m=5,±1 (which we use Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 4c to motivate not controlling and accordingly note in the subfigure caption), in this context we use “negligible” to mean that for a given probed Zernike mode (1) it shows close-to-linear behavior for both positive and negative input amplitudes, and (2) cross terms do not reach amplitudes beyond the probed dominant mode over the linear range of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' That being said, we acknowledge that decreased performance for both dominant mode and cross term non-linearities will result in decreased achievable closed-loop WFE and may require a modified calibration and/or control scheme, which are also discussed further in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4 and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 reconstructed output ( m WFE, PV) n,m=1,-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=1,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 input ( m WFE, PV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=2,0 y=x n,m=1,-1 n,m=1,1 n,m=2,0 (a) Linearity for tip, tilt, and focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 reconstructed output ( m WFE, PV) n,m=2,-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=2,2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=3,-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=3,-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=3,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 input ( m WFE, PV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=3,3 y=x n,m=2,-2 n,m=2,2 n,m=3,-3 n,m=3,-1 n,m=3,1 n,m=3,3 (b) Linearity for n = 2 →3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 reconstructed output ( m WFE, PV) n,m=4,-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=4,-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=4,0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=4,2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=5,-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=5,-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=5,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=5,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 input ( m WFE, PV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='25 n,m=5,5 y=x n,m=4,-4 n,m=4,-2 n,m=4,0 n,m=4,2 n,m=4,4 n,m=5,-5 n,m=5,-3 n,m=5,-1 n,m=5,1 n,m=5,3 n,m=5,5 (c) Linearity for n = 4 →5, informing our decision to not control the highly non-linear modes n,m=5,±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Linearity curves for Zernike modes n = 1 → 5, separated into different modal groups (a-c), which are separately optimized by interaction matrix amplitudes and different command matrix SVD cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Chopper phase jitter analysis, showing the cumulative chopped pupil frame intensity as a fraction relative to the sequence-averaged cumulative un-chopped pupil frame intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The light source is saturating all illuminated pixels during this dataset, ensuring that variations shown here are due to chopper phase jitter, which are shown here to be present at the ±1% level, ultimately preventing high order wavefront reconstruction for spatial frequencies ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 c/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' On-air Stabilization On-air closed-loop results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Although the stabilized environment from our 0 200 400 600 800 1000 iteration number 10 3 10 2 low order WFE ( m rms) TTF n=1 5 loop closed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 log10[low order WFE ( m rms)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 normalized kernel density TTF n=1 5 open loop closed loop (a) (b) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Real-time correction of on-air wavefront error (measured by the chopper WFS at a frame rate of about 50 Hz), showing both tip/tilt/focus (TTF) and all controlled modes (n = 1 → 5), both in the time domain (a) and by WFE distribution (computed using the kdeplot function of the seaborn package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Waskom 2021) comparing the open and closed loop sequences (b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', respectively to the left and right of the dotted line in panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' granite optical table allows only a moderate gain, we still demonstrate a closed-loop WFE reduction mean +/-std=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='805+/-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='83 100 choppedpupililluminationfraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='82 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='81 10-4 PSD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='80 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='79 10-8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='78 0 1000 2000 3000 4000 5000 100 101 choppedpupilframenumber(at11oHz) temporalfrequency(Hz)10 of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2x for all controlled modes relative to ambient air (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', the median of the solid vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' dotted orange distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 6b is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2), illustrating the potential of this technique to stabilize quasi- static temporal disturbances on-sky (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', thermal variations, flexure, and additional sources of slow beam wander).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Closed-loop control was manually tuned for optimal performance using a integrator controller with gains of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='003, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='008 for the TTF, low order Zernike, and high order Zernike modal groups, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' such low gains were necessary for loop stability due to SEAL’s highly stabilized ∼nm-level on-air WFE (Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2022a), which as shown is mostly below the noise floor of individual frames of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Note these results help address concerns that the spinning chopper wheel would be generating additional turbulence, since telemetry from closing the loop on-air clearly shows improvement compared to open-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' It is important to note here that we obtained the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 6 with a high-speed referencing setup designed to minimize chopper blade phase jitter limitations, as discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3, and/or integrated effects of on-air turbulence that the chopper may be generating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Specifically, both chop{φref} from equation 1 and all command matrices are obtained just before the real-time open and closed loop telemetry sequence begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This strategy ensures that chop{φref} and the command matrices do not suffer from the loss of information due to wavefront “blurring” effects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' such as chopper blade phase jitter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' on-air turbulence evolving,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' and/or quasi-static evolution of optics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' since we found that a reference flat frame and 18 Zernike mode probes (each of which are taken relative to a reference flat before the next mode is probed) acquired at ∼50 Hz sufficiently freezes the on-air turbulence and chopper blade phase jitter effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Other strategies we tried to obtain chop{φref} did not work as well, such as longer exposure acquisition in attempt to “average out” these turbulent on-air effects, which, corroborated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 5, did not work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The disadvantage of this fast referencing technique, however, is that single frames can be noisier than other approaches, preventing the deepest convergent WFE this approach can provide;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' we discuss this further in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Real-time AO Residual Turbulence Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 7 summarizes our results for closing the loop on DM-induced AO residual turbulence, clearly demonstrating a performance improvement in both WFS telemetry and observed Strehl ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Input turbulence is normalized to a 100 nm rms, -2 power law phase screen and translated assuming a single 10 m/s frozen flow ground layer for a 10m telescope, with chopper pair images acquired at 50 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We also use the same high-speed referencing and calibration technique for simulated AO residual turbulence here as described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1 for on-air real-time control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Closed-loop control was manually tuned for optimal performance using a leaky integrator controller with gains of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3 for the TTF, low order Zernike, and high order Zernike modal groups, respectively, and with a leak of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='95 for all modal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 7a and b shows a total WFE reduction of ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2x, roughly consistent with the measured Strehl ratio enhancement from 73 to 92% (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 7c) via the Mar´ecehal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' A clear reduction of all controlled modes empirically demonstrates sufficiently minimal cross-talk between different modal groups that use different IM amplitudes and SVD cutoffs, as initially motivated in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Although it could still be the case that non-linearities dominate the closed-loop error budget in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 7, such an error budget analysis is beyond the scope of this first introductory paper of the pupil chopping concept, and regardless our results show that these non-linearities are at least ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2x below an extreme AO residual WFE, which is an important result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Although the closed loop vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' open loop WFE histogram mean values clearly decrease, their distribution widths remain unchanged, suggesting 11 0 500 1000 1500 2000 iteration number 10 2 10 1 low order WFE ( m rms) TTF n=1 5 loop closed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 log10[low order WFE ( m rms)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='5 normalized kernel density TTF n=1 5 open loop closed loop (a) (b) open loop closed loop 0 2 4 radial separation ( /D) 10 2 10 1 100 median normalized intensity OL CL flat (c) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Analogous to Figure 6, (a) and (b) show WFE vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' time and open and closed loop distributions for AO residual turbulence applied on the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Note that open loop WFE values for the n = 1 → 5 group may be under- or over-estimated as these levels approach the linear ranges of some modes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' However, panel (c) shows the stack of un-chopped images over the open and closed loop sequences and the corresponding intensity profile for both sequences compared to a static best flat image with no turbulence applied, showing a ∼20% Strehl improvement by closing the loop on AO residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' either our chosen control parameters are not yet fully optimized for robust closed-loop stability and/or similar sources of noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', chopper phase jitter) account for the distribution width in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Although these results assume use of a second stage “cascade” AO system (Cerpa-Urra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', a separate DM and WFS have produced the residual AO phases used as input into these experiments, this is an important first step in demonstrating the potential of this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The clear benefit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 7 of closed loop control in the stack of un-chopped images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', science frames) and corresponding radial profiles already illustrates how our existing lab setup could be deployed as a second stage AO system to enhance performance, albeit with AO control limitations as outlined in Cerpa-Urra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' It is also possible to develop an RTC for two common path WFSs 12 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g, a SHWFS and chopper focal plane WFS) to control one or more common path DM(s), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', using the framework developed in Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This multi-WFS architecture is particularly interesting to explore further for our chopper-based focal plane WFS proposed here, as temporal modulation from the chopper wheel is non-common path to a first-stage WFS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', unlike DM-based phase diversity approaches that would use a common path DM, which further decrease science duty cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' DISCUSSION AND FUTURE WORK Expanding further on the chopper phase jitter limitations presented in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3, the first option to improving chopper performance would be to reduce the phase jitter by at least 10x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Our inquiries to vendors such as Thorlabs and Scitec Instruments indicate that off-the-self optical choppers with such capabilities are not immediately available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' More custom options have nonetheless demonstrated the ability to reach such limits (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2022) and should be explored further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' However, the DM-based chopping approach presented in Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2022b) does not suffer from a similar phase jitter problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Of particular importance to highlight after these demonstrations is the potential for non-coronagraphic focal plane WFS applications, including single conjugate AO, laser guide star (LGS) AO (compatible with off-axis sensing), wavefront sensing in crowded fields, and tomographic reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' All of these applications are enabled simply by recording consecutive chopped and un-chopped focal plane images, with the latter used for science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' LGS and off-axis sensing applications would additionally require the detector to image the off-axis guide star (and for LGSs placed to the appropriate position to image the focal plane), Crowded field sensing would require point-source detection algorithms to accommodate a different position distributions of stars for a given on-sky pointing and/or for LGS tomography pre-defined image positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Multiple sources in crowded fields could also potentially limit the number of controllable modes, as in principle the same pixels on a detector cannot be used for wavefront reconstruction from two different incoherent sources without a multi-star wavefront control strategy (Sirbu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Tomographic reconstruction with such multiple sources could then enable wide-field AO correction, either for ground-layer AO for a single pupil-conjugated DM or for multi-congugate AO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Relatedly, guiding on resolved astrophysical objects should be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' There are many areas to further explore and develop such that this pupil chopping technique can be operational on-sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Perhaps the most pressing is overcoming the ∼10 c/p sensing limit presented in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3, currently limiting higher order wavefront control to DMs with less than 20 × 20 actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' For the DM-based chopper technique, it is possible that more complex chopper “ridge- line”3 geometries could enable improved high order measurement sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Although this would be a challenging endavor for an optical chopper blade, it is a simple application for a DM, and particularly for a segmented DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Expanding on the discussion in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1, generating chop{φref} from simulation and/or using a fully synthetic interaction matrix would be interesting to explore further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Like the fast referencing approach previously discussed, this approach does not suffer from wavefront blurring effects, but furthermore it is not photon and/or detector noise-limited, potentially allowing deeper achievable closed-loop convergent WFEs but with the added risk of additional model-based errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Such model errors could be less problematic for this method as presented in this manuscript 3 By “ridge-line,” we mean the shape of the line defining the transition between the modulated and un-modulated pupil fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In this paper we have just considered this to be a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 13 compared to coronagraphic focal plane wavefront sensing techniques that have additional degrees of freedom to model (Potier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Coronagraphic applications of this technique should also be explored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' in principle the chopper temporal modulation concept can remain identical to what we presented in this paper, but additional coronagraph-specific questions should be investigated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', chopping in an apodizer vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Lyot stop plane, and WFS linearity and sensitivity dependence on different coronagraph designs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Higher duty cycle chopper operations could also be explored further;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' the 2% duty used in laboratory testing presented in this paper (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1) is effectively generating a PSF with an obscured aperture at a single chop fraction, but higher duty cycles would cover a range of fractions over a single camera integration due to the continuous motion of the chopper blade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Although these effects could be simulated and tested, again the DM-based chopping approach introduced in Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2022b) does not have this problem and can in principle reach higher than 50 % duty cycles with a custom camera read out scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Another interesting area to further explore is absolute phase retrieval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' we have only presented this technique so far as differential relative to a pre-calibrated best flat enabled by some other method, but this method could in principle enable non-linear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', Gerchberg–Saxton) absolute wavefront reconstruction (either with a single chopper pair for the chopper blade “amplitude diversity” approach and/or a single chopper image for the DM-based chopper “phase diversity” approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Performance over a large spectral bandpass should be explored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' although PSF images inherently suffer from chromatic magnification without a Wynne corrector, in this technique the wavefront diversity is applied in the pupil plane, potentially enabling robustness to higher spectral bandwidth operations with this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' CONCLUSION We have presented a new focal plane wavefront sensing technique, which uses an optical chopper designed to partially block and then unblock the pupil in consecutive frames while recording syn- chronized focal plane images (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This pupil chopping provides sufficient amplitude diversity to reconstruct wavefront modes ≲10 c/p (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' It is also optimal for high speed wavefront control, requiring only two consecutive images (one of which is the science image) and a MVM to generate DM commands for residual AO correction (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In addition to simulations of this new concept, we presented laboratory results using SEAL (§3), summarized as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2 we measured good linearity for reconstructed low order Zernike modes (n < 6), but higher order Zernikes were not measurable due to chopper phase jitter (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' but see §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We first closed the loop on air at ∼50 Hz (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', consecutive image acquisition at 100 Hz) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='1, clearly improving the WFE and demonstrating the potential for real-time correction of quasi-static errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' We next closed the loop in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='2 on DM-induced AO residual turbulence, also at 50 Hz, showing a clear performance gain, including by a measured 20% Strehl ratio increase in closed vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' open loop cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' More topics need to be addressed before this technique can be deployed at observatories on-sky (§4), but thus far we foresee no showstoppers towards enabling this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The power of this technique— particularly in the DM-based chopping approach presented first in Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' (2022b) and in a forthcoming more-detailed paper (Soto, Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', in prep)—is its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Hardware simplicity 14 (only requiring a focal plane imager in addition to a conventional AO system),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' software simplicity (op- erating with a linear MVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' in comparison to other iterative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' non-linear focal plane WFS techniques),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' and broad compatibility (with applications non-coronagraphic and/or coronagraphic systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' LGS and/or natural guide star systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' and correction of high-speed atmospheric residuals and/or quasi- static WFEs) illusrate a promising and powerful potential for this new technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We gratefully acknowledge research support of the University of California Observatories for funding this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Author B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Gerard thanks the 2018 SCExAO team for hosting discussions that led to the pupil chopping concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' This work performed under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The document number is LLNL-JRNL-843401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The authors thank the anonymous reviewer for their detailed consideration and feedback of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' LINEAR LEAST-SQUARES RECONSTRUCTOR SUMMARY Below we summarize the widely-used calibration routine to convert WFS measurememts in real- time into closed-loop DM commands as applied to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' For k DM modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', Zernike modes), w from equation 1 is recorded, for low order modes selecting pixel values within a given control radius of the detector focal plane optical axis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', if controlling up to 5 radial orders of Zernike modes, only using w pixel values within a 5 λ/D radius from the optical axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Thus, for DM each mode we generate a vector wk of dimensions 1 × j, where j, is the number of w pixel values used for a given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' an Interaction Matrix (IM) concatenates wk for all k modes into a k × j matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' A reference vector of DM actuator commands for all modes, R, is also saved during this modal calibration, generating a k × p matrix in units of DM command space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=', volts), where p is the number of actuators within the two-dimensional DM pupil footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' The command matrix (CM), used to convert w space to DM command space assuming linearity between the two, is then CM = IM† · −R, (A1) where · and † represent a matrix dot product and a matrix pseudo inverse process such as Singular Value Decomposition (SVD), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Open loop DM commands are then reconstructed by an on-sky input w via DMCopen-loop = won-sky · CM (A2) where won-sky is a 1×j vector, representing a real-time w value, and DMCon-sky is a 1×p vector of real-time DM commands (DMCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFIT4oBgHgl3EQfgCvc/content/2301.11282v1.pdf'} +page_content=' Finally, closed-loop real-time control with a leaky integrator is given by DMCn = l DMCn−1 + g DMCopen-loop, (A3) 15 where DMCn is the current frame of DMCs, DMCn−1 is the previous frame of DMCs, and g and l are the integrator controller gain and leak parameters, respectively.' metadata={'source': 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Kishk, Member, IEEE and Mohamed-Slim Alouini, +Fellow, IEEE +Abstract +Using Unmanned Aerial Vehicles (UAVs) to enhance network coverage has proven a variety of +benefits compared to terrestrial counterparts. One of the commonly used mathematical tools to model +the locations of the UAVs is stochastic geometry (SG). However, in the existing studies, both users and +UAVs are often modeled as homogeneous point processes. In this paper, we consider an inhomogeneous +Poisson point process (PPP)-based model for the locations of the users that captures the degradation in the +density of active users as we move away from the town center. In addition, we propose the deployment of +aerial vehicles following the same inhomogeneity of the users to maximize the performance. In addition, +a multi-tier network model is also considered to make better use of the rich space resources. Then, the +analytical expressions of the coverage probability for a typical user and the total coverage probability +are derived. Finally, we optimize the coverage probability with limitations of the total number of UAVs +and the minimum local coverage probability. Finally we give the optimal UAV distribution parameters +when the maximum overall coverage probability is reached. +Index Terms +Coverage probability, urban model, multi-tier UAV network, stochastic geometry, inhomogeneous +Poisson point process. +I. INTRODUCTION +In the next generation mobile network (5G, beyond 5G), UAVs have many application sce- +narios [1]–[3], among which UAV-aided ubiquitous coverage becomes an important topic [4]. +Because UAVs are easy to deploy, highly mobile, and have 3D deployment, they are often used to +Ruibo Wang and Mohamed-Slim Alouini are with King Abdullah University of Science and Technology (KAUST), +CEMSE division, Thuwal 23955-6900, Saudi Arabia. Mustafa A. Kishk is with the Department of Electronic Engineering, +National University of Ireland, Maynooth, W23 F2H6, Ireland. (e-mail: ruibo.wang@kaust.edu.sa; mustafa.kishk@mu.ie; +slim.alouini@kaust.edu.sa). +arXiv:2301.00879v1 [cs.NI] 2 Jan 2023 + +2 +build temporary or dynamic networks and provide ubiquitous coverage. Especially, UAV is widely +used to relieve the pressure of large crowds gathering in small areas [4], [5]. They are proven +to provide reliable system coverage in hot spots and provide additional system performance [6]. +Due to the demand for high rate signals, multi-tier vertical heterogeneous networks (VHetNet) +is proposed to make use of space resources in city centers [7], [8]. +One of the main unanswered questions in the realm of UAV-enabled wireless networks is +where and how high the UAVs should be deployed [9]. The common assumption in SG-based +literature is that the user’s spatial distribution is homogeneous. Consequently, existing literature +typically assumes that the density of the UAVs is spatially invariant. However, according to recent +studies on resident population densities, a more proper assumption would be for the density of +the users (and consequently the UAVs) drops as their distance from the town center increases +[10]. Analyzing the influence of such a setup on the wireless network’s performance is the main +objective of this paper. More details on the contributions of this paper are provided later in +Sec. I-B. +A. Related Work +SG is a powerful mathematical method of analyzing communication networks with irregular +topology [11]. Furthermore, the SG framework is suitable for modeling and analyzing devices in +motion, such as UAVs, cars [12], and LEO satellites [13], [14]. The SG-based analytical results +of the network coverage probability can provide accurate approximations to the actual network +[15], [16]. Next, the authors in [17] proposed an air-to-ground line-of-sight (LoS) probability +model suitable for town centers. In this model, the probability of the UAV being blocked by the +building decreases with the increase of the elevation angle of the UAV to the typical user. This +model divides UAVs into LoS UAVs and non-line-of-sight (NLoS) UAVs. Since the model is +related to density, area, and height of building [18], it is suitable for various scenarios. Based on +the LoS probability model, there has been some literature on UAV networking in town centers +[6], [19]. +In the existing research, some resident population density models have been considered. +Different user distributions in several urban environments are proposed in [20]. A disjoint +clustered model for large resident population density is set up in [21]. A central model is +provided in [10] and is adopted in this paper. In the central model, user density decreases +with the distance from the user to the center. However, the above articles pay more attention + +3 +to the modeling of users, while the UAVs are simply deployed. UAVs are deployed as a +homogeneous PPP in [10], [21], while the locations of UAVs are determined by clustering +in [20]. Therefore, the deployment of UAVs is also worth exploring. However, with regard to +analyzing downlink network coverage performance, changing the distribution of UAVs brings +much more difficulty in technical derivation than changing the distribution of users. Given that +UAVs form a homogeneous PPP, the downlink coverage performance of users at any location +is the same. Nevertheless, when the density of the UAV is not constant, the distributions of the +distance between the serving UAV and the interfering UAV to the user are different for the users +at different locations, which makes the analysis challenging. +To effectively utilize the deployable space of UAVs, developing the vertical deployment mode +of UAVs is also worth studying, in addition to designing the horizontal distribution of UAVs. +Based on the SG framework, authors in [8], [9], [22]–[25] have put forward multi-tier VHetNets +consisting of ground base stations (BSs), UAVs and high altitude platforms (HAPs) and low earth +orbit (LEO)-satellites. The above researches all introduced the concept of association probability +to describe the probability of users choosing a communication device in a specific tier (instead +of other tiers) to provide services. Unfortunately, the above analytical framework is unsuitable +for our study because the UAVs are not uniformly distributed in our paper. Designing a different +method to obtain the association probability is another challenge. +B. Contribution +The contributions of this paper can be summarized as follows: +• We study a resident population density-inspired model of the urban area. The density of +users decreases with the distance to the town center. UAVs follow a similar distribution to +the distribution of users and are deployed at different altitudes with different densities. +• We derive the analytical result of coverage probability under the specific model and prove +that it is consistent with the Monte-Carlo simulation. In addition, the existing coverage +probability analysis framework is extended to data rate and energy efficiency. +• The coverage performance of multi-tier networks and single-tier networks are compared. +We also compare the coverage performance of the population density-inspired distribution +and the homogeneous distribution of UAV. +• By adjusting the distribution of UAVs in each tier, we optimize the coverage probability +under different user distributions. Furthermore, remarks on the parameter design criterion + +4 +for UAV distribution are given. +II. SYSTEM MODEL +A. Network Model +User +LoS UAV +NLoS UAV +𝒙 +𝒚 +𝒛 +ℎ! +ℎ!"# +𝑇𝑖𝑒𝑟 𝑘 + 1 +… +… +Typical user +Tagged LoS UAV +Tagged NLoS UAV +𝑇𝑖𝑒𝑟 𝑘 +Fig. 1: Illustration of the system model. +As shown in Fig. 1, we consider a scenario in which ground users are distributed according to +an inhomogeneous PPP, which is inspired by the resident population distribution model proposed +in [10]. We assume that the UAVs in the VHetNet are deployed based on the ground users +density and, hence, their locations also follow an inhomogeneous PPP. Assuming that the center +of the town is located at the origin, the densities of the users and the UAVs near the origin +are relatively high, while the density goes down as we move away from the origin. Assume the +users are located at the ground, with horizontal distance zu to the origin, the density distribution +of the users Λu (zu) can be represented as follows, +Λu (zu) = λue−βuzu, +(1) +where λu determines the total density of the plane, βu is a measure of homogeneity. When the +value of βu is large, the users are spatially condensed at the origin. When βu = 0 the process +degenerates to a homogeneous PPP. Without loss of generality, we focus on a typical user located +on the positive X-axis. +We assume that K tiers of UAVs are distributed at a set of some fixed heights hk independently. +Their location distribution of each tier form a 2D inhomogeneous PPP, denoted by Φk +∆= {xi,k} + +UserhTler1TypicalTagg +1Tlerh5 +where xi,k refers to the 3D location of UAV i in tier k. We prefer polar coordinates (zi, θi, hk) +to represent xi,k, where zi is the horizontal distance between the UAV and the origin, θi is +the angle between the X-axis and the line which connects the projection of the UAV and the +origin. Because designers tend to place more UAVs in densely populated areas, it is reasonable +to assume that they will be in the same distribution as the users. Thus, in tier k, the density +distribution ΛUAV,k can be described as, +ΛUAV,k (zi) = λke−βkzi, +(2) +where λk and βk are parameters of the UAVs in tier k, which have the same meaning as the +users’ parameters in the density distribution. Furthermore, we assume that each tier of UAVs +have the same transmitting power ρk. The quad Tk = {hk, λk, βk, ρk} , k = 1, 2, ..., K is used to +represent the parameters of the kth tier. +B. Channel Model +To model the air-to-ground channel between a user and a UAV, we need to take into con- +sideration the LoS and NLoS scenarios [17]. Considering a UAV in tier k, given the horizontal +distance z between the UAV’s projection on the ground and the user, the probability of setting +up an LoS link between the typical user and the UAV is [17], [26], +P LoS +k +(z) = +1 +1 + a exp +� +−b +� 180 +π tan−1 � hk +z +� +−a +��, +(3) +where a and b are environment-dependent parameters. From the perspective of the typical user, +the inhomogeous PPP process corresponding to the K-tier UAVs can be split into two disjoint +PPPs, that is Φk=ΦLoS,k ∪ ΦNLoS,k and ΦLoS,k ∩ ΦNLoS,k = φ, where ΦLoS,k and ΦNLoS,k denote +the set of UAVs which establish LoS and NLoS conditions for the typical user respectively, φ +is the empty set. +In this article, UAVs with an LoS link to the typical user are abbreviated as LoS UAVs, while +the rest are abbreviated as NLoS UAVs. Therefore, the 3D VHetNet is split into 2K disjoint +two-dimensional PPPs, with the density P Q +k ΛUAV,k, where Q = {LoS, NLoS}. After LoS and +NLoS states are defined, the channel fading model can be established, which is described by +small-scale fading and large-scale fading. +For small scale fading, we denote channel fading power gains in terms of independent random +variables GLoS and GNLoS, under LoS and NLoS conditions for the typical user, respectively. + +6 +In order to represent several fading scenarios, Nakagami-m fading is experienced with shape +parameters and scale parameters (mLoS, +1 +mLoS) and (mNLoS, +1 +mNLoS) for LoS and NLoS links, +respectively. As a result, the probability density functions (PDF) of the power gains GQ is given +by [27] +fGQ (g) = mQmQrmQ−1 +Γ (mQ) +e−mQ g, +(4) +where Γ (mQ) = +� ∞ +0 xmQ−1e−xdx is the Gamma function, Q = {LoS, NLoS}. +For large scale fading, ηLoS and ηNLoS are mean additional gain for LoS and NLoS trans- +missions [17], with ηLoS > ηNLoS satisfied. Combining small scale and large scale fading, the +received power of the typical user, transmitted by a UAV in tier k, is given by, +Sk (r) = +� +� +� +ηLoSρkGLoSr−αLoS +in case of LoS +ηNLoSρkGNLoSr−αNLoS +in case of NLoS +. +(5) +where ρk is the transmission power of UAVs in tier k, αLoS and αNLoS are path-loss exponents for +LoS and NLoS transmissions, with αLoS < αNLoS satisfied, r is the Euclidean distance between +the UAV and the typical user, which is computed by polar coordinates (zi, θi, hk) of the UAV +and the horizontal distance zu from the user to the origin +rxi,k = +� +(zicosθi − zu)2 + (zisinθi)2 + h2 +k. +(6) +C. Interference +In each tier, the closest LoS and NLoS UAVs are called tagged UAVs. According to the +strongest average received power association strategy [9], the typical user will associate with the +UAV with the strongest average received power among these 2K tagged UAVs. Furthermore, +denote the location of the associated UAV as xo. Note that the typical user may not associate +with the closest UAV, since in different tiers, the transmitted power of the UAVs is different. +In urban areas where UAVs are densely distributed, it is necessary to consider interference +between UAVs. Considering a worst-case scenario, except for the associated UAV, other LoS +and NLoS UAVs interfere with the typical user, which we refer to as interfering UAVs. Given +that the horizontal distance from the typical user to the origin is zu and the associated UAV is +in tier j, the total interference can be expressed as a function of rxo, which is the Euclidean + +7 +distance between the associated UAV and the typical user, +Ij (rxo, zu) = +K +� +k=1 +� +� +xi,k∈ΦLoS,k\{xo} +ηLoSρkGLoSr−αLoS +xi,k ++ +� +xi,k∈ΦNLoS,k\{xo} +ηNLoSρkGNLoSr−αNLoS +xi,k +� +. +(7) +As shown in the above formulation, the value of the total interference is related to zu, rxo and +the tier where the associated UAV is located. rxo and the transmission power of tier j determine +the received power from the associated UAV. +D. Performance Analysis +Assuming the associated UAV is located in tier j, the instantaneous signal-to-interference plus +noise ratio (SINR) at the typical user is given by the following equation, +SINR = +� +� +� +ηLoSρjGLoSr +−αLoS +xo +Ij(rxo|zu )+σ2 +xo ∈ ΦLoS,k +ηNLoSρjGNLoSr +−αNLoS +xo +Ij(rxo|zu )+σ2 +xo ∈ ΦNLoS,k +. +(8) +where σ2 is the additive white Gaussian noise (AWGN) power, and Ij (rxo |zu) is the total +interference power. +The reliability of the service provided can be evaluated by the average performance of +SINR. In consequence, the coverage probability, which represents the probability that the system +can provide reliable connections is defined as the probability that the SINR is greater than a +predefined threshold γ: +P C ∆= P [SINR > γ] . +(9) +III. PROBLEM FORMULATION +In this section, our objective is to obtain the analytical expression of coverage probability. +From the definition of coverage probability in (9), we know that distinguishing the associated +UAV and the interfering UAVs is a prerequisite for computing the coverage probability. Taking an +LoS associated UAV for example, the following steps are used to obtain the analytic expression +for the coverage probability: (i) derive the PDF of the distance distribution of tagged UAV in +each tier k, denoted as fRLoS,k (r, zu), (ii) calculate the probability of the tagged UAV in tier +k being associated with the typical user, defined as association probability P A +LoS,k (r, zu), with +fRLoS,k (r, zu) P A +LoS,k (r, zu) being the PDF of the distance between the associated UAV in tier k +and the typical user, (iii) for a specific distance r between the typical user and its associated + +8 +UAV, denote the probability that the SINR is greater than the threshold γ as the conditional +coverage probability P C (γ, zu |r). The average coverage probability is obtained by taking the +expectation of the conditional coverage probability P C (γ, r, zu) with respect to the PDF of r in +step (ii). Steps (i) - (iii) will be explained in Sec. III-B Sec. III-C, and Sec. III-D, respectively. +A. Nearest Interfering UAVs +A clear understanding of the location range of interfering UAVs is necessary when analyzing +tagged or associated drone distribution. Obviously, for tier k, the nearest interfering UAV should +locate at a distance larger than hk for the typical user. Another possible lower bound of the +distance for an interfering UAV is to ensure a lower average receiving power than the associated +UAV. In the following lemmas, the distance between the typical user and the nearest interfering +UAVs is given. +Lemma 1. Given that the typical user is associated with a LoS UAV located at distance r in tier +j, the closest interfering LoS and NLoS UAV in tier k are at least at distances dLoS−LoS,j,k (r) +and dLoS−NLoS,j,k (r) , given by +dLoS−LoS,j,k (r) = max +� +hk, +�ρk +ρj +� +1 +αLoS r +� +, +(10) +dLoS−NLoS,j,k (r) = max +� +hk, +�ηNLoSρkE[GNLoS] +ηLoSρjE[GLoS] +� +1 +αNLoS r +αLoS +αNLoS +� +. +(11) +Proof. According to the received power in (5), the average received power of the associated UAV +at distancerin tier j is Sj = ηLoSρjGLoSr−αLoS. The closest interfering LoS UAV in tier k is at +least at distances dLoS−LoS,j,k, which can be obtained by solving the equality ηLoSρjGLoSr−αLoS = +ηLoSρkGLoSd−αLoS +LoS−LoS,j,k. Similarly, for NLoS interfering UAVs, dLoS−NLoS,j,k can be obtained by +solving the equality ηLoSρjGLoSr−αLoS = ηNLoSρkGNLoSd−αNLoS +LoS−NLoS,j,k. +Lemma 2. Given that the typical user is associated with a NLoS UAV located at distance r in tier +j, the closest interfering LoS and NLoS UAV in tier k are at least at distances dNLoS−LoS,j,k (r) +and dNLoS−NLoS,j,k (r) , given by +dNLoS−LoS,j,k (r) = max +� +hk, +� ηLoSρkE[GLoS] +ηNLoSρjE[GNLoS] +� +1 +αLoS r +αNLoS +αLoS +� +, +(12) + +9 +dNLoS−NLoS,j,k (r) = max +� +hk, +�ρk +ρj +� +1 +αNLoS r +� +. +(13) +Proof. The proof is similar to that of lemma 1. +In the subsequent analysis, the horizontal distance is more practical than the Euclidean distance +in this model. The horizontal distances zQ,j,k (r) corresponding to lemma 1 and lemma 2 are +defined as +zQ,j,k (r) = +� +d2 +Q,j,k (r) − h2 +k, +(14) +where Q = {LoS − LoS, LoS − NLoS, NLoS − LoS, NLoS − NLoS}. +B. Distance Distribution of Tagged UAV +Before deriving the PDF of the distance of associated UAV, obtaining the distance distributions +of tagged UAVs is necessary. The distance distributions are given in the following lemmas. +Lemma 3. Given the distance between the typical user and the origin is zu, the CDF of the +distance between the tagged LoS UAV in tier k and the typical user is given by, +FRLoS,k (r, zu) = 1 − exp +� +− +� zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +� ϕLoS−LoS(l,r,zu) +−ϕLoS−LoS(l,r,zu) +vLoS +k +(zu, l, θ) dθdl +� +, +(15) +where +ϕQ,j,k (l, r, zu) = arccos +�l2 + z2 +u − z2 +Q,j,k (r) +2 l zu +� +, +(16) +vQ +k (zu, l, θ) = |l| ΛUAV,k (l) P Q +k (du2U (zu, l, θ)) , +(17) +where Q = {LoS − LoS, LoS − NLoS, NLoS − LoS, NLoS − NLoS}, the horizontal distances +zQ,j,k (r) are defined in (14), ΛUAV,k (r) and P LoS +k +(z) are given in (2) and (3), respectively. The +distance between the potential interfering UAV and the typical user du2U (zu, l, θ) in (17) is given +by, +du2U (zu, l, θ) = +� +(zu − l cos θ)2 + (l sin θ)2. +(18) +Proof. See Appendix A. +Lemma 4. Given the distance between the typical user and the origin is zu, the CDF of distance + +10 +between the tagged NLoS UAV in tier k and the typical user is given by +FRNLoS,k (r, zu) = 1 − exp +� +− +� zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +� ϕNLoS−NLoS(l,r,zu) +−ϕNLoS−NLoS(l,r,zu) +vNLoS +k +(zu, l, θ) dθdl +� +, +(19) +where vQ +k (zu, l, θ) and ϕQ,j,k (l, r, zu) are given in (17) and (16), respectively. +Proof. The proof is similar to that of Lemma 3, therefore omitted here. +Lemma 5. Given the distance between the typical user and the origin is zu, the PDF of distance +between the tagged Q UAV in tier k and the typical user is given by, +fRQ,k (r, zu) = exp +� +− +� zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +� ϕQ−Q(l,r,zu) +−ϕQ−Q(l,r,zu) +vQ +k (zu, l, θ) dθdl +� +× +� � zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +− 4r 1 (r > hk) vQ +k (zu, l, ϕQ−Q) +� +4 l2 z2 +u − (l2 + z2 +u − r2 + h2 +k)2dl ++ +� ϕQ−Q +� +zu+√ +r2−h2 +k,r,zu +� +−ϕQ−Q +� +zu+√ +r2−h2 +k,r,zu +� +r vQ +k +� +zu, zu + +� +r2 − h2 +k, θ +� +� +r2 − h2 +k +dθ ++ +� ϕQ−Q +� +zu−√ +r2−h2 +k,r,zu +� +−ϕQ−Q +� +zu−√ +r2−h2 +k,r,zu +� +r vQ +k +� +zu, zu − +� +r2 − h2 +k, θ +� +� +r2 − h2 +k +dθ +� +, +(20) +where 1 (r > hk) is an indicator function, its value is 1 when r > hk is satisfied, otherwise 0, +vQ +k (zu, l, θ) and ϕQ,j,k (l, r, zu) are given in (17) and (16), respectively. For a LoS tagged UAV, +Q in (20) is replaced with LoS, while Q is replaced with NLoS for an NLoS tagged UAV. +Proof. See Appendix B. +C. Association Probabilities +Association probability is used to describe the probability that a tagged UAV will eventually +be selected as the associated UAV. For the tagged LoS UAV in tier k, there will be no LoS +UAVs providing stronger power in tier k than the tagged UAV, while the NLoS UAVs in tier +k may provide stronger average received power, and in other tiers, both LoS and NLoS UAVs +may provide stronger power. As a result, the association probabilities are given in the following +lemmas. + +11 +Lemma 6. Given the distance between the typical user and the origin is zu, for the LoS tagged +UAV from tier j at Euclidean distance r from the typical user, the probability that the typical +user is associated with this specific UAV is given by +P A +LoS,j (r, zu) = +K +� +k=1,j̸=k +exp +� +− +� zu+zLoS−LoS,j,k(r) +zu−zLoS−LoS,j,k(r) +� ϕLoS−LoS(l,r,zu) +−ϕLoS−LoS(l,r,zu) +vLoS +k +(zu, l, θ) dθdl +� +× +K +� +k=1 +exp +� +− +� zu+zLoS−NLoS,j,k(r) +zu−zLoS−NLoS,j,k(r) +� ϕLoS−NLoS(l,r,zu) +−ϕLoS−NLoS(l,r,zu) +vNLoS +k +(zu, l, θ) dθdl +� +, +(21) +where vQ +k (zu, l, θ) and ϕQ,j,k (l, r, zu) are given in (17) and (16), respectively. +Proof. See Appendix C. +Lemma 7. Given the distance between the typical user and the origin is zu, for the NLoS tagged +UAV from tier j at Euclidean distance r from the typical user, the probability that the typical +user is associated with this specific UAV is given by +P A +NLoS,j (r, zu) = +K +� +k=1 +exp +� +− +� zu+zNLoS−LoS,j,k(r) +zu−zNLoS−LoS,j,k(r) +� ϕNLoS−LoS(l,r,zu) +−ϕNLoS−LoS(l,r,zu) +vLoS +k +(zu, l, θ) dθdl +� +× +K +� +k=1,j̸=k +exp +� +− +� zu+zNLoS−NLoS,j,k(r) +zu−zNLoS−NLoS,j,k(r) +� ϕNLoS−NLoS(l,r,zu) +−ϕNLoS−NLoS(l,r,zu) +vNLoS +k +(zu, l, θ) dθdl +� +, +(22) +where vQ +k (zu, l, θ) and ϕQ,j,k (l, r, zu) are given in (17) and (16), respectively. +Proof. The proof is similar to that of Lemma 6, therefore omitted here. +D. Coverage Probability +As an indispensable intermediate result to enable computing coverage probability, the Laplace +Transform of interference is given in the following lemma. +Lemma 8. Given that the distance between the typical user and the origin is zu, the Laplace +transform of the interference power conditioned on the associated UAV in tier j with Euclidean +distance r from the typical user is given by, +LIQ1,j (s, r, zu) = +K +� +k=1 +� +LIQ1−LoS,j,k (s, r, zu) × LIQ1−NLoS,j,k (s, r, zu) +� +, +(23) + +12 +where Q1 is replaced with LoS when the typical user is associated with LoS UAV, Q1 is replaced +with NLoS when NLoS UAV is associated, and LIQ1−Q2,j,k (s, r, zu), Q2 = {LoS, NLoS} is given +by, +LIQ1−Q2,j,k (s, r, zu) = exp +� +− +� max{0,zu−zQ1−Q2,j,k(r)} +0 +� π +−π +vQ2 +k (zu, l, θ) wQ2,k (s, zu, l, θ) dθdl +� +× exp +� +− +� +∞ +zu+zQ1−Q2,j,k(r) +� π +−π +vQ2 +k (zu, l, θ) wQ2,k (s, zu, l, θ) dθdl +� +× exp +� +− 2 +� zu+zQ1−Q2,j,k(r) +zu−zQ1−Q2,j,k(r) +� π +ϕQ1−Q2,j,k(l,r,zu) +vQ2 +k (zu, l, θ) wQ2,k (s, zu, l, θ)dθdl +� +, +(24) +where +wQ2,k (s, r) = 1 − +� +� +mQ2 +mQ2 + sηQ2ρkGQ2(d2 +u2U (zu, l, θ) + h2 +k) +−αQ2 +2 +� +� +mQ2 +2 +, +(25) +vQ +k (zu, l, θ) and ϕQ,j,k (l, r, zu) are given in (17) and (16), respectively. lemma that Q1 repre- +sents the type of associated UAV, while Q2 represents the type of interfering UAVs. +Proof. See Appendix D. +As the distance distributions of tagged UAVs and the association probabilities have been +derived, we are ready to calculate the local coverage probability. The definition and derivation +of the local coverage probability are given as follows. +Definition 1 (Local coverage probability). The local coverage probability P C (zu, γ) is the +probability that the SINR of the typical user at distance zu from the origin is greater than +threshold γ. +Theorem 1. The exact coverage probability P C (zu, γ) for the typical user is given by, +P C (zu, γ) = +K +� +k=1 +� +∞ +hk +fRLoS,k (r, zu)P A +LoS,k (r, zu) +mLoS−1 +� +n=0 +�(−s)n +n! +∂n +∂sn LULoS,k (s, r, zu) +� +s=µLoS,k(r,γ) +dr ++ +K +� +k=1 +� +∞ +hk +fRNLoS,k (r, zu) P A +NLoS,k (r, zu) +mNLoS−1 +� +n=0 +�(−s)n +n! +∂n +∂sn LUNLoS,k (s, r, zu) +� +s=µNLoS(r,γ) +dr, +(26) + +13 +LUQ,k (s, r, zu) = exp +� +−σ2s +� +LIQ,k (s, r, zu) , +(27) +µQ,k (r) =mQγη−1 +Q ρ−1 +k rαQ, +(28) +Q = {LoS, NLoS} in (27) and (28), fRQ,k (r, zu), P A +LoS,k (r, zu) and P A +NLoS,k (r, zu) are defined +in (20), (21) and (22). +Proof. See Appendix E. +Based on the local coverage probability, the definition and derivation of the overall coverage +probability are given as follows. +Definition 2 (Overall coverage probability). The overall coverage probability is the average +coverage probability of all users. +From the definition, the overall coverage probability for the typical user is the normalized +expectation of the local coverage probability with regard to zu. +Corollary 1. The overall exact coverage probability with the SINR threshold γ is given by, +P C +Overall (γ) = +� +∞ +0 +Λu (zu) P C (zu, γ) zudzu +� +∞ +0 +Λu (zu) zudzu +. +(29) +As is shown in (26), higher-order derivatives of the Laplace transform are needed while +deriving the exact coverage probability. Because the computational complexity increases rapidly +as the order of the derivative increases, the amount of computation is not acceptable under large +shape parameters mLoS and mNLoS. Therefore, we provide an approximate evaluation of the +coverage probability using the upper bound of the CDF of the Gamma distribution [28]. +Theorem 2. The approximate coverage probability �P C (zu, γ) for the typical user is given by, +�P C (zu, γ) = +K +� +k=1 +� +∞ +hk +fRLoS,k (r, zu)P A +LoS,k (r, zu) +mLoS +� +n=1 +�mLoS +n +� +(−1)n+1LULoS,k(n ωLoS µLoS,k (r, γ), r, zu)dr ++ +K +� +k=1 +� +∞ +hk +fRNLoS,k (r, zu) P A +NLoS,k (r, zu) +mNLoS +� +n=1 +�mNLoS +n +� +(−1)n+1LUNLoS,k(n ωNLoS µNLoS(r, γ),r,zu)dr, +(30) +where +ωQ = (mQ!) +− +1 +mQ , Q = {LoS, NLoS}, +(31) + +14 +TABLE I: Table of Parameters +hk [m] +β +P C +Overall +Density of UAVs +One-tier +50 +3.2 ×10−3 +0.9026 +1 UAV/km2 +One-tier +100 +3.2 ×10−3 +0.9203 +1 UAV/km2 +One-tier +150 +3.2 ×10−3 +0.9367 +1 UAV/km2 +Three-tier +(50,100,150) +(4.5, 5.8, 7.6) ×10−3 +0.9713 +1 UAV/km2 +Uniform Distribution +100 +0 +0.3845 +1 UAV/km2 +fRQ,k (r, zu), P A +LoS,k (r, zu), P A +NLoS,k (r, zu) and µQ,k are defined in (20), (21), (22) and (28). +Proof. See Appendix F. +The same as the overall exact coverage probability, the overall approximate coverage proba- +bility is given in the following corollary. +Corollary 2. The overall approximate coverage probability is given by, +�P C +Overall (γ) = +� +∞ +0 +Λu (zu) �P C (zu, γ) zudzu +� +∞ +0 +Λu (zu) zudzu +. +(32) +IV. NUMERICAL RESULTS +In this section, we compare the coverage performance of different systems and optimize the +overall coverage probability by changing the distribution of UAVs in different tiers. Referring +to [9], [17], [29], we assume the channel parameters as follows: the LoS and NLoS path- +loss exponents are αLoS = 2 and αNLoS = 3, the mean additional gains for LoS and NLoS +transmissions are ηLoS = 0dB and ηNLoS = −20dB, m parameters of Nakagami-m fading for +LoS and NLoS UAVs are mLoS = 2 and mNLoS = 1, the noise power is σ2 = 10−7W, the +parameters for the probability of establishing an LoS link in (3) are a = 4.88 and b = 0.429. +The deterministic parameters of users’ distribution in (1) are λu = 10−3m−2 and βu = 5 × 10−3. +As a non-homogeneous PPP, the distribution of users can be realized by thinning property [11] of +homogeneous PPP. Finally, we assume three tiers of UAVs are deployed in a small town center +square with sides of 5km, at 50,100 and 150 meters height, with the corresponding transmission +power 2, 7, and 12 dBm, respectively, and the same value of λ1 = λ2 = λ3 = 4 × 10−5m−2. +In Fig. 2, a curve of local coverage probability for the typical user as a function of the distance +between the origin and the typical user is plotted. We compare the coverage performance of +population density-inspired UAV systems (one-tier and three-tier) with that of the uniformly + +15 +distributed UAV system. All of the above systems have the same UAV deployment density +on average as 25 UAVs/km2 (i.e., λh = 10−6m−2 and βh = 0 for uniform distribution). The +distribution parameters β of the other three systems are shown in the table, λ = 4 × 10−5m−2 +as mentioned above. +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +Distance between the Typical User and the Origin [m] +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Coverage Probability for Typical User + One-tier, 50m + One-tier, 100m + One-tier, 150m + Three-tier + Uniform Distribution +Fig. 2: Local Coverage Probability for the Typical User. +As shown in Table I, different β are chosen to keep the density of the UAVs as 1 UAV/km2. +Assume the threshold of coverage probability is γ = −15dB. It can be seen that the results +of the Monte-Carlo simulation (lines) coincide well with the results of the theoretical analysis +(points). With the same number of UAVs, no matter how far away the typical user is from +the origin, the performance of the three-tier network is always better than that of the one- +tier networks. As shown in Table I, the three-tier network has significant advantages in terms +of overall coverage. At the edge, the advantage of the three-tier network is further expanded, +indicating that the lower limit of network coverage or SINR can be guaranteed. In addition, the +coverage performance of all three systems is declining from center to edge due to the reduced +density of UAV deployment. Compared with the proposed distribution, the uniform distribution +system only has a slight advantage in the edge area, but the overall performance is far inferior + +16 +to the other four systems. The clustering effect brings the non-uniform distribution advantages +over the uniform distribution. According to Fig. 2, when the user is close to the center, the +coverage probability of the UAV network under resident population density-inspired distribution +is significantly greater than that under the uniform distribution. Most users are clustered in the +area close to the center, so the proposed distribution has significant advantages in the overall +coverage probability. +We study the following optimization problems and record the results in Fig. 3 and Fig. 4. We +want to maximize the overall coverage probability by changing the distribution of UAVs, under +the premise that the total number of UAVs is limited (the first constraint) and the coverage of users +in any location is guaranteed (the second constraint). Therefore, the mathematical representation +of the optimization problem is as follows, +arg max +β1,β2 +P C +Overall (γ1) +s.t. +K +� +k=1 +� +∞ +0 +ΛUAV,k (z) dz ≤ Nmax, +P [SINR ≥ γ2] ≥ 0.95, +∀zu ≥ 0, +(33) +where the threshold of overall coverage probability is γ1 = −8dB and the threshold of local +coverage for the typical user is γ2 = −20dB. The total number of UAVs is limited to Nmax = +1000, that is, the maximum density of UAV is 40 UAVs/km2. We deploy UAVs at the first the +second tiers (h = 50, 100 m). +A paradoxical but interesting conclusion in Fig. 3 is, the UAV distribution preferred by the +system is a non-uniform one influenced by the resident density distribution, but the optimal +distribution is not closely related to the residents numerically. The optimal overall coverage +probability P C∗ +Overall and the corresponding β∗ +1 and β∗ +2 value are marked in the figure. It can +be seen that there is a considerable difference among β∗ +1, β∗ +2 and βu. For a large β, the first +constraint cannot be satisfied due to the high density of UAVs. Under this condition, the coverage +probability is set to 0, so the dark blue area at the bottom left appears. When β increases to +10−3, there is a large amount of interference near the centre because the system still tends to +be uniformly distributed and the density is relatively high. With the increase of β, the overall +coverage probability is improved rapidly. Near the optimal area, the coverage performance of +the system is no longer very sensitive to both β1 and β2. This is interesting because in such a + +17 +𝜷𝟏 +∗ = 𝟏. 𝟔×𝟏𝟎#𝟐 +𝜷𝟐 +∗ = 𝟏×𝟏𝟎#𝟑 +𝑷𝑶𝒗𝒆𝒓𝒂𝒍𝒍 +𝑪∗ += 𝟎. 𝟖𝟓𝟎𝟑 +Fig. 3: Overall coverage probability under different UAV distributions. +tolerant system, we do not have to select an accurate set of optimal parameters to determine the +distribution of UAVs, but only to estimate a range. For a large β, the small number of drones +are almost all concentrated in the central area, making it difficult for users in the edge area to +maintain good communication conditions. Therefore, the second constraint cannot be satisfied, +and the dark blue area appears at the top right of the image. Finally, with the same number of +UAVs as the optimal distribution, the overall coverage probability of the uniform distribution in +the same condition (h = 50, 100 m, λ1 = λ2 = 4 × 10−6 m−2) is only 0.2883, which is much +lower than 0.8503. +Although the optimization problem (33) has been carefully studied in Fig. 3, it is still necessary +to study the behavior of the system hidden in the dark blue area. Fig. 4 shows the influence of +the UAV distribution in a single tier on the overall coverage probability. We observe the special +case of β2 = 10−2 in Fig. 3 and broaden the range of β1. +First, it is easy to find that the overall coverage probability increases at the beginning and then +decreases with the increase of the value of β1. This can simply be explained by the fact that too + +0.610~20.8 +0.710-3 +10~20.300.418 +10-7 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Overall Coverage Probability +Doesn't +satisfy +the second +constraint +Doesn't +satisfy +the first +constraint +Feasible +Region +Fig. 4: The influence of the UAV distribution in single tier on the overall coverage probability. +many UAVs will cause too much interference in the central area, while having too few UAVs +will make it difficult for the user to find a close UAV to establish an LoS link. When β1 ≤ 10−5, +the number of UAVs in the first tier is much larger than that in the second tier, so the overall +coverage probability is stable and tends to be similar to that of uniform distribution in the first +tier. When β1 ≥ 5.6 × 10−2, the number of UAVs in the first tier is rapidly decreasing, which +means there are no available LoS UAVs nearby for some users. It is not hard to predict that the +final result will converge to the scenario where only the second tier of UAVs are providing the +service. +V. FURTHER REMARKS +A. Analytic Framework Extension +This subsection presents how to extend the existing analysis framework to other scenarios and +network models. Enhancing the coverage is one of the application scenarios for UAV networks. +UAV networks can also relieve the pressure of insufficient channel capacity in town centers. +Remark 1. Based on Shannon’s theorem and the definition of coverage probability in (9), the +channel capacity can be expressed as [30], +P [B log2 (1 + SINR) > R] = P +� +SINR > 2 +R +B − 1 +� += P [SINR > �γ] , +(34) + +DfirstRD19 +where �γ = 2 +R +B − 1 is the rate threshold. By replacing �γ into SINR threshold γ, the local +probability and overall probability that the channel capacity is greater than the rate threshold +can be obtained by �P C (zu, �γ) given by (30) and �P C +Overall (�γ) given by (32), respectively. +Next, we study green communications in a small hot spot area centered on a base station. +Energy efficiency, the number of bits that can be transmitted per unit of energy consumed, is +used as a performance metric for green communication. For convenience, we calculate the energy +efficiency as the ratio of the number of bits transmitted per unit time (channel capacity) to the +energy consumed per unit time (transmission power). The introduction of the UAV network +allows the central base station to reduce its coverage area, thereby reducing transmission power +and enhancing energy efficiency. The following remark illustrates how the coverage probability +analytic framework can be applied to the above scenario. +Remark 2. From the definition of energy efficiency, it can be calculated by the ratio of channel +capacity to transmission power, +P +� B +ρk +log2 (1 + SINR) > E +� += P +� +SINR > 2 +Eρk +B − 1 +� += P [SINR > �γ] , +(35) +By replacing �γ = 2 +Eρk +B − 1 into SINR threshold γ, the local probability and overall probability +that the energy efficiency is greater than the rate threshold can be obtained by �P C (zu, �γ) given +by (30) and �P C +Overall (�γ) given by (32), respectively. +Furthermore, there is no need for dense deployment of UAVs near the base station in this +case. Fortunately, the network model can be easily extended to the above scenario by adjusting +the density distribution of UAVs in (2). Under the premise that the density (whether of the user +or the UAV) is only related to the distance to the town center, the analytical framework of this +paper is applicable to any distribution model. +B. Distribution Parameter Design +According to the above theorems, it can be seen that the relationship between the coverage +probability and the spatial distribution of UAVs is not straightforward, and obtaining the optimal +parameters by optimization tools is challenging. Therefore, the following qualitative criteria for +parameters about UAVs’ vertical and horizontal distributions are given. Notice that all of the +remarks have been verified by simulation. + +20 +Remark 3. Remarks on altitudes h1, h2, . . . , hK are given as follow. +• In most cases, UAVs have an optimal altitude, and it is better to deploy the UAVs near the +optimal altitude. +• While facing a low communication quality, the UAVs are suggested to be distributed at a +low altitude so that UAVs are closer to users and users in the LoS region can be covered. +• In a good communication environment, by increasing the deployment altitude, more users +can establish LoS links with UAVs, therefore, increasing the coverage probability. +Remark 4. Remarks on the number of tiers K are given as follows. +• With the increase of tiers, more parameters can be optimized so that the coverage perfor- +mance can be improved to some extent. A network with fewer tiers can be considered a +special case of a network with more tiers. However, the improvement in coverage perfor- +mance is limited when more than three tiers are applied. +• Consider a more general case where the receivers (users) can be divided into M classes +according to different gain and demodulation capabilities. Multi-tier distribution has sig- +nificant advantages over single-tier distribution, and the number of deployment tiers K +is recommended to be larger than M. Assuming that the optimal UAV deployment alti- +tude for the receiver of the m-th class is h∗ +m, the height of UAVs is suggested to satisfy +min {h∗ +1, h∗ +2, . . . , h∗ +M} < hk < max {h∗ +1, h∗ +2, . . . , h∗ +M} , ∀k. +Remark 5. Remarks on homogeneity β are given as follows. +• The value of β is related to the strength of interference power relative to noise. When the +interference power is significantly stronger than the environmental noise, it is suggested to +choose a smaller β to make the distribution of UAVs more homogeneous and vice versa. +Considering that the value of βk will affect the average number of UAVs in hot areas when +βk is changed, λk is adjusted to keep the average number of UAVs unchanged. +• We use the exhaustive search to solve the optimization problem about β given in (33), which +results in the calculation complexity increases exponentially with K. An improved alternate +maximization method can be a substitution for the exhaustive search as described in [31]. +The complexity of this method is O (NK2), where N is the preset maximum number of +rounds. The set of suboptimal parameters is obtained by optimizing from β1, β2, . . . to βK +in order. When optimizing βk, the βk is repeatedly reduced by the predefined step size for +at most N times, and one of the β in the set {β1, β2, . . . , βk−1} is increased, so that the + +21 +TABLE II: Optimization of K and β. +1 UAV/km2 +One-tier +Two-tier +Three-tier +Five-tier +h1 = 50m +N/A +N/A +β1 = 4.5 × 10−3 +β1 = 5.4 × 10−3 +h2 = 75m +N/A +N/A +N/A +β2 = 6.5 × 10−3 +h3 = 100m +N/A +β3 = 4.2 × 10−3 +β3 = 5.8 × 10−3 +β3 = 7.8 × 10−3 +h4 = 125m +N/A +N/A +N/A +β4 = 8.2 × 10−3 +h5 = 150m +β5 = 3.2 × 10−3 +β5 = 5.4 × 10−3 +β5 = 7.6 × 10−3 +β5 = 9.8 × 10−3 +P C +Overall +0.9367 +0.9557 +0.9713 +0.9786 +coverage probability is maximized when the UAV density is unchanged. The optimization of +βk ends when the coverage probability no longer increases. +The example in Table II provides further explanation for the above remarks. In Table II, we +compare the coverage performance under different numbers of tiers K. The total density and the +density of UAVs in each tier are fixed as 1 UAV/km2 and λk = 4×10−5. The set of homogeneity +{β1, β2, . . . , βK} is obtained by alternate maximization method. Overall, increasing the number +of tiers allows more parameters to be optimized, thus achieving better coverage performance. +The UAV deployment in the one-iter network (β5 = 3.2×10−3) can be regarded as a special case +of that of a two-tier network ({β3, β5} = {+∞, 3.2 × 10−3}), but it is not optimal. Finally, the +gain in coverage probability from deploying more than three tiers of UAV networks is limited. +VI. CONCLUSION AND FUTURE WORK +In this paper, We studied the coverage performance of multi-tier UAV networks in a centralized +urban model. We first derived the distance distribution of tagged UAVs and association probability +for the selected typical user. Based on this, the analytical expression of downlink coverage +probability is given and proved to be consistent with the Monte-Carlo simulation results. As +a result, the coverage probability for the typical user and intermediate products are all related +to the distance zu. Both the local and total coverage performance are significantly improved +by increasing the number of UAV network tiers. The urban population density-inspired model +has a huge advantage over the uniform distribution performs. However, too much concentration +of UAVs in the central area will bring more noise to the town center and fail to maintain +communication for users at the edge. Therefore, how to design the distribution of each tier of +UAVs is crucial. + +22 +One future research direction is introducing interference and noise mitigation technologies into +the framework based on the proposed resident population density-inspired model. In urban areas, +the relatively dense deployment of UAVs may cause strong interference. Strong environmental +noise in town centers is also one factor limiting the performance of wireless communication. +Under the SG framework, orthogonal channel [32] and directional antenna gain [33] can be +introduced into the system model respectively to reduce interference and noise power. In addition, +we model the users as a PPP, which means that the user’s movement is undirected and random. +Considering that there is a directional flow of people in the town [34], analyzing the coverage +probability of the urban system based on SG will be challenging and application-oriented. +APPENDIX A +PROOF OF LEMMA 3 +When the distance between the typical user and the origin is fixed, given that the distance +between the tagged LoS UAV in tier k and user RLoS,k is a random valuable, the Cumulative +Distribution Function (CDF) of RLoS,k is given by +FRLoS,k (r, zu) = P [RLoS,k < r] = 1 − P [RLoS,k > r] += 1 − P [N (Ak (r)) = 0] +(a) += 1 − exp +� +− +� +Ak(r) +ΛUAV,k (l) ldldθ +� +, +(36) +where ΛUAV,k (l) is defined in (2), N (Ak (r)) in (36) counts the number of the UAVs in region +Ak (r), which is a circle at the height of hk centered directly above the typical user with radius +� +r2 − h2 +k, and (a) is given by the property of the general PPP [35], +P [N (Ak (r)) = n] = exp +� +− +� +Ak(r) +ΛUAV,k (l) l dldθ +� exp +� +− +� +Ak(r) ΛUAV,k (l) l dldθ +�n +n! +. (37) +where zu is the horizontal distance from the typical user to the origin, λu determines the total +density of the plane, βu is a measure of homogeneity. +To integrate formulation in (36) over the region of Ak, the area is divided into infinite +concentric circular arcs centered at the point which is directly above the origin at the height of +hk. When the radius l of the concentric circular arc is fixed, the density function ΛUAV,k is a +constant, the coordinates of the points on the arc can be uniquely represented by θ. The bold +part of the bottom half of the Fig. 5 is one of the concentric arcs. The upper bound of θ can +be obtained from the geometric relations in the Fig. 5, denoted as ϕLoS−LoS, defined in (16), + +23 +𝑥 +𝑦 +𝑧 +𝝋 +𝜽 +𝒍 +u2U +𝒅 +𝒛𝑸𝟏%𝑸𝟐,𝒋,𝒌 +𝒛𝒖 +Fig. 5: Vertical Viewed System Schematic Figure. +and the lower bound of θ is −ϕLoS−LoS because of the symmetry of the circle. Furthermore, the +horizontal distance du2U (zu, l, θ) between the typical user and the point on the arc is defined +in (18), which can also be obtained from simple geometrical relationships. Hence, we have the +following equation +� +Ak(r) +ΛUAV,k (l) ldldθ = +� zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +� ϕLoS−LoS(l,r,zu) +−ϕLoS−LoS(l,r,zu) +vLoS +k +(zu, l, θ) dθdl, +(38) +where ΛUAV,k (l) and vQ +k (zu, l, θ) are defined in 2) and (17), respectively. It is important to note +that l may be negative when the value of zLoS−LoS,j,k is greater than the horizontal distance zu. +Therefore, |l| is used in the outer integral. +APPENDIX B +PROOF OF LEMMA 5 +As in Lemma 5, Q is used to represent the type of tagged UAVs, i.e., Q is replaced with LoS +when an LoS UAV is tagged or NLoS otherwise. By taking the derivative of FRQ,k (r, zu), the +distribution of the nearest UAVs in tier k with a distance r from the user is obtained, which is + +24 +denoted as fRQ,k (r, zu), +fRQ,k (r, zu) = ∂ +∂rFRQ,k (r, zu) += ∂ +∂r +� +1 − exp +� +− +� zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +� ϕQ−Q(l,r,zu) +−ϕQ−Q(l,r,zu) +vQ +k (zu, l, θ) dθdl +�� +(a) += exp +� +− +� zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +� ϕQ−Q(l,r,zu) +−ϕQ−Q(l,r,zu) +vQ +k (zu, l, θ) dθdl +� +× +� � zu+√ +r2−h2 +k +zu−√ +r2−h2 +k +∂ +∂rfin,k (l, zu, r, θ) +� +�� +� +The derivative of the integrand +dl ++ +∂ +� +zu + +� +r2 − h2 +k +� +∂r +fin,k +� +zu + +� +r2 − h2 +k, zu, r, θ +� +� +�� +� +The derivative of the integral upper bound +− +∂ +� +zu − +� +r2 − h2 +k +� +∂r +fin,k +� +zu − +� +r2 − h2 +k, zu, r, θ +� +� +�� +� +The derivative of the integral upper bound +� +, +(39) +where vQ +k (zu, l, θ) and ϕQ−Q (l, r, zu) are defined in (17) and (16), respectively, and (a) follows +Leibnitz’s rule, +fin,k (l, zu, r, θ) is the integrand of the outer integral, given by +fin,k (l, zu, r, θ) = +� ϕQ−Q(l,r,zu) +−ϕQ−Q(l,r,zu) +vQ +k (zu, l, θ) dθ. +(40) +For the derivative of the integral upper bound in (39), +∂ +� +zu + +� +r2 − h2 +k +� +∂r +fin,k +� +zu + +� +r2 − h2 +k, zu, r, θ +� += +r +� +r2 − h2 +k +� ϕQ−Q +� +zu+√ +r2−h2 +k,r,zu +� +−ϕQ−Q +� +zu+√ +r2−h2 +k,r,zu +� vQ +k +� +zu, zu + +� +r2 − h2 +k, θ +� +dθ. +(41) +The derivative of the integral lower bound is similar to that of (41), therefore omitted here. + +25 +For the derivative of the intergrad, +∂ +∂rfin,k (l, zu, r, θ) = ∂ +∂r +� ϕQ−Q(l,r,zu) +−ϕQ−Q(l,r,zu) +vQ +k (zu, l, θ) dθ +(a) += vQ +k (zu, l, ϕQ−Q) ∂ϕQ−Q (l, r, zu) +∂r +− vQ +k (zu, l, −ϕQ−Q) ∂ (−ϕQ−Q (l, r, zu)) +∂r +� +(b) += 2vQ +k (zu, l, ϕQ−Q) ∂ϕQ−Q (l, r, zu) +∂r +(c) += 1 (r > hk) +4 r vQ +k (zu, l, ϕQ−Q) +� +4 l2 z2 +u − (l2 + z2 +u − r2 + h2 +k)2, +(42) +where (a) follows Leibniz’s rule for internal integral, and the expression in (b) is simplified by +the fact du2U (zu, l, −ϕQ−Q) = du2U (zu, l, ϕQ−Q), which can be easily obtained by (16), +1 (r > hk) is the indicator function defined in (5), and (c) is obtained by substitute ∂ϕQ−Q(l,r,zu) +∂r +, +which is given by, +∂ϕQ−Q (l, r, zu) +∂r += ∂ +∂u arccos +�l2 + z2 +u − u2 +2 l zu +� +∂ +∂v +� +v2 − h2 +k +∂v +∂r += +2u +� +4 l2 z2 +u − (l2 + z2 +u − u2)2 +2v +2 +� +v2 − h2 +k +· 1 +��ρk +ρk +� +1 +αLoS r > hk +� += 1 (r > hk) +2r +� +4 l2 z2 +u − (l2 + z2 +u − r2 + h2 +k)2, +(43) +where u = zLoS−LoS,j,k (r) |j=k = 1 (r > hk) +� +r2 − h2 +k, and v = dLoS−LoS,j,k (r) |j=k = max {hk, r}. +Substitute (41) and (42) into (39), the final result is derived. +𝑧& +𝑧'$('𝟐,*,+ +𝑧'$('𝟐,*,+ +𝑧& +Origin +The typical +user +User's non- +interference circle +Interfering +UAVs' circle +Fig. 6: Two relationships between user’s non-interference circle and interfering UAVs’ circle. + +126 +APPENDIX C +PROOF OF LEMMA 6 +When the distance between the typical user and the origin is fixed, the probability that the +typical user is associated with the tagged LoS UAV in tier j is equal to the probability that the +average received power of other 2K − 1 tagged UAVs is lower than it, where K is the number +of tiers. Using the solution of lemma 1, we have +P A +LoS,j(r, zu) = +K +� +k=1,j̸=k +P [RLoS,k > dLoS−LoS,j,k (r)] × +K +� +k=1 +P [RNLoS,k > dLoS−NLoS,j,k (r)] += +K +� +k=1,j̸=k +P [N (ALoS,k (r)) = 0] × +K +� +k=1 +P [N (ANLoS,k (r)) = 0] +(a) += +K +� +k=1,j̸=k +exp +� +− +� +ALoS,k(r) +vLoS +k +(zu, l, θ)dl dθ +� +× +K +� +k=1 +exp +� +− +� +ANLoS,k(r) +vNLoS +k +(zu, l, θ)dl dθ +� +, +(44) +where RQ,k is the distance between the tagged UAV in tier k and typical user, vQ +k (zu, l, θ) +is defined in (17), N (AQ,k (r)) counts the number of the UAVs in region AQ,k (r), which is +a circle at the height of hk centered directly above the typical user with radius zLoS−Q,j,k (r), +Q = {LoS, NLoS}, and (b) is given by the property of the general PPP in (37). The following +two equations can be obtained in a similar way to (38), +� +AQ,k(r) +vQ +k (zu, l, θ)dl dθ = +� zu+zQ−Q,j,k +zu−zQ−Q,j,k +� ϕQ−Q(l,r,zu) +−ϕQ−Q(l,r,zu) +vQ +k (zu, l, θ)dθdl, +(45) +where Q = {LoS, NLoS}. The final result is derived by substituting (45) into (44). + +27 +APPENDIX D +PROOF OF LEMMA 8 +For LoS associated UAV in tier j, the Laplace transform of the interference power can be +expressed as, +LILoS,j (s, r, zu) +(a) += EILoS,j +� +e−sILoS,j� += EΦ,G +� +exp +� +− s +K +� +k=1 +� +� +x∈ΦLoS,k\xo +ηLoSρkGLoSr−αLoS + +� +x∈ΦNLoS,k +ηNLoSρkGNLoSr−αNLoS +��� +(b) += +K +� +k=1 +EΦLoS,k +� +� +x∈ΦLoS,k\xo +EGLoS +� +exp +� +−s ηLoSρkGLoSr−αLoS�� +� +× +K +� +k=1 +EΦNLoS,k +� +� +x∈ΦNLoS,k +EGNLoS +� +exp +� +−s ηNLoSρkGNLoSr−αNLoS�� +� +, +(46) +where (a) follows the definition of Laplace transform, +ΦLoS,k\Φo are all of the LoS UAVs in tier k except for the associated one, and (b) follows +the independence of the point process and the small scale fading, (c) is obtained. For Laplace +transform of the interference caused by LoS UAVs in tier k, +In equation (47) shown at the top of the next page, (a) follows the PGFL of inhomogeneous +PPP [35], AC +LoS,k(r) is the complement of ALoS,k(r) in the two dimensional plane at the height +of hk, the definition of ALoS,k(r) is described in Appendix A, vQ +k (zu, l, θ) and ϕQ−Q (l, r, zu) +are defined in (17) and (16) respectively, wLoS,k (s, r) defined in (25) is used to simplify the +expression in (b). +As is shown in Fig. 6, there should be non-interfering UAVs inside the green circle centered +at the typical user, the green circle is called the user’s non-interference circle. The difference +between the left and right images is whether the origin is included by user’s non-interference +circle. For a fixed radius l, the circles centered at the origin is used to cover the possible locations +of interfering UAVs with horizontal distance l to the origin, called the interfering UAV’s circle. +These two circles may be separated or intersected, and sometimes one circle may contain another, +shown in step (b) of (47). The Laplace transform of the interference in other conditions is similar +to the process in (47), therefore omitted here. + +28 +EΦLoS,k +� +� +� +x∈ΦLoS,k\xo +EGLoS +� +exp +� +−s ηLoSρkGLoSr−αLoS�� +� +� +(a) += exp(− +� +AC +LoS,k(r) +vLoS +k +(zu, l, θ) +� +1 − EGLoS +� +exp +� +−s ηLoSρkGLoS +� +d2 +u2U (zu, l, θ) + h2 +k +�−αLoS/2��� +dθdl) += exp +� +− +� +AC +LoS,k(r) +vLoS +k +(zu, l, θ) +� +1 − +� +mLoS +mLoS + s ηLoSρkGLoS(d2 +u2U (zu, l, θ) + h2 +k)−αLoS/2 +�mLoS� +l dθdl +� +(b) += exp +� +− +� max{0,zu−zLoS−LoS,j,k(r)} +0 +� π +−π +vLoS +k +(zu, l, θ) wLoS,k (s, zu, l, θ) dθdl +� +�� +� +User′s non−interference circle separates from interfering UAVs′ circle +� +× exp +� +− +� +∞ +zu+zLoS−LoS,j,k(r) +� π +−π +vLoS +k +(zu, l, θ) wLoS,k (s, zu, l, θ) dθdl +� +�� +� +User′s non−interference circle contained by interfering UAVs′ circle +� +× exp +� +− 2 +� zu+zLoS−LoS,j,k(r) +zu−zLoS−LoS,j,k(r) +� π +ϕLoS−LoS,j,k(l,r,zu) +vLoS +k +(zu, l, θ) wLoS,k (s, zu, l, θ) dθdl +� +�� +� +User′s non−interference circle and interfering UAVs′ circle intersect +� +(47) +APPENDIX E +PROOF OF THEOREM 1 +By the definition of coverage probability in (9), SINR becomes a deterministic expression only +when: (i) the tier where the associated UAV is located; (ii) LoS or NLoS link constructed by the +typical user and associated UAV; (iii) the distance between the typical user and the origin; (iv) +the Euclidean distance between the typical user and the associated UAV. Therefore, the coverage +probability of the typical user is given by (48) at the top of next page. where P A +LoS,k (r, zu), +P A +NLoS,k (r, zu) and fRQ,k (r, zu) are given in (21), (22) and (20), respectively, (a) is obtained +by substituting UQ,k (r, zu) = IQ,k (r, zu) + σ2 and µQ,k (r, γ) are define in (28) into the former +result, (b) is obtained from the expectation of r. In order to get the final analytical result, the + +29 +P C (zu, γ) = +K +� +k=1 +Er,I +� +P A +LoS,k (r, zu) P +�ηLoSρkGLoSr−αLoS +ILoS,k (r, zu) + σ2 > γ +�� ++ +K +� +k=1 +Er,I +� +P A +NLoS,k (r, zu) P +�ηNLoSρkGNLoSr−αNLoS +INLoS,k (r, zu) + σ2 +> γ +�� +(a) += +K +� +k=1 +Er,U +� +P A +LoS,k (r, zu) P [GLoS > µLoS,k (r, γ) ULoS,k (r, zu)] +� ++ +K +� +k=1 +Er,U +� +P A +NLoS,k (r, zu) P [GNLoS > µNLoS (r, γ) UNLoS,k (r, zu)] +� +(b) += +K +� +k=1 +� +∞ +hk +EU [P [GLoS > µLoS,k (r, γ) ULoS,k (r, zu)]] P A +LoS,k (r, zu) fRLoS,k (r, zu) dr ++ +K +� +k=1 +� +∞ +hk +EU [P [GNLoS > µNLoS (r, γ) UNLoS,k (r, zu)]] P A +NLoS,k (r, zu) fRNLoS,k (r, zu) dr, +(48) +next steps are taken, +EU [P [GLoS > µLoS,k (r, γ) ULoS,k (r, zu)]] +(a) += EU +�Γu (mLoS, mLoSµLoS,k (r, γ) ULoS,k (r, zu)) +Γ (mLoS) +� +(b) += EU +� +exp (−µLoS,k (r, γ) U (r, zu)) +mLoS−1 +� +n=0 +(µLoS,k (r, γ) ULoS,k (r, zu))n +n! +� += +mLoS−1 +� +n=0 +(µLoS,k (r, γ))n +n! +EU +� +exp (−µLoS,k (r, γ) ULoS,k (r, zu)) (ULoS,k (r, zu))n� +(c) += +mLoS−1 +� +n=0 +�(−s)n +n! +∂n +∂snLULoS,k (s, r, zu) +� +s=µLoS,k(r,γ) +, +(49) +where (a) follows the complementary cumulative distribution function (CCDF) of the Gamma +distribution F G (g) = Γu(m,mg) +Γ(m) +, where Γu (m, mg) = +� +∞ +mg tm−1e−tdt is the upper incomplete +Gamma function, and (b) follows the definition Γu(m,mg) +Γ(m) += exp (−g) +m−1 +� +n=0 +gn +n! [36], by the linearity +of the expectation operator and +EU [exp (−sULoS,k (r, zu)) ULoS,k(r, zu)n] = (−1)n ∂n +∂snLULoS,k (s, r, zu) , +(50) + +30 +(c) is obtained. The steps of NLoS UAVs are similar to that of LoS UAVs, therefore omitted +here. +APPENDIX F +PROOF OF THEOREM 2 +Because the first several steps of the proof of approximate coverage probability are similar to +that of exact coverage probability, we start from formulation (49) step (a), +EU +�Γu (mLoS, mLoSµLoS,k (r, γ) ULoS,k (r |ru)) +Γ (mLoS) +� +(a) += 1 − EU +�Γl (mLoS, mLoSµLoS,k (r, γ) ULoS,k (r |ru)) +Γ (mLoS) +� +(b) +≈ 1 − EU [(1 − exp (−βLoSµLoS,k (r, γ) ULoS,k (r, zu)))mLoS] +(c) += EU +� mLoS +� +n=1 +�mLoS +n +� +(−1)n+1 exp (−nωLoSµLoS,k (r, γ) ULoS,k (r, zu)) +� += +mLoS +� +n=1 +�mLoS +n +� +(−1)n+1LULoS,k (nωLoSµLoS,k (r, γ)), +(51) +where LULoS,k (s, r, k) is given in (27), and s = nωLoSµLoS,k (r, γ), Γl (m, mg) = +� mg +0 +tm−1e−tdt +in step (a) is the lower incomplete Gamma function, which satisfies Γu(m,mg) +Γ(m) += 1 − Γl(m,mg) +Γ(m) +. +(b) follows from the tight approximation to coverage probability, where ωLoS = (mLoS!) +−1 +mLoS . 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Heath, “Coverage and rate analysis for millimeter-wave cellular networks,” IEEE Transactions on Wireless +Communications, vol. 14, no. 2, pp. 1100–1114, 2014. + diff --git a/ZdAyT4oBgHgl3EQf9vpC/content/tmp_files/load_file.txt b/ZdAyT4oBgHgl3EQf9vpC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ebbd927b0abcc65851509df89fcae2058dc7d44e --- /dev/null +++ b/ZdAyT4oBgHgl3EQf9vpC/content/tmp_files/load_file.txt @@ -0,0 +1,1288 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf,len=1287 +page_content='1 Resident Population Density-Inspired Deployment of K-tier Aerial Cellular Network Ruibo Wang, Mustafa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Kishk, Member, IEEE and Mohamed-Slim Alouini, Fellow, IEEE Abstract Using Unmanned Aerial Vehicles (UAVs) to enhance network coverage has proven a variety of benefits compared to terrestrial counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' One of the commonly used mathematical tools to model the locations of the UAVs is stochastic geometry (SG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' However, in the existing studies, both users and UAVs are often modeled as homogeneous point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In this paper, we consider an inhomogeneous Poisson point process (PPP)-based model for the locations of the users that captures the degradation in the density of active users as we move away from the town center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In addition, we propose the deployment of aerial vehicles following the same inhomogeneity of the users to maximize the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In addition, a multi-tier network model is also considered to make better use of the rich space resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Then, the analytical expressions of the coverage probability for a typical user and the total coverage probability are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Finally, we optimize the coverage probability with limitations of the total number of UAVs and the minimum local coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Finally we give the optimal UAV distribution parameters when the maximum overall coverage probability is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Index Terms Coverage probability, urban model, multi-tier UAV network, stochastic geometry, inhomogeneous Poisson point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' INTRODUCTION In the next generation mobile network (5G, beyond 5G), UAVs have many application sce- narios [1]–[3], among which UAV-aided ubiquitous coverage becomes an important topic [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Because UAVs are easy to deploy, highly mobile, and have 3D deployment, they are often used to Ruibo Wang and Mohamed-Slim Alouini are with King Abdullah University of Science and Technology (KAUST), CEMSE division, Thuwal 23955-6900, Saudi Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Mustafa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Kishk is with the Department of Electronic Engineering, National University of Ireland, Maynooth, W23 F2H6, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (e-mail: ruibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='wang@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='sa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' mustafa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='kishk@mu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='ie;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' slim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='alouini@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='sa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='00879v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='NI] 2 Jan 2023 2 build temporary or dynamic networks and provide ubiquitous coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Especially, UAV is widely used to relieve the pressure of large crowds gathering in small areas [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' They are proven to provide reliable system coverage in hot spots and provide additional system performance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Due to the demand for high rate signals, multi-tier vertical heterogeneous networks (VHetNet) is proposed to make use of space resources in city centers [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' One of the main unanswered questions in the realm of UAV-enabled wireless networks is where and how high the UAVs should be deployed [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The common assumption in SG-based literature is that the user’s spatial distribution is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Consequently, existing literature typically assumes that the density of the UAVs is spatially invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' However, according to recent studies on resident population densities, a more proper assumption would be for the density of the users (and consequently the UAVs) drops as their distance from the town center increases [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Analyzing the influence of such a setup on the wireless network’s performance is the main objective of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' More details on the contributions of this paper are provided later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' I-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Related Work SG is a powerful mathematical method of analyzing communication networks with irregular topology [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Furthermore, the SG framework is suitable for modeling and analyzing devices in motion, such as UAVs, cars [12], and LEO satellites [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The SG-based analytical results of the network coverage probability can provide accurate approximations to the actual network [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Next, the authors in [17] proposed an air-to-ground line-of-sight (LoS) probability model suitable for town centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In this model, the probability of the UAV being blocked by the building decreases with the increase of the elevation angle of the UAV to the typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' This model divides UAVs into LoS UAVs and non-line-of-sight (NLoS) UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Since the model is related to density, area, and height of building [18], it is suitable for various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Based on the LoS probability model, there has been some literature on UAV networking in town centers [6], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In the existing research, some resident population density models have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Different user distributions in several urban environments are proposed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' A disjoint clustered model for large resident population density is set up in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' A central model is provided in [10] and is adopted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In the central model, user density decreases with the distance from the user to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' However, the above articles pay more attention 3 to the modeling of users, while the UAVs are simply deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' UAVs are deployed as a homogeneous PPP in [10], [21], while the locations of UAVs are determined by clustering in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, the deployment of UAVs is also worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' However, with regard to analyzing downlink network coverage performance, changing the distribution of UAVs brings much more difficulty in technical derivation than changing the distribution of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given that UAVs form a homogeneous PPP, the downlink coverage performance of users at any location is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Nevertheless, when the density of the UAV is not constant, the distributions of the distance between the serving UAV and the interfering UAV to the user are different for the users at different locations, which makes the analysis challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' To effectively utilize the deployable space of UAVs, developing the vertical deployment mode of UAVs is also worth studying, in addition to designing the horizontal distribution of UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Based on the SG framework, authors in [8], [9], [22]–[25] have put forward multi-tier VHetNets consisting of ground base stations (BSs), UAVs and high altitude platforms (HAPs) and low earth orbit (LEO)-satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The above researches all introduced the concept of association probability to describe the probability of users choosing a communication device in a specific tier (instead of other tiers) to provide services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Unfortunately, the above analytical framework is unsuitable for our study because the UAVs are not uniformly distributed in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Designing a different method to obtain the association probability is another challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Contribution The contributions of this paper can be summarized as follows: We study a resident population density-inspired model of the urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The density of users decreases with the distance to the town center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' UAVs follow a similar distribution to the distribution of users and are deployed at different altitudes with different densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We derive the analytical result of coverage probability under the specific model and prove that it is consistent with the Monte-Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In addition, the existing coverage probability analysis framework is extended to data rate and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The coverage performance of multi-tier networks and single-tier networks are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We also compare the coverage performance of the population density-inspired distribution and the homogeneous distribution of UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' By adjusting the distribution of UAVs in each tier, we optimize the coverage probability under different user distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Furthermore, remarks on the parameter design criterion 4 for UAV distribution are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Network Model User LoS UAV NLoS UAV 𝒙 𝒚 𝒛 ℎ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ℎ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' "# 𝑇𝑖𝑒𝑟 𝑘 + 1 … … Typical user Tagged LoS UAV Tagged NLoS UAV 𝑇𝑖𝑒𝑟 𝑘 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 1: Illustration of the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 1, we consider a scenario in which ground users are distributed according to an inhomogeneous PPP, which is inspired by the resident population distribution model proposed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We assume that the UAVs in the VHetNet are deployed based on the ground users density and, hence, their locations also follow an inhomogeneous PPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Assuming that the center of the town is located at the origin, the densities of the users and the UAVs near the origin are relatively high, while the density goes down as we move away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Assume the users are located at the ground, with horizontal distance zu to the origin, the density distribution of the users Λu (zu) can be represented as follows, Λu (zu) = λue−βuzu, (1) where λu determines the total density of the plane, βu is a measure of homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When the value of βu is large, the users are spatially condensed at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When βu = 0 the process degenerates to a homogeneous PPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Without loss of generality, we focus on a typical user located on the positive X-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We assume that K tiers of UAVs are distributed at a set of some fixed heights hk independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Their location distribution of each tier form a 2D inhomogeneous PPP, denoted by Φk ∆= {xi,k} UserhTler1TypicalTagg 1Tlerh5 where xi,k refers to the 3D location of UAV i in tier k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We prefer polar coordinates (zi, θi, hk) to represent xi,k, where zi is the horizontal distance between the UAV and the origin, θi is the angle between the X-axis and the line which connects the projection of the UAV and the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Because designers tend to place more UAVs in densely populated areas, it is reasonable to assume that they will be in the same distribution as the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Thus, in tier k, the density distribution ΛUAV,k can be described as, ΛUAV,k (zi) = λke−βkzi, (2) where λk and βk are parameters of the UAVs in tier k, which have the same meaning as the users’ parameters in the density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Furthermore, we assume that each tier of UAVs have the same transmitting power ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The quad Tk = {hk, λk, βk, ρk} , k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=', K is used to represent the parameters of the kth tier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Channel Model To model the air-to-ground channel between a user and a UAV, we need to take into con- sideration the LoS and NLoS scenarios [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Considering a UAV in tier k, given the horizontal distance z between the UAV’s projection on the ground and the user, the probability of setting up an LoS link between the typical user and the UAV is [17], [26], P LoS k (z) = 1 1 + a exp � −b � 180 π tan−1 � hk z � −a ��, (3) where a and b are environment-dependent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' From the perspective of the typical user, the inhomogeous PPP process corresponding to the K-tier UAVs can be split into two disjoint PPPs, that is Φk=ΦLoS,k ∪ ΦNLoS,k and ΦLoS,k ∩ ΦNLoS,k = φ, where ΦLoS,k and ΦNLoS,k denote the set of UAVs which establish LoS and NLoS conditions for the typical user respectively, φ is the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In this article, UAVs with an LoS link to the typical user are abbreviated as LoS UAVs, while the rest are abbreviated as NLoS UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, the 3D VHetNet is split into 2K disjoint two-dimensional PPPs, with the density P Q k ΛUAV,k, where Q = {LoS, NLoS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' After LoS and NLoS states are defined, the channel fading model can be established, which is described by small-scale fading and large-scale fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For small scale fading, we denote channel fading power gains in terms of independent random variables GLoS and GNLoS, under LoS and NLoS conditions for the typical user, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 6 In order to represent several fading scenarios, Nakagami-m fading is experienced with shape parameters and scale parameters (mLoS, 1 mLoS) and (mNLoS, 1 mNLoS) for LoS and NLoS links, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As a result, the probability density functions (PDF) of the power gains GQ is given by [27] fGQ (g) = mQmQrmQ−1 Γ (mQ) e−mQ g, (4) where Γ (mQ) = � ∞ 0 xmQ−1e−xdx is the Gamma function, Q = {LoS, NLoS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For large scale fading, ηLoS and ηNLoS are mean additional gain for LoS and NLoS trans- missions [17], with ηLoS > ηNLoS satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Combining small scale and large scale fading, the received power of the typical user, transmitted by a UAV in tier k, is given by, Sk (r) = � � � ηLoSρkGLoSr−αLoS in case of LoS ηNLoSρkGNLoSr−αNLoS in case of NLoS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (5) where ρk is the transmission power of UAVs in tier k, αLoS and αNLoS are path-loss exponents for LoS and NLoS transmissions, with αLoS < αNLoS satisfied, r is the Euclidean distance between the UAV and the typical user, which is computed by polar coordinates (zi, θi, hk) of the UAV and the horizontal distance zu from the user to the origin rxi,k = � (zicosθi − zu)2 + (zisinθi)2 + h2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (6) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Interference In each tier, the closest LoS and NLoS UAVs are called tagged UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' According to the strongest average received power association strategy [9], the typical user will associate with the UAV with the strongest average received power among these 2K tagged UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Furthermore, denote the location of the associated UAV as xo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Note that the typical user may not associate with the closest UAV, since in different tiers, the transmitted power of the UAVs is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In urban areas where UAVs are densely distributed, it is necessary to consider interference between UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Considering a worst-case scenario, except for the associated UAV, other LoS and NLoS UAVs interfere with the typical user, which we refer to as interfering UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given that the horizontal distance from the typical user to the origin is zu and the associated UAV is in tier j, the total interference can be expressed as a function of rxo, which is the Euclidean 7 distance between the associated UAV and the typical user, Ij (rxo, zu) = K � k=1 � � xi,k∈ΦLoS,k\\{xo} ηLoSρkGLoSr−αLoS xi,k + � xi,k∈ΦNLoS,k\\{xo} ηNLoSρkGNLoSr−αNLoS xi,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (7) As shown in the above formulation, the value of the total interference is related to zu, rxo and the tier where the associated UAV is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' rxo and the transmission power of tier j determine the received power from the associated UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Performance Analysis Assuming the associated UAV is located in tier j, the instantaneous signal-to-interference plus noise ratio (SINR) at the typical user is given by the following equation, SINR = � � � ηLoSρjGLoSr −αLoS xo Ij(rxo|zu )+σ2 xo ∈ ΦLoS,k ηNLoSρjGNLoSr −αNLoS xo Ij(rxo|zu )+σ2 xo ∈ ΦNLoS,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (8) where σ2 is the additive white Gaussian noise (AWGN) power, and Ij (rxo |zu) is the total interference power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The reliability of the service provided can be evaluated by the average performance of SINR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In consequence, the coverage probability, which represents the probability that the system can provide reliable connections is defined as the probability that the SINR is greater than a predefined threshold γ: P C ∆= P [SINR > γ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (9) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' PROBLEM FORMULATION In this section, our objective is to obtain the analytical expression of coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' From the definition of coverage probability in (9), we know that distinguishing the associated UAV and the interfering UAVs is a prerequisite for computing the coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Taking an LoS associated UAV for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the following steps are used to obtain the analytic expression for the coverage probability: (i) derive the PDF of the distance distribution of tagged UAV in each tier k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' denoted as fRLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (ii) calculate the probability of the tagged UAV in tier k being associated with the typical user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' defined as association probability P A LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' with fRLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) P A LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) being the PDF of the distance between the associated UAV in tier k and the typical user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (iii) for a specific distance r between the typical user and its associated 8 UAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' denote the probability that the SINR is greater than the threshold γ as the conditional coverage probability P C (γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu |r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The average coverage probability is obtained by taking the expectation of the conditional coverage probability P C (γ, r, zu) with respect to the PDF of r in step (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Steps (i) - (iii) will be explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' III-B Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' III-C, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' III-D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Nearest Interfering UAVs A clear understanding of the location range of interfering UAVs is necessary when analyzing tagged or associated drone distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Obviously, for tier k, the nearest interfering UAV should locate at a distance larger than hk for the typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Another possible lower bound of the distance for an interfering UAV is to ensure a lower average receiving power than the associated UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In the following lemmas, the distance between the typical user and the nearest interfering UAVs is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given that the typical user is associated with a LoS UAV located at distance r in tier j, the closest interfering LoS and NLoS UAV in tier k are at least at distances dLoS−LoS,j,k (r) and dLoS−NLoS,j,k (r) , given by dLoS−LoS,j,k (r) = max � hk, �ρk ρj � 1 αLoS r � , (10) dLoS−NLoS,j,k (r) = max � hk, �ηNLoSρkE[GNLoS] ηLoSρjE[GLoS] � 1 αNLoS r αLoS αNLoS � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' According to the received power in (5), the average received power of the associated UAV at distancerin tier j is Sj = ηLoSρjGLoSr−αLoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The closest interfering LoS UAV in tier k is at least at distances dLoS−LoS,j,k, which can be obtained by solving the equality ηLoSρjGLoSr−αLoS = ηLoSρkGLoSd−αLoS LoS−LoS,j,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Similarly, for NLoS interfering UAVs, dLoS−NLoS,j,k can be obtained by solving the equality ηLoSρjGLoSr−αLoS = ηNLoSρkGNLoSd−αNLoS LoS−NLoS,j,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given that the typical user is associated with a NLoS UAV located at distance r in tier j, the closest interfering LoS and NLoS UAV in tier k are at least at distances dNLoS−LoS,j,k (r) and dNLoS−NLoS,j,k (r) , given by dNLoS−LoS,j,k (r) = max � hk, � ηLoSρkE[GLoS] ηNLoSρjE[GNLoS] � 1 αLoS r αNLoS αLoS � , (12) 9 dNLoS−NLoS,j,k (r) = max � hk, �ρk ρj � 1 αNLoS r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The proof is similar to that of lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In the subsequent analysis, the horizontal distance is more practical than the Euclidean distance in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The horizontal distances zQ,j,k (r) corresponding to lemma 1 and lemma 2 are defined as zQ,j,k (r) = � d2 Q,j,k (r) − h2 k, (14) where Q = {LoS − LoS, LoS − NLoS, NLoS − LoS, NLoS − NLoS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Distance Distribution of Tagged UAV Before deriving the PDF of the distance of associated UAV, obtaining the distance distributions of tagged UAVs is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The distance distributions are given in the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given the distance between the typical user and the origin is zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the CDF of the distance between the tagged LoS UAV in tier k and the typical user is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' FRLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = 1 − exp � − � zu+√ r2−h2 k zu−√ r2−h2 k � ϕLoS−LoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕLoS−LoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (15) where ϕQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = arccos �l2 + z2 u − z2 Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r) 2 l zu � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (16) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) = |l| ΛUAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l) P Q k (du2U (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (17) where Q = {LoS − LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' LoS − NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' NLoS − LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' NLoS − NLoS},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the horizontal distances zQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r) are defined in (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ΛUAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r) and P LoS k (z) are given in (2) and (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The distance between the potential interfering UAV and the typical user du2U (zu, l, θ) in (17) is given by, du2U (zu, l, θ) = � (zu − l cos θ)2 + (l sin θ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given the distance between the typical user and the origin is zu, the CDF of distance 10 between the tagged NLoS UAV in tier k and the typical user is given by FRNLoS,k (r, zu) = 1 − exp � − � zu+√ r2−h2 k zu−√ r2−h2 k � ϕNLoS−NLoS(l,r,zu) −ϕNLoS−NLoS(l,r,zu) vNLoS k (zu, l, θ) dθdl � , (19) where vQ k (zu, l, θ) and ϕQ,j,k (l, r, zu) are given in (17) and (16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The proof is similar to that of Lemma 3, therefore omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given the distance between the typical user and the origin is zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the PDF of distance between the tagged Q UAV in tier k and the typical user is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' fRQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = exp � − � zu+√ r2−h2 k zu−√ r2−h2 k � ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � × � � zu+√ r2−h2 k zu−√ r2−h2 k − 4r 1 (r > hk) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ϕQ−Q) � 4 l2 z2 u − (l2 + z2 u − r2 + h2 k)2dl + � ϕQ−Q � zu+√ r2−h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu � −ϕQ−Q � zu+√ r2−h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu � r vQ k � zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu + � r2 − h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ � � r2 − h2 k dθ + � ϕQ−Q � zu−√ r2−h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu � −ϕQ−Q � zu−√ r2−h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu � r vQ k � zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu − � r2 − h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ � � r2 − h2 k dθ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (20) where 1 (r > hk) is an indicator function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' its value is 1 when r > hk is satisfied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' otherwise 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) and ϕQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) are given in (17) and (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For a LoS tagged UAV, Q in (20) is replaced with LoS, while Q is replaced with NLoS for an NLoS tagged UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Association Probabilities Association probability is used to describe the probability that a tagged UAV will eventually be selected as the associated UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For the tagged LoS UAV in tier k, there will be no LoS UAVs providing stronger power in tier k than the tagged UAV, while the NLoS UAVs in tier k may provide stronger average received power, and in other tiers, both LoS and NLoS UAVs may provide stronger power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As a result, the association probabilities are given in the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 11 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given the distance between the typical user and the origin is zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' for the LoS tagged UAV from tier j at Euclidean distance r from the typical user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the probability that the typical user is associated with this specific UAV is given by P A LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = K � k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j̸=k exp � − � zu+zLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) zu−zLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � ϕLoS−LoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕLoS−LoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � × K � k=1 exp � − � zu+zLoS−NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) zu−zLoS−NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � ϕLoS−NLoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕLoS−NLoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vNLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (21) where vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) and ϕQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) are given in (17) and (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given the distance between the typical user and the origin is zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' for the NLoS tagged UAV from tier j at Euclidean distance r from the typical user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the probability that the typical user is associated with this specific UAV is given by P A NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = K � k=1 exp � − � zu+zNLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) zu−zNLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � ϕNLoS−LoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕNLoS−LoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � × K � k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j̸=k exp � − � zu+zNLoS−NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) zu−zNLoS−NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � ϕNLoS−NLoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕNLoS−NLoS(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vNLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (22) where vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) and ϕQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) are given in (17) and (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The proof is similar to that of Lemma 6, therefore omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Coverage Probability As an indispensable intermediate result to enable computing coverage probability, the Laplace Transform of interference is given in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Given that the distance between the typical user and the origin is zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the Laplace transform of the interference power conditioned on the associated UAV in tier j with Euclidean distance r from the typical user is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' LIQ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = K � k=1 � LIQ1−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) × LIQ1−NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (23) 12 where Q1 is replaced with LoS when the typical user is associated with LoS UAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Q1 is replaced with NLoS when NLoS UAV is associated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and LIQ1−Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Q2 = {LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' NLoS} is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' LIQ1−Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = exp � − � max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu−zQ1−Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r)} 0 � π −π vQ2 k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) wQ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � × exp � − � +∞ zu+zQ1−Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � π −π vQ2 k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) wQ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � × exp � − 2 � zu+zQ1−Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) zu−zQ1−Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � π ϕQ1−Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vQ2 k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) wQ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ)dθdl � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (24) where wQ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r) = 1 − � � mQ2 mQ2 + sηQ2ρkGQ2(d2 u2U (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) + h2 k) −αQ2 2 � � mQ2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (25) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) and ϕQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) are given in (17) and (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' lemma that Q1 repre- sents the type of associated UAV, while Q2 represents the type of interfering UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As the distance distributions of tagged UAVs and the association probabilities have been derived, we are ready to calculate the local coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The definition and derivation of the local coverage probability are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Definition 1 (Local coverage probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The local coverage probability P C (zu, γ) is the probability that the SINR of the typical user at distance zu from the origin is greater than threshold γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The exact coverage probability P C (zu, γ) for the typical user is given by, P C (zu, γ) = K � k=1 � +∞ hk fRLoS,k (r, zu)P A LoS,k (r, zu) mLoS−1 � n=0 �(−s)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ∂n ∂sn LULoS,k (s, r, zu) � s=µLoS,k(r,γ) dr + K � k=1 � +∞ hk fRNLoS,k (r, zu) P A NLoS,k (r, zu) mNLoS−1 � n=0 �(−s)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ∂n ∂sn LUNLoS,k (s, r, zu) � s=µNLoS(r,γ) dr, (26) 13 LUQ,k (s, r, zu) = exp � −σ2s � LIQ,k (s, r, zu) , (27) µQ,k (r) =mQγη−1 Q ρ−1 k rαQ, (28) Q = {LoS, NLoS} in (27) and (28), fRQ,k (r, zu), P A LoS,k (r, zu) and P A NLoS,k (r, zu) are defined in (20), (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Based on the local coverage probability, the definition and derivation of the overall coverage probability are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Definition 2 (Overall coverage probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The overall coverage probability is the average coverage probability of all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' From the definition, the overall coverage probability for the typical user is the normalized expectation of the local coverage probability with regard to zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The overall exact coverage probability with the SINR threshold γ is given by, P C Overall (γ) = � +∞ 0 Λu (zu) P C (zu, γ) zudzu � +∞ 0 Λu (zu) zudzu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (29) As is shown in (26), higher-order derivatives of the Laplace transform are needed while deriving the exact coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Because the computational complexity increases rapidly as the order of the derivative increases, the amount of computation is not acceptable under large shape parameters mLoS and mNLoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, we provide an approximate evaluation of the coverage probability using the upper bound of the CDF of the Gamma distribution [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The approximate coverage probability �P C (zu, γ) for the typical user is given by, �P C (zu, γ) = K � k=1 � +∞ hk fRLoS,k (r, zu)P A LoS,k (r, zu) mLoS � n=1 �mLoS n � (−1)n+1LULoS,k(n ωLoS µLoS,k (r, γ), r, zu)dr + K � k=1 � +∞ hk fRNLoS,k (r, zu) P A NLoS,k (r, zu) mNLoS � n=1 �mNLoS n � (−1)n+1LUNLoS,k(n ωNLoS µNLoS(r, γ),r,zu)dr, (30) where ωQ = (mQ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=') − 1 mQ , Q = {LoS, NLoS}, (31) 14 TABLE I: Table of Parameters hk [m] β P C Overall Density of UAVs One-tier 50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 ×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9026 1 UAV/km2 One-tier 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 ×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9203 1 UAV/km2 One-tier 150 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 ×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9367 1 UAV/km2 Three-tier (50,100,150) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='6) ×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9713 1 UAV/km2 Uniform Distribution 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='3845 1 UAV/km2 fRQ,k (r, zu), P A LoS,k (r, zu), P A NLoS,k (r, zu) and µQ,k are defined in (20), (21), (22) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' See Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The same as the overall exact coverage probability, the overall approximate coverage proba- bility is given in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The overall approximate coverage probability is given by, �P C Overall (γ) = � +∞ 0 Λu (zu) �P C (zu, γ) zudzu � +∞ 0 Λu (zu) zudzu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (32) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we compare the coverage performance of different systems and optimize the overall coverage probability by changing the distribution of UAVs in different tiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Referring to [9], [17], [29], we assume the channel parameters as follows: the LoS and NLoS path- loss exponents are αLoS = 2 and αNLoS = 3, the mean additional gains for LoS and NLoS transmissions are ηLoS = 0dB and ηNLoS = −20dB, m parameters of Nakagami-m fading for LoS and NLoS UAVs are mLoS = 2 and mNLoS = 1, the noise power is σ2 = 10−7W, the parameters for the probability of establishing an LoS link in (3) are a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='88 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The deterministic parameters of users’ distribution in (1) are λu = 10−3m−2 and βu = 5 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As a non-homogeneous PPP, the distribution of users can be realized by thinning property [11] of homogeneous PPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Finally, we assume three tiers of UAVs are deployed in a small town center square with sides of 5km, at 50,100 and 150 meters height, with the corresponding transmission power 2, 7, and 12 dBm, respectively, and the same value of λ1 = λ2 = λ3 = 4 × 10−5m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 2, a curve of local coverage probability for the typical user as a function of the distance between the origin and the typical user is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We compare the coverage performance of population density-inspired UAV systems (one-tier and three-tier) with that of the uniformly 15 distributed UAV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' All of the above systems have the same UAV deployment density on average as 25 UAVs/km2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=', λh = 10−6m−2 and βh = 0 for uniform distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The distribution parameters β of the other three systems are shown in the table, λ = 4 × 10−5m−2 as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 0 200 400 600 800 1000 1200 1400 1600 Distance between the Typical User and the Origin [m] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9 1 Coverage Probability for Typical User One-tier, 50m One-tier, 100m One-tier, 150m Three-tier Uniform Distribution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 2: Local Coverage Probability for the Typical User.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As shown in Table I, different β are chosen to keep the density of the UAVs as 1 UAV/km2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Assume the threshold of coverage probability is γ = −15dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' It can be seen that the results of the Monte-Carlo simulation (lines) coincide well with the results of the theoretical analysis (points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' With the same number of UAVs, no matter how far away the typical user is from the origin, the performance of the three-tier network is always better than that of the one- tier networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As shown in Table I, the three-tier network has significant advantages in terms of overall coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' At the edge, the advantage of the three-tier network is further expanded, indicating that the lower limit of network coverage or SINR can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In addition, the coverage performance of all three systems is declining from center to edge due to the reduced density of UAV deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Compared with the proposed distribution, the uniform distribution system only has a slight advantage in the edge area, but the overall performance is far inferior 16 to the other four systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The clustering effect brings the non-uniform distribution advantages over the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 2, when the user is close to the center, the coverage probability of the UAV network under resident population density-inspired distribution is significantly greater than that under the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Most users are clustered in the area close to the center, so the proposed distribution has significant advantages in the overall coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We study the following optimization problems and record the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We want to maximize the overall coverage probability by changing the distribution of UAVs, under the premise that the total number of UAVs is limited (the first constraint) and the coverage of users in any location is guaranteed (the second constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, the mathematical representation of the optimization problem is as follows, arg max β1,β2 P C Overall (γ1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' K � k=1 � +∞ 0 ΛUAV,k (z) dz ≤ Nmax, P [SINR ≥ γ2] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='95, ∀zu ≥ 0, (33) where the threshold of overall coverage probability is γ1 = −8dB and the threshold of local coverage for the typical user is γ2 = −20dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The total number of UAVs is limited to Nmax = 1000, that is, the maximum density of UAV is 40 UAVs/km2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We deploy UAVs at the first the second tiers (h = 50, 100 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' A paradoxical but interesting conclusion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 3 is, the UAV distribution preferred by the system is a non-uniform one influenced by the resident density distribution, but the optimal distribution is not closely related to the residents numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The optimal overall coverage probability P C∗ Overall and the corresponding β∗ 1 and β∗ 2 value are marked in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' It can be seen that there is a considerable difference among β∗ 1, β∗ 2 and βu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For a large β, the first constraint cannot be satisfied due to the high density of UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Under this condition, the coverage probability is set to 0, so the dark blue area at the bottom left appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When β increases to 10−3, there is a large amount of interference near the centre because the system still tends to be uniformly distributed and the density is relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' With the increase of β, the overall coverage probability is improved rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Near the optimal area, the coverage performance of the system is no longer very sensitive to both β1 and β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' This is interesting because in such a 17 𝜷𝟏 ∗ = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 𝟔×𝟏𝟎#𝟐 𝜷𝟐 ∗ = 𝟏×𝟏𝟎#𝟑 𝑷𝑶𝒗𝒆𝒓𝒂𝒍𝒍 𝑪∗ = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 𝟖𝟓𝟎𝟑 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 3: Overall coverage probability under different UAV distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' tolerant system, we do not have to select an accurate set of optimal parameters to determine the distribution of UAVs, but only to estimate a range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For a large β, the small number of drones are almost all concentrated in the central area, making it difficult for users in the edge area to maintain good communication conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, the second constraint cannot be satisfied, and the dark blue area appears at the top right of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Finally, with the same number of UAVs as the optimal distribution, the overall coverage probability of the uniform distribution in the same condition (h = 50, 100 m, λ1 = λ2 = 4 × 10−6 m−2) is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2883, which is much lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Although the optimization problem (33) has been carefully studied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 3, it is still necessary to study the behavior of the system hidden in the dark blue area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 4 shows the influence of the UAV distribution in a single tier on the overall coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We observe the special case of β2 = 10−2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 3 and broaden the range of β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' First, it is easy to find that the overall coverage probability increases at the beginning and then decreases with the increase of the value of β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' This can simply be explained by the fact that too 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='610~20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='710-3 10~20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='418 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content="9 1 Overall Coverage Probability Doesn't satisfy the second constraint Doesn't satisfy the first constraint Feasible Region Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 4: The influence of the UAV distribution in single tier on the overall coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' many UAVs will cause too much interference in the central area, while having too few UAVs will make it difficult for the user to find a close UAV to establish an LoS link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When β1 ≤ 10−5, the number of UAVs in the first tier is much larger than that in the second tier, so the overall coverage probability is stable and tends to be similar to that of uniform distribution in the first tier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When β1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='6 × 10−2, the number of UAVs in the first tier is rapidly decreasing, which means there are no available LoS UAVs nearby for some users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' It is not hard to predict that the final result will converge to the scenario where only the second tier of UAVs are providing the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' FURTHER REMARKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Analytic Framework Extension This subsection presents how to extend the existing analysis framework to other scenarios and network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Enhancing the coverage is one of the application scenarios for UAV networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' UAV networks can also relieve the pressure of insufficient channel capacity in town centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Based on Shannon’s theorem and the definition of coverage probability in (9), the channel capacity can be expressed as [30], P [B log2 (1 + SINR) > R] = P � SINR > 2 R B − 1 � = P [SINR > �γ] , (34) DfirstRD19 where �γ = 2 R B − 1 is the rate threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' By replacing �γ into SINR threshold γ, the local probability and overall probability that the channel capacity is greater than the rate threshold can be obtained by �P C (zu, �γ) given by (30) and �P C Overall (�γ) given by (32), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Next, we study green communications in a small hot spot area centered on a base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Energy efficiency, the number of bits that can be transmitted per unit of energy consumed, is used as a performance metric for green communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For convenience, we calculate the energy efficiency as the ratio of the number of bits transmitted per unit time (channel capacity) to the energy consumed per unit time (transmission power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The introduction of the UAV network allows the central base station to reduce its coverage area, thereby reducing transmission power and enhancing energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The following remark illustrates how the coverage probability analytic framework can be applied to the above scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' From the definition of energy efficiency, it can be calculated by the ratio of channel capacity to transmission power, P � B ρk log2 (1 + SINR) > E � = P � SINR > 2 Eρk B − 1 � = P [SINR > �γ] , (35) By replacing �γ = 2 Eρk B − 1 into SINR threshold γ, the local probability and overall probability that the energy efficiency is greater than the rate threshold can be obtained by �P C (zu, �γ) given by (30) and �P C Overall (�γ) given by (32), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Furthermore, there is no need for dense deployment of UAVs near the base station in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Fortunately, the network model can be easily extended to the above scenario by adjusting the density distribution of UAVs in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Under the premise that the density (whether of the user or the UAV) is only related to the distance to the town center, the analytical framework of this paper is applicable to any distribution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Distribution Parameter Design According to the above theorems, it can be seen that the relationship between the coverage probability and the spatial distribution of UAVs is not straightforward, and obtaining the optimal parameters by optimization tools is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, the following qualitative criteria for parameters about UAVs’ vertical and horizontal distributions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Notice that all of the remarks have been verified by simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 20 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Remarks on altitudes h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' , hK are given as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In most cases, UAVs have an optimal altitude, and it is better to deploy the UAVs near the optimal altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' While facing a low communication quality, the UAVs are suggested to be distributed at a low altitude so that UAVs are closer to users and users in the LoS region can be covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In a good communication environment, by increasing the deployment altitude, more users can establish LoS links with UAVs, therefore, increasing the coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Remarks on the number of tiers K are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' With the increase of tiers, more parameters can be optimized so that the coverage perfor- mance can be improved to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' A network with fewer tiers can be considered a special case of a network with more tiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' However, the improvement in coverage perfor- mance is limited when more than three tiers are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Consider a more general case where the receivers (users) can be divided into M classes according to different gain and demodulation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Multi-tier distribution has sig- nificant advantages over single-tier distribution, and the number of deployment tiers K is recommended to be larger than M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Assuming that the optimal UAV deployment alti- tude for the receiver of the m-th class is h∗ m, the height of UAVs is suggested to satisfy min {h∗ 1, h∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' , h∗ M} < hk < max {h∗ 1, h∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' , h∗ M} , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Remarks on homogeneity β are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The value of β is related to the strength of interference power relative to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When the interference power is significantly stronger than the environmental noise, it is suggested to choose a smaller β to make the distribution of UAVs more homogeneous and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Considering that the value of βk will affect the average number of UAVs in hot areas when βk is changed, λk is adjusted to keep the average number of UAVs unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We use the exhaustive search to solve the optimization problem about β given in (33), which results in the calculation complexity increases exponentially with K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' An improved alternate maximization method can be a substitution for the exhaustive search as described in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The complexity of this method is O (NK2), where N is the preset maximum number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The set of suboptimal parameters is obtained by optimizing from β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' to βK in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When optimizing βk, the βk is repeatedly reduced by the predefined step size for at most N times, and one of the β in the set {β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' , βk−1} is increased, so that the 21 TABLE II: Optimization of K and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 1 UAV/km2 One-tier Two-tier Three-tier Five-tier h1 = 50m N/A N/A β1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='5 × 10−3 β1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='4 × 10−3 h2 = 75m N/A N/A N/A β2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='5 × 10−3 h3 = 100m N/A β3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 × 10−3 β3 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8 × 10−3 β3 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8 × 10−3 h4 = 125m N/A N/A N/A β4 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 × 10−3 h5 = 150m β5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 × 10−3 β5 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='4 × 10−3 β5 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='6 × 10−3 β5 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='8 × 10−3 P C Overall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='9786 coverage probability is maximized when the UAV density is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The optimization of βk ends when the coverage probability no longer increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The example in Table II provides further explanation for the above remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In Table II, we compare the coverage performance under different numbers of tiers K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The total density and the density of UAVs in each tier are fixed as 1 UAV/km2 and λk = 4×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The set of homogeneity {β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' , βK} is obtained by alternate maximization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Overall, increasing the number of tiers allows more parameters to be optimized, thus achieving better coverage performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The UAV deployment in the one-iter network (β5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2×10−3) can be regarded as a special case of that of a two-tier network ({β3, β5} = {+∞, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='2 × 10−3}), but it is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Finally, the gain in coverage probability from deploying more than three tiers of UAV networks is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this paper, We studied the coverage performance of multi-tier UAV networks in a centralized urban model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' We first derived the distance distribution of tagged UAVs and association probability for the selected typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Based on this, the analytical expression of downlink coverage probability is given and proved to be consistent with the Monte-Carlo simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As a result, the coverage probability for the typical user and intermediate products are all related to the distance zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Both the local and total coverage performance are significantly improved by increasing the number of UAV network tiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The urban population density-inspired model has a huge advantage over the uniform distribution performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' However, too much concentration of UAVs in the central area will bring more noise to the town center and fail to maintain communication for users at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, how to design the distribution of each tier of UAVs is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 22 One future research direction is introducing interference and noise mitigation technologies into the framework based on the proposed resident population density-inspired model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In urban areas, the relatively dense deployment of UAVs may cause strong interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Strong environmental noise in town centers is also one factor limiting the performance of wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Under the SG framework, orthogonal channel [32] and directional antenna gain [33] can be introduced into the system model respectively to reduce interference and noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In addition, we model the users as a PPP, which means that the user’s movement is undirected and random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Considering that there is a directional flow of people in the town [34], analyzing the coverage probability of the urban system based on SG will be challenging and application-oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' APPENDIX A PROOF OF LEMMA 3 When the distance between the typical user and the origin is fixed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' given that the distance between the tagged LoS UAV in tier k and user RLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k is a random valuable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the Cumulative Distribution Function (CDF) of RLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k is given by FRLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = P [RLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k < r] = 1 − P [RLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k > r] = 1 − P [N (Ak (r)) = 0] (a) = 1 − exp � − � Ak(r) ΛUAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l) ldldθ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (36) where ΛUAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l) is defined in (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' N (Ak (r)) in (36) counts the number of the UAVs in region Ak (r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' which is a circle at the height of hk centered directly above the typical user with radius � r2 − h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and (a) is given by the property of the general PPP [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' P [N (Ak (r)) = n] = exp � − � Ak(r) ΛUAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l) l dldθ � exp � − � Ak(r) ΛUAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l) l dldθ �n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (37) where zu is the horizontal distance from the typical user to the origin, λu determines the total density of the plane, βu is a measure of homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' To integrate formulation in (36) over the region of Ak, the area is divided into infinite concentric circular arcs centered at the point which is directly above the origin at the height of hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' When the radius l of the concentric circular arc is fixed, the density function ΛUAV,k is a constant, the coordinates of the points on the arc can be uniquely represented by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The bold part of the bottom half of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 5 is one of the concentric arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The upper bound of θ can be obtained from the geometric relations in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 5, denoted as ϕLoS−LoS, defined in (16), 23 𝑥 𝑦 𝑧 𝝋 𝜽 𝒍 u2U 𝒅 𝒛𝑸𝟏%𝑸𝟐,𝒋,𝒌 𝒛𝒖 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 5: Vertical Viewed System Schematic Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and the lower bound of θ is −ϕLoS−LoS because of the symmetry of the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Furthermore, the horizontal distance du2U (zu, l, θ) between the typical user and the point on the arc is defined in (18), which can also be obtained from simple geometrical relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Hence, we have the following equation � Ak(r) ΛUAV,k (l) ldldθ = � zu+√ r2−h2 k zu−√ r2−h2 k � ϕLoS−LoS(l,r,zu) −ϕLoS−LoS(l,r,zu) vLoS k (zu, l, θ) dθdl, (38) where ΛUAV,k (l) and vQ k (zu, l, θ) are defined in 2) and (17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' It is important to note that l may be negative when the value of zLoS−LoS,j,k is greater than the horizontal distance zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, |l| is used in the outer integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' APPENDIX B PROOF OF LEMMA 5 As in Lemma 5, Q is used to represent the type of tagged UAVs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=', Q is replaced with LoS when an LoS UAV is tagged or NLoS otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' By taking the derivative of FRQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the distribution of the nearest UAVs in tier k with a distance r from the user is obtained,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' which is 24 denoted as fRQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' fRQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = ∂ ∂rFRQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = ∂ ∂r � 1 − exp � − � zu+√ r2−h2 k zu−√ r2−h2 k � ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl �� (a) = exp � − � zu+√ r2−h2 k zu−√ r2−h2 k � ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � × � � zu+√ r2−h2 k zu−√ r2−h2 k ∂ ∂rfin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) � �� � The derivative of the integrand dl + ∂ � zu + � r2 − h2 k � ∂r fin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k � zu + � r2 − h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ � � �� � The derivative of the integral upper bound − ∂ � zu − � r2 − h2 k � ∂r fin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k � zu − � r2 − h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ � � �� � The derivative of the integral upper bound � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (39) where vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) and ϕQ−Q (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) are defined in (17) and (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and (a) follows Leibnitz’s rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' fin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) is the integrand of the outer integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' given by fin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) = � ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (40) For the derivative of the integral upper bound in (39), ∂ � zu + � r2 − h2 k � ∂r fin,k � zu + � r2 − h2 k, zu, r, θ � = r � r2 − h2 k � ϕQ−Q � zu+√ r2−h2 k,r,zu � −ϕQ−Q � zu+√ r2−h2 k,r,zu � vQ k � zu, zu + � r2 − h2 k, θ � dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (41) The derivative of the integral lower bound is similar to that of (41), therefore omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 25 For the derivative of the intergrad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ∂ ∂rfin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) = ∂ ∂r � ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) −ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθ (a) = vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ϕQ−Q) ∂ϕQ−Q (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) ∂r − vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' −ϕQ−Q) ∂ (−ϕQ−Q (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)) ∂r � (b) = 2vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ϕQ−Q) ∂ϕQ−Q (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) ∂r (c) = 1 (r > hk) 4 r vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ϕQ−Q) � 4 l2 z2 u − (l2 + z2 u − r2 + h2 k)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (42) where (a) follows Leibniz’s rule for internal integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and the expression in (b) is simplified by the fact du2U (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' −ϕQ−Q) = du2U (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ϕQ−Q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' which can be easily obtained by (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 1 (r > hk) is the indicator function defined in (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and (c) is obtained by substitute ∂ϕQ−Q(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) ∂r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' which is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ∂ϕQ−Q (l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) ∂r = ∂ ∂u arccos �l2 + z2 u − u2 2 l zu � ∂ ∂v � v2 − h2 k ∂v ∂r = 2u � 4 l2 z2 u − (l2 + z2 u − u2)2 2v 2 � v2 − h2 k 1 ��ρk ρk � 1 αLoS r > hk � = 1 (r > hk) 2r � 4 l2 z2 u − (l2 + z2 u − r2 + h2 k)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (43) where u = zLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r) |j=k = 1 (r > hk) � r2 − h2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and v = dLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r) |j=k = max {hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Substitute (41) and (42) into (39), the final result is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=" 𝑧& 𝑧'$('𝟐,*,+ 𝑧'$('𝟐,*,+ 𝑧& Origin The typical user User's non- interference circle Interfering UAVs' circle Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 6: Two relationships between user’s non-interference circle and interfering UAVs’ circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 126 APPENDIX C PROOF OF LEMMA 6 When the distance between the typical user and the origin is fixed, the probability that the typical user is associated with the tagged LoS UAV in tier j is equal to the probability that the average received power of other 2K − 1 tagged UAVs is lower than it, where K is the number of tiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Using the solution of lemma 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' we have P A LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) = K � k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j̸=k P [RLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k > dLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r)] × K � k=1 P [RNLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k > dLoS−NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r)] = K � k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j̸=k P [N (ALoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r)) = 0] × K � k=1 P [N (ANLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r)) = 0] (a) = K � k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j̸=k exp � − � ALoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ)dl dθ � × K � k=1 exp � − � ANLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) vNLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ)dl dθ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (44) where RQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k is the distance between the tagged UAV in tier k and typical user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' vQ k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) is defined in (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' N (AQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r)) counts the number of the UAVs in region AQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' which is a circle at the height of hk centered directly above the typical user with radius zLoS−Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Q = {LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' NLoS},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and (b) is given by the property of the general PPP in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The following two equations can be obtained in a similar way to (38), � AQ,k(r) vQ k (zu, l, θ)dl dθ = � zu+zQ−Q,j,k zu−zQ−Q,j,k � ϕQ−Q(l,r,zu) −ϕQ−Q(l,r,zu) vQ k (zu, l, θ)dθdl, (45) where Q = {LoS, NLoS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The final result is derived by substituting (45) into (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 27 APPENDIX D PROOF OF LEMMA 8 For LoS associated UAV in tier j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the Laplace transform of the interference power can be expressed as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' LILoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) (a) = EILoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j � e−sILoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j� = EΦ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='G � exp � − s K � k=1 � � x∈ΦLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k\\xo ηLoSρkGLoSr−αLoS + � x∈ΦNLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k ηNLoSρkGNLoSr−αNLoS ��� (b) = K � k=1 EΦLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k � � x∈ΦLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k\\xo EGLoS � exp � −s ηLoSρkGLoSr−αLoS�� � × K � k=1 EΦNLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k � � x∈ΦNLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k EGNLoS � exp � −s ηNLoSρkGNLoSr−αNLoS�� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (46) where (a) follows the definition of Laplace transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ΦLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k\\Φo are all of the LoS UAVs in tier k except for the associated one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and (b) follows the independence of the point process and the small scale fading,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (c) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For Laplace transform of the interference caused by LoS UAVs in tier k, In equation (47) shown at the top of the next page, (a) follows the PGFL of inhomogeneous PPP [35], AC LoS,k(r) is the complement of ALoS,k(r) in the two dimensional plane at the height of hk, the definition of ALoS,k(r) is described in Appendix A, vQ k (zu, l, θ) and ϕQ−Q (l, r, zu) are defined in (17) and (16) respectively, wLoS,k (s, r) defined in (25) is used to simplify the expression in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 6, there should be non-interfering UAVs inside the green circle centered at the typical user, the green circle is called the user’s non-interference circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The difference between the left and right images is whether the origin is included by user’s non-interference circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' For a fixed radius l, the circles centered at the origin is used to cover the possible locations of interfering UAVs with horizontal distance l to the origin, called the interfering UAV’s circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' These two circles may be separated or intersected, and sometimes one circle may contain another, shown in step (b) of (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The Laplace transform of the interference in other conditions is similar to the process in (47), therefore omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' 28 EΦLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k � � � x∈ΦLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k\\xo EGLoS � exp � −s ηLoSρkGLoSr−αLoS�� � � (a) = exp(− � AC LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) � 1 − EGLoS � exp � −s ηLoSρkGLoS � d2 u2U (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) + h2 k �−αLoS/2��� dθdl) = exp � − � AC LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) � 1 − � mLoS mLoS + s ηLoSρkGLoS(d2 u2U (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) + h2 k)−αLoS/2 �mLoS� l dθdl � (b) = exp � − � max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu−zLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r)} 0 � π −π vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) wLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � �� � User′s non−interference circle separates from interfering UAVs′ circle � × exp � − � +∞ zu+zLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � π −π vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) wLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � �� � User′s non−interference circle contained by interfering UAVs′ circle � × exp � − 2 � zu+zLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) zu−zLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(r) � π ϕLoS−LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k(l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='zu) vLoS k (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) wLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' θ) dθdl � �� � User′s non−interference circle and interfering UAVs′ circle intersect � (47) APPENDIX E PROOF OF THEOREM 1 By the definition of coverage probability in (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' SINR becomes a deterministic expression only when: (i) the tier where the associated UAV is located;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (ii) LoS or NLoS link constructed by the typical user and associated UAV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (iii) the distance between the typical user and the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (iv) the Euclidean distance between the typical user and the associated UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Therefore, the coverage probability of the typical user is given by (48) at the top of next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' where P A LoS,k (r, zu), P A NLoS,k (r, zu) and fRQ,k (r, zu) are given in (21), (22) and (20), respectively, (a) is obtained by substituting UQ,k (r, zu) = IQ,k (r, zu) + σ2 and µQ,k (r, γ) are define in (28) into the former result, (b) is obtained from the expectation of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' In order to get the final analytical result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' the 29 P C (zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) = K � k=1 Er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='I � P A LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) P �ηLoSρkGLoSr−αLoS ILoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) + σ2 > γ �� + K � k=1 Er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='I � P A NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) P �ηNLoSρkGNLoSr−αNLoS INLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) + σ2 > γ �� (a) = K � k=1 Er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='U � P A LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) P [GLoS > µLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)] � + K � k=1 Er,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='U � P A NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) P [GNLoS > µNLoS (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) UNLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)] � (b) = K � k=1 � +∞ hk EU [P [GLoS > µLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)]] P A LoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) fRLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) dr + K � k=1 � +∞ hk EU [P [GNLoS > µNLoS (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) UNLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)]] P A NLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) fRNLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu) dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (48) next steps are taken,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' EU [P [GLoS > µLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)]] (a) = EU �Γu (mLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' mLoSµLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)) Γ (mLoS) � (b) = EU � exp (−µLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) U (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)) mLoS−1 � n=0 (µLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' � = mLoS−1 � n=0 (µLoS,k (r, γ))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' EU � exp (−µLoS,k (r, γ) ULoS,k (r, zu)) (ULoS,k (r, zu))n� (c) = mLoS−1 � n=0 �(−s)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' ∂n ∂snLULoS,k (s, r, zu) � s=µLoS,k(r,γ) , (49) where (a) follows the complementary cumulative distribution function (CCDF) of the Gamma distribution F G (g) = Γu(m,mg) Γ(m) , where Γu (m, mg) = � +∞ mg tm−1e−tdt is the upper incomplete Gamma function, and (b) follows the definition Γu(m,mg) Γ(m) = exp (−g) m−1 � n=0 gn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' [36], by the linearity of the expectation operator and EU [exp (−sULoS,k (r, zu)) ULoS,k(r, zu)n] = (−1)n ∂n ∂snLULoS,k (s, r, zu) , (50) 30 (c) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' The steps of NLoS UAVs are similar to that of LoS UAVs, therefore omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' APPENDIX F PROOF OF THEOREM 2 Because the first several steps of the proof of approximate coverage probability are similar to that of exact coverage probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' we start from formulation (49) step (a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' EU �Γu (mLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' mLoSµLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r |ru)) Γ (mLoS) � (a) = 1 − EU �Γl (mLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' mLoSµLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r |ru)) Γ (mLoS) � (b) ≈ 1 − EU [(1 − exp (−βLoSµLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)))mLoS] (c) = EU � mLoS � n=1 �mLoS n � (−1)n+1 exp (−nωLoSµLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ) ULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' zu)) � = mLoS � n=1 �mLoS n � (−1)n+1LULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (nωLoSµLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (51) where LULoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' k) is given in (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' and s = nωLoSµLoS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='k (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Γl (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' mg) = � mg 0 tm−1e−tdt in step (a) is the lower incomplete Gamma function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' which satisfies Γu(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='mg) Γ(m) = 1 − Γl(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content='mg) Γ(m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' (b) follows from the tight approximation to coverage probability, where ωLoS = (mLoS!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=') −1 mLoS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' It has been proved in [37] that the tighter upper bound provides an accurate approximation of the CDF of the Gamma distribution, which is bounded by � 1 − e−ω1mg�m < Γl (m, mg) Γ (m) < � 1 − e−ω2mg�m, (52) where m ̸= 1, and ω1 = � � � 1, if m > 1 (m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=') −1 m , if m < 1 ω2 = � � � (m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=') −1 m , if m > 1 1, if m < 1 , (53) and step (c) is given by the binomial theorem, and it is necessary to assume that mLoS is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQf9vpC/content/2301.00879v1.pdf'} +page_content=' Sekander, H.' metadata={'source': 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specific type of inverse problem. One of such +problems is applied in Electrical Impedance Tomography +(EIT). +In general, EIT can be posed as a minimization prob- +lem and solved by iterative methods, which require knowl- +edge of derivatives of the objective function. In practice, +this can be challenging because analytical closed-form so- +lutions for them are hard to derive and implement effi- +ciently. +In this paper, we study the effectiveness of automatic +differentiation (AD) to solve EIT in a minimization frame- +work. We devise a case study where we compare solutions +of the inverse problem obtained with AD methods and +with the manually-derived formulation of the derivative +against the true solution. +Furthermore, we study the viability of AD for large scale +inverse problems by checking the memory and load re- +quirements of AD as the resolution of the model increases. +With powerful infrastructure, AD can pave the way for +faster and simpler inverse solvers and provide better re- +sults than classical methods. +1 +Introduction +Electrical Impedance Tomography (EIT) is a non-invasive +imaging method that produces images by determining the +electrical conductivity inside a subject using only electrical +measurements obtained at its surface. More specifically, +sinusoidal currents are applied to the subject through elec- +trodes placed in certain locations at the surface of the ob- +ject. The resulting voltages are then measured, making it +possible to infer internal properties of the objects. EIT is +a low-cost method and harmless for human being, since +it only applies low amplitude currents. +Additionally, it +allows for real-time monitoring of various subjects even +in the most difficult conditions. There are applications of +this technology for medical purposes, in scenarios such as +ventilation monitoring, detecting brain hemorrhages and +breast cancer. EIT is also used in geophysical imaging, +flow analysis and other industrial purposes. For further +insight into the applications see [14] and [1]. +Figure 1: Example of a target conductivity over the do- +main Ω that represents a simple model of breast cancer +where tumors have higher conductivity than the back- +ground. The domain Ω is represented by the black cir- +cumference which has a conductivity of σout. In a blue +circle it is represented a region with different conductivity +σin from the background one σout. +A particularly relevant application of EIT is in the +early determination of breast cancer, specifically for young +women where the risk of the ionizing X-rays of mammo- +graphies outweigh the benefits of regular check-ups. Fig. +1 describes one simplified EIT scenario where the blue +region represents cancer inside the breast, denoted as a +domain Ω. The assumption is that healthy and cancerous +tissue have different conductivity values σ1, σ2, respec- +tively. The goal is to locate a potential region affected by +cancer from measurements on the breast surface, which is +the boundary of the domain Ω and denoted as ∂Ω. +1 +arXiv:2301.11410v1 [math.NA] 26 Jan 2023 + +TargetconductivityoverdomainΩ +Gout +QinThe measurements are obtained by injecting into the +domain Ω a fixed set of different electrical current pat- +terns Ij. Each Ij is defined by injecting electrical current +through all electrodes in a particular manner, i.e., for L +electrodes we have Ij = (Ij,1, ..., Ij,L). Simultaneously, we +measure the resulting voltages Vj for each current pat- +tern, obtaining a voltage measurement at each electrode, +denoted as Vj = (Vj,1, ..., Vj,L). +This leads to a set of +true measurements denoted by mj = (Ij, Vj). Then, the +corresponding inverse problem is to determine the electri- +cal conductivity over Ω that leads to these measurements. +In the particular case of Fig. 1 we want to determine the +conductivity outside and inside the anomaly, σout and σin, +respectively, and the location of the anomaly (in blue). +This is a hard problem because in general there is no +analytical expression that maps a set of electrical mea- +surements back to the respective conductivity profile that +generates them. +To solve this inverse problem we first need to under- +stand how to solve the direct problem, that is, computing +electrical measurements Vj for a given set of currents Ij +and conductivity σ. The direct problem has an easier so- +lution, since the propagation of electrical current through +the domain obeys the well-known Maxwell equations. +Many methods for solving the direct problem are de- +scribed in the literature, e.g., Finite Element Method +(FEM) [12], Boundary Element Method (BEM) [4], and, +more recently Deep Learning methods (DL) [10]. +Independently of the numerical method used to solve +the direct problem, such a procedure is commonly desig- +nated as simulation. Hence, for a given conductivity pro- +file we can obtain through a simulation method the electri- +cal measurements denoted as mSim +j += (Ij, V Sim +j +), for each +different current pattern with j = 1, ..., N. We can thus +define an operator that maps conductivity into voltage +measurements, here termed by direct operatorand given +as: +Sim : σ �→ V Sim = (V Sim +1,1 , .., V Sim +j,l , ..., V Sim +N,L) ∈ RL·N (1) +where V Sim +j,l +represent voltages measured at the l-th elec- +trode for the j-th current pattern. +Our goal is to find a conductivity profile σ that matches +measurements m = (m1, ..., mN). Thus, we can formulate +EIT as the following minimization problem by making use +of the direct operator Sim: +min +σ +1 +2 +��Sim(σ) − mtrue��2 +2 . +(2) +We use the L2-norm here for simplicity, but, in general, +we could use any other norm as long as it is differentiable. +Most classical methods for solving this minimization +problem are based on iteratively improving the solution. +The update requires computing the derivative of both the +loss function in (2) and the Sim operator. +To solve the inverse problem under an optimization +framework we opted for the Levenberg-Marquardt algo- +rithm [6,7]. It is a simple quasi-Newton method that only +requires the Jacobian computation of the Sim operator. +Further details about the method are given in Appendix +B. +In essence, the main challenges to solve the minimiza- +tion problem (2) with iterative classic methods are: +• to ensure that the simulator is once-differentiable +with respect to a conductivity parameterization; +• devise a method to compute the respective derivatives +of the simulator. +Our study explores a simulation operator obtained +through FEM, which is already well established for EIT, +see [8]. +Figure 2: Circular anomaly defined over a triangular FEM +mesh in 2D. Electrodes are attached to the boundary, +black lines. +When the Sim operator is given by FEM we can deduce +an analytical closed-form of the derivative with respect to +the conductivity variation. It is simply obtained with re- +spect to a conductivity discretization over the FEM mesh, +see Fig. 2. As such, it requires derivative computations +with respect to conductivity values over all elements of the +mesh. If the conductivity is defined through a different pa- +rameterization we can obtain the respective derivatives by +the chain rule of differentiation. For such endeavor, the +analytical formulation needs to be adapted and derived +for each particular parameterization of the conductivity. +As a result this formula is hard to derive and implement, +see [5] and [13]. +Automatic differentiation (AD) is a method that au- +tomatically evaluates exact derivatives for complex pro- +grams. +It exploits the simple mathematical operations +the programs are built on, to automatically compute the +derivative through the chain rule. While the initial con- +cept was developed in the sixties [15], only recently with +advancements in hardware and efficient implementations, +like JAX [2], it has gained traction for application in gen- +eral problems. +2 + +Piece-wise conductivity +withcircularanomaly +1.4 +1.0 +1.3 +0.5 +1.2 +1.1 +> 0.0 +S/m +1.0 +-0.5 +0.9 +0.8 +-1.0 +0.7 +1.0 +0.5 +0.0 +0.5 +1.0 +x +ElectrodesIn this paper, we explore automatic differentiation as +an alternative to manual methods for computing the Ja- +cobian of differentiable simulators. In particular, the goal +is to validate its effectiveness in solving the EIT inverse +problem. By effectiveness we mean that it is as success- +ful in solving the inverse problem as previous methods, +namely, through analytical formulation. By doing so we +show its versatility compared with analytic formulation +and moreover verify its viability for high resolution im- +ages where . +The validation is done by comparing the absolute error +between solutions obtained by solving the minimization +problem with both methods to compute the derivatives +and the absolute error compared with the true solution. In +particular, we evaluate the maximum difference between +both Jacobian computations to check if they are evaluating +to the same result. Then as a second set of checks, we +explore the memory consumption of AD and show that +it is still in reasonable terms as the problem scales with +higher resolution. +Our end goal is to show feasibility and practicality of AD +as a tool for lowering the entry barrier for other inverse +problems in Partial Differential Equations, where AD can +also be applied. +In the following, we first introduce the EIT case study +we are using for comparison. +In Section 3, we explain +how the required derivatives are computed with both the +analytical and AD method. In Sec. 4, we introduce our +experimental setup. Results comparing the effectiveness +of both methods and viability of AD are given in Sec. 5, +and conclusions are drawn out in Sec. 6. +2 +Establishing a case study +In this section we establish a case study in order to make +a clear comparison between both methods for computing +derivatives. +2.1 +EIT scenario +To demonstrate our claims we focus on a two-dimensional +setup. We remark that this is not physically accurate since +electrical current propagates in three dimensions However, +it simplifies the construction of our case study. +EIT is an ill-posed inverse problem [8] and thus we need +to take into account the possible instability of the prob- +lem, i.e., small variations in the measurements may imply +large variation on parameters solution. In practice, this +makes it hard to solve the inverse problem since true mea- +surements, captured with real-world measuring devices, +always contain noise. Therefore, real solutions for noisy +input data can be drastically distinct from the true solu- +tion. +Due to this, it becomes hard to accurately determine +a very large number of parameters, e.g., the value of the +conductivity at all mesh elements (see Fig. +2), from a +small number of measurements. An example would be a +conductivity defined over a fine mesh which has a value at +each mesh element, see Fig. 2. +To mitigate this problem we want to make as many +measurements as possible. However, the possible number +of distinct measurements is constrained by the quantity +of electrodes. This occurs since for L electrodes there are +only L−1 linearly independent current patterns for which +the voltage measurements yield independent information +of the conductivity, see [8]. +The best way to mitigate this issue is to work on simpler +cases. By doing so we can reduce the parameter space and +have less variability on the solutions, like in Fig. 1. Even +though instability issues do not become completely fixed, +the space of measurements has a lower, more tractable, +dimensionality. +For the sake of comparison we wish to make, it is enough +to focus on conductivity profiles with a circular region of +distinct conductivity value from the background, see Fig. +1 and 2. These anomalies are parameterized by their cen- +ter (cx, cy) inside the domain Ω, radius r and conductivity +value inside and outside σin, σout, respectively. +We work with this simplification because it is easier to +obtain a solution to the inverse problem due to the pa- +rameterization of such region being given by only a few +parameters. Further, we remark that we need to make sure +that the parameterization is differentiable. Our choice of +circular regions is based on this, since it is easy to define +a smooth parameterization. For regions with corners two +smoothing procedures would be required, one to smoothen +the corners and another to smooth the parameterization. +By the reasons above, in our experiments we assume the +existence of a single circular anomaly with conductivity +value different from the background, like in Fig. +1and +denoted the parameterization variables as +σ = (r, cx, cy, σin, σout). +(3) +We introduce now the EIT model, the conductivity pa- +rameterization definition and the measurement setup we +use to proceed with out comparison. +2.2 +Voltage measuring setup +We introduce here the measuring setup that is applied for +the direct problem. +In this simple 2D setup, we define the Sim operator +in (1) according to the case study and the measurement +setup. +Recall that with L electrodes at the surface ∂Ω, we can +at most apply L− 1 linearly independent current patterns +Ij ∈ RL with j = 1, ..., L − 1. The Sim operator is ob- +tained by solving the direct problem for each Ij and deter- +mine the respective voltages Vj ∈ RL over the electrodes. +The more measurements we can perform the better +we are able to potentially reconstruct the conductivity. +Therefore, we need to choose L − 1 linearly independent +3 + +current patterns. This choice is non-trivial. One possibil- +ity presented in the literature [8] is obtained by injecting +currents in a wave pattern through the electrodes accord- +ing to +Ij,l = +� +A cos(jθl), +j = 1, ..., L +2 , +A sin +� +(j − L +2 )θl +� +, +j = L +2 + 1, ..., L − 1 +(4) +with θl = +2π +L l and A the constant current amplitude. +These patterns have been shown to obtain th best result on +the detection of conductivities profiles with small anoma- +lies in the regions furthest from the boundary, [8]. +The experiments are performed in the following setting: +• Ω is a circular domain with radius rΩ10cm; +• Current amplitude of A = 3mA, which is a reason- +able value for human subjects, and the voltages are +measured in (mV); +• Attach L = 16 electrodes equally spaced at the +boundary with each having fixed length π/64. +We refer to Figure 2 for a visual representation of the +setting. +Under the above setup, the simulator in equation (1) is +given as +Sim : R5 → RL(L−1) +(5) +(r, cx, cy, σin, σout) �→ (V Sim +1 +, ..., V Sim +j,l +, ..., V Sim +L−1,L) +with V Sim +j,l +∈ R being the voltage measurement on the l-th +electrode obtained by the direct problem solution for the +trigonometric current pattern Ij. +3 +Modeling EIT +3.1 +Direct problem +Currents propagating in human tissues and organs can be +satisfactorily modeled by the Complete Electrode Model +[3]. +It accounts for the finite nature of electrodes, for +the current injection through them and for the electro- +chemical effects happening between skin and electrode sur- +face. +Let Ω describe the subject region we are evaluating. To +establish a measurement setup, we attach L electrodes at +the subject boundary ∂Ω. Through them we apply an elec- +trical current pattern I = (I1, ..., IL) into Ω. The objective +is to find the electrical potential u inside and the voltages +at electrodes V = (V1, ..., VL) that fulfill the system of +equations describing the Complete Electrode Model: +� +� +� +� +� +� +� +� +� +∇ · (σ∇u) = 0, +in Ω, +� +El σ ∂u +∂ν dS = Il, +l = 1, 2, ..., L +σ ∂u +∂ν = 0, +in ∂Ω \ ∪L +l=1El +u + zlσ ∂u +∂ν +�� +El = Vl, +l = 1, 2, ..., L +(6) +where ν is the outward pointing normal vector at ∂Ω, dS +is measuring length of the boundary and σ is the conduc- +tivity distribution. +The first equation represents electrical current diffu- +sion. The second and third define the insertion of cur- +rent through electrodes, meaning current spreads through +the whole electrode before being inserted into the domain +and in regions without electrodes there isn’t current flow- +ing. +Finally, the last equations model the electrochem- +ical effects at interface of skin-electrode, with zl termed +as contact impedance representing the resistance at that +interface. +To ensure the existence and uniqueness of a solution, +the current pattern must satisfy Kirchoff’s law and we fix +a reference voltage condition: +L +� +l=1 +Il = 0, +and +L +� +l=1 +Vl = 0. +(7) +3.2 +Modeling the circular anomaly +In this section, we define the conductivity parameteriza- +tion formally introduced in Section 2. +The parameterization is done through a level-set, i.e., +a function that has positive sign inside the region it de- +scribes, negative on the outside and equal to zero on the +region boundary. In particular, a circle level-set LS(x, y) +can be defined through a center c = (cx, cy) and a radius +r as follows +LS(x, y) = r2 − +� +(x − cx)2 + (y − cy)2� +. +(8) +The level-set function is positively valued if the point +(x, y) is inside the circular anomaly, negative if it is outside +and zero if its precisely at the boundary of the anomaly. +As such, we can use the Heaviside function H(z) that +equals 1 if z > 0 and 0 otherwise, to fully describe the +conductivity profile of interest through +σ(x, y) = σinH(LS(x, y)) + σout (1 − H(LS(x, y))) . +(9) +Under this formulation σ is not differentiable due to the +discontinuity of H at z = 0. In order to attain differen- +tiability, we use a smooth approximation of the Heaviside +function given as +Hϵ(z) = 1 +π arctan +�z +ϵ +� ++ 1 +2. +The conductivity σ is instead established in terms of Hϵ, +where ϵ > 0 works as a smoothing parameter. The smaller +it is the closer Hϵ is to H. +This smoothing procedure is necessary both for the an- +alytical computation as well as AD. In fact, we need to +take into account the mathematical differentiability for a +proper implementation of derivatives through AD. For ex- +ample, JAX AD applies the derivative to H by follow- +ing the conditional operations if else, which implies a +derivative of 0 everywhere, which is not true for z = 0. +4 + +4 +Derivatives computation +In order to solve the inverse problem in a minimization +framework, we need to compute derivatives of the Sim +operator. In this section, we deduce the analytical formula +and explain how to apply AD to Sim, in order to obtain +the derivatives with respect to the parameters of interest. +We recall that the direct solver and Sim are indepen- +dent of the derivative computation method. +4.1 +Analytical Computation +We recall that by Eq. (5) we have that the FEM simulator +operator is given by +Sim : R5 → RL(L−1) +(cx, cy, r, σin, σout) �→ +� +V Sim +1 +, ..., V Sim +j,l +, ..., V Sim +L−1,L +� +. (10) +To avoid heavy notation, we denote the vector of voltage +measurements by V Sim ∈ RL(L−1) and Vn ∈ RL are the +voltages measured j-th current pattern. +The Jacobian matrix J ∈ RL(L−1)×5 is given by +J = +� +∂V Sim +∂cx +∂V Sim +∂cy +∂V Sim +∂r +∂V +∂σin +∂V Sim +∂σout +� +(11) +In order to provide an analytical formulation, we specif- +ically focus on the computation of derivatives for each Vn +with respect to a single parameter, which if done for all +n = 1, ..., L − 1 determines one column of the Jacobian. +Furthermore, we need to specify a method to simulate +the measurements. +In this paper, we have used FEM applied to the Com- +plete Electrode Model described before. The FEM solu- +tion is θ = (α, β) ∈ RN+L−1, where α describes the elec- +trical potential inside Ω and β the voltages at the elec- +trodes. Accordingly, we denote for each current pattern +Ij the FEM solution by θj = [αj, βj] ∈ RN+L−1 with re- +spect to ˜Ij on the right-hand side of the FEM system of +equations (a variation of Ij). +With this in mind, the voltages are computed by Vj = +Mβj where M is a matrix defining the basis functions +used by FEM at the electrodes. For further detail about +the FEM solution we point to Appendix A. +Now, if we define ˜ +M = [ˆ0 M] ∈ RL×(N+L−1) then we +have +Vn = ˜ +Mθn = ˜ +MA−1 ˜In. +(12) +As +such, +it +holds +for +any +parameter +w +of +{cx, cy, r, σin, σout} that: +∂Vn +∂w = +∂ +� +˜ +MA−1 ˜In +� +∂w +. +Since neither ˜ +M and ˜In depend on the conductivity σ +and, therefore, for any of the parameters, it holds that +∂Vn +∂w = ˜ +M ∂A−1 +∂w +˜In = − ˜ +MA−1 ∂A +∂w A−1 ˜In +(13) +with the last equality following from matrix calculus prop- +erties. +Thus, in essence, the computation resumes to the stiff- +ness matrix derivative and noticing that A−1 ˜In = θn. Set- +ting γ = ˜ +MA−1 the computation of the derivative in Eq. +(13) simplifies to +∂Vn +∂w = −γT ∂A +∂w θn. +(14) +As such, the focus is on the computation of ∂A +∂w. The +stiffness matrix A is composed of four blocks, like, +� +B1 + B2 +C +CT +D +� +. +The block B1 is the only one depending on the conductiv- +ity. Due to its definition there is a clear way of computing +the derivatives of B1 with respect to the conductivity value +σk over each mesh element (see the Appendix for further +details on its definition): +∂B1 +ij +∂σk += +�� +Tk ∇φi · ∇φj dx, if i, j ∈ Tk +0, otherwise. +(15) +Furthermore, the resulting matrix is independent of σ +therefore it can be precomputed at the start and re-used. +Through the chain rule we have that +∂B1 +ij +∂w = +K +� +k=0 +∂B1 +ij +∂σk +∂σk +∂w . +(16) +We note that due to sparsity of the matrix defined in +Eq. (15) it can be assembled very efficiently. However, +this optimal performance is an extra layer of complexity +that needs to be solved manually and AD takes care of +that automatically. +The remaining object to be computed from Eq. (14) is +γ. Since, A is a very large sparse matrix the best way to +do determine it is by solving the adjoint system equivalent +to γ = ˜ +MA−1 given as +AT γ = ˜ +M T with γ ∈ RN+(L−1)×L. +(17) +Since A depends on the conductivity σ this system needs +to be solve once at each iteration of the inverse solver. +Finally, a formula for the derivatives in Eq. (11) is ob- +tained after solving the adjoint system (17) and comput- +ing the derivative of B1 as in (16). The derivatives are +compactly given through the formula +∂Vn +∂w = −γT +� ∂B1 +∂w +0 +0 +0 +� +θn. +(18) +Through this demonstration, we have seen that it can +be very tedious to deduce and implement the analytical +derivatives for complex problems, like ours. +For simple +functions, an analytical derivative in compact form takes +the lead in efficiency, however we want to experiment with +the case of more complex functions. +5 + +4.2 +Automatic differentiation method +In this section, we introduce how to apply JAX automatic +differentiation toolbox [2] to obtain the Jacobian. +Fur- +ther details about the inner workings of AD and JAX are +explained in Appendix C. +Since our direct operator Sim has more output variables +than input variables we note that the most-efficient AD +mode is the forward-mode. +The implementation of a differentiable simulator Sim +means we can simply use JAX AD to compute the deriva- +tives. Our Sim operator is differentiable with respect to +the parameterization variables (r, cx, cy, σin, σout) that de- +fine the anomaly, as introduced in section 3.2. +This preparation are a requirement for both derivative +methods, but now the derivative computation with AD is +simply implemented through JAX. +To do so, we implement a routine that defines the di- +rect operator Sim given in Eq. (5). The implementation +is established through the solution of the direct problem +through FEM, that we here hide as the simulator method. +Listing 1 provides the routine with all of these in mind. +import jax +def direct_operator(anomaly_parameters): +"""Simulate measurements for given +input function with JAX. +Args: +anomaly_parameters: Array of shape +(5,) with parametrization variables +of circular anomalies. +Returns: +measurements: Array of shape +(nmb_electrodes(nmb_electrodes-1),) that +contains the voltage measurements for all +current patterns. +""" +# Compute measurements +measurements = simulator(anomaly_parameters) +return measurements +Listing 1: Definition of the direct operator through a gen- +eral simulator method. +In +order +to +compute +the +Jacobian +defined +in +Eq. +(11) +with +JAX +one +only +needs +to +call +jax.jacfwd(direct operator) for our direct operator as +in Listing 2. +To establish the inverse solver these function definitions +are redundant and we can immediately call simulator +and jax.jacfwd(direct operator) in the inverse solver +routine. This definition is just for visualization purposes +in this section. +def jacobian(anomaly_parameters): +"""Compute Jacobian with JAX AD +Args: +anomaly_parameters: 1d array of shape +(5,) with parametrization variables +of circular anomalies. +Returns: +Jacobian matrix of shape +(nmb_electrodes(nmb_electrodes-1), 5). +""" +# Define the jacobian through forward-mode +jacobian = jax.jacfwd(direct_operator) +return jacobian(anomaly_parameters) +Listing 2: Computation of the Jacobian matrix through +JAX automatic differentiation toolbox. +5 +Experimental setup +To compare both analytic and automatic differentiation +methods, we explore their evaluation at different conduc- +tivities, and how they fit in to solve the inverse problem. +For the latter, we consider two particular cases for the in- +verse problem. The first case, that we label as the case +of fixed conductivities is simpler. We want to deter- +mine only the location parameters (r, cx, cy) and we as- +sume the conductivity values inside σin and outside σout +are fixed. This scenario can represent breast cancer, for +example, where we know a priori conductivity values of +different tissues, and we are only concerned in determining +the anomaly location. +The second case, that we label as the case of gen- +eral conductivities, we want to determine all param- +eters (r, cx, cy, σin, σout). This is a more general scenario +where we only know there is a circular anomaly and want +to characterize it in terms of location, radius and conduc- +tivity. +Recall that we fix a voltage measurement setup to sim- +plify the comparison. Our only interest is to show that AD +is as good as analytical methods in terms of solution ac- +curacy. Further, we show that the memory requirements +for AD scale reasonably well with the mesh resolution, +to show that AD can be effectively implemented in more +realistic cases involving more complex scenarios and 3D +meshes. +All of the experiments have been run in a machine with +the following hardware specifications: +• CPU Intel Core i5-12400F (released in Q1 2022, 12th +gen., 4.4 GHz, 6 cores, 12 threads, 64 GB RAM); +• GPU NVIDIA GeForce RTX 3070 (released in Q4 +2020, 6144 CUDA cores, 8 GB memory). +6 + +We chose this machine because it has typical med-range +specs and can be considered as a good example of an af- +fordable solution for the numerical computation, compat- +ible with the lower cost of EIT. We remark that besides +automatic differentiation, JAX excels in optimizing the +performance for a given hardware. Therefore, we have not +performed any specific optimization, but appropriate care +as been taken throughout the implementation. +5.1 +Establish a ground truth +In order to have a “lab” setup, i.e., one we can control +the experiment from start to finish, we define a voltage +measurements dataset through simulation. For such, we +randomly initialize our conductivity parameterization un- +der a certain range of parameters and determine their re- +spective voltage measurements m. +To test new inverse solvers we need to generate mea- +surements with the highest resolution possible to avoid +the so-called inverse crimes. Such crimes occur by using +the same resolution to obtain m and Sim operator com- +putationally. By doing it, we do not account for errors +arising from the approximate nature of the direct solver, +which occurs when using true measurements obtained by +a real-world measuring device, which adequately we can +think as having infinite resolution. As such, we need to +choose a higher mesh resolution for m than for Sim oper- +ator, since they are obtained both through FEM. +With this in mind we generate our ground truth dataset +of voltage measurements with the highest possible resolu- +tion for our hardware specifications. In our work, it was +established with a FEM mesh of 5815 elements that is set +accordingly to have each element with a edge length of +h = 0.035 relative to the domain size. +Furthermore, we generate the dataset through the fol- +lowing random initialization of the anomaly parameters: +• Uniformly generate conductivity centers anywhere in- +side the disk domain Ω = B1(0) with radius 1. Hence, +we use polar coordinates to generate the centers. To +start we uniformly generate an angle between [0, 2π]. +Then, we uniformly generate a value in [0, 1] to obtain +a radius sample by taking square root of it. Joining +both through polar coordinates gives an almost uni- +formly sampled set of 2D points inside Ω; +• Uniformly generate an anomaly radius, taking into +consideration the center position generated on the +previous point, so that anomalies are strictly in Ω. +As such, for each center we select the anomaly radius +uniformly from [0.1, 1 − |c|], where |c| is the distance +from center to origin; +• Uniformly generate conductivity values inside σin +from [1, 1.6] S/m and outside σout from [0.6, 1.] S/m. +Such values do not encapsulate any particular medical +or industrial scenario. +Our model assumes that contact impedances on each +electrode are fixed and have value z = 5 × 10−6Ω·. +In fact, we generate two separate datasets each with +1000 cases. +One for the case of fixed conductivities +where we randomly generate 1000 anomalies and compute +the respective measurements with fixed conductivity value +inside of σin = 1.4 S/m and outside of σout = 0.7 S/m. +Another for the case of general conductivities where +we randomly generate 1000 anomalies and compute their +measurements as described above. +Furthermore, we provide an initial sanity check for the +general dataset. We verified that the Jacobian computed +through both methods matches with minimal error mar- +gin, which may arise due to round off errors. This analysis +is presented in Appendix D. +6 +Results +In order to solve the inverse problem for the two cases +described above, we use a FEM mesh with 5210 elements +set by h = 0.037 to define the Sim operator, in order +to avoid inverse crimes. Our chosen inverse solver is the +Levenberg-Marquardt method with a line search algorithm +on each iteration. Further, we establish two stopping cri- +teria based on a maximum number of iterations equal to +20 and a relative mean squared loss +1 +2 +∥Sim(σ) − mtrue∥2 +2 +∥mtrue∥2 +2 +< ξ +(19) +with a feasible threshold of ξ = 0.001. This choice was +established empirically, since after that it becomes hard +to improve the anomaly reconstruction. +Let σAD and σAN be the solutions obtained through +the inverse solver with the different methods to compute +the derivative. In order to verify the effectiveness of AD in +solving the EIT inverse problem we evaluate how σAD and +σAN compare with the true solution σtrue and how they +compare with each other. This evaluation is based on the +mean squared error between the anomalies, i.e., for two +different anomaly parameterizations σ1, σ2 we evaluate +MSE(σ1, σ2) := ∥σ1 − σ2∥2. +In +essence, +we +compute +MSE(σtrue, σAD), +MSE(σtrue, σAN), +MSE(σAD, σAN). +Then, +we +per- +form an analysis of the mean squared errors by computing +simple statistics of the mean, variance, maximum and +minimum error, and by plotting the histogram with a +logarithmic scale in the x-axis. +We remark that the following analysis is focused on a +general analysis on the reconstructions obtained through +the different methods and does not verifies the nature of +the errors obtained, i.e., we do not check if the errors +are occurring for one specific parameter or for small/large +values of those same parameters. +7 + +6.1 +Case 1: Fixed Conductivities +In this case our goal is to determine the anomaly param- +eterized by σtrue = (r, cx, cy), since we know a priori that +the conductivity inside and outside are σin = 1.4 S/m +and σout = 0.7 S/m, respectively. Here, we denote σtrue +as the conductivity we aim to discover and mtrue for the +respective measurements. +We start from our measurements dataset for the fixed +conductivities with the set of 1000 voltage measure- +ments corresponding to different anomalies. +This num- +ber of experiments was constrained by time and hardware +capabilities. +The statistical analysis for this case is given in Table 1 +and the histogram for the different mean squared errors +are in Fig. 3 and 4. +Mean +S2 +Max. +Min. +MSE(σtrue, σAD) +0.0456 +0.0059 +0.4177 +0.0020 +MSE(σtrue, σAN) +0.0455 +0.0057 +0.4007 +0.0020 +MSE(σAD, σAN) +0.002 +2.64e-4 +0.2702 +1.51e-5 +Table 1: Statistics of mean squared errors of fixed conduc- +tivities, case 1, that compares the reconstructed conduc- +tivities obtained through the different derivative methods +with the true anomalies. +Figure 3: +Histogram of the mean squared errors of +fixed conductivities, case 1, comparing the reconstructed +anomalies obtained through the different derivative meth- +ods with the true anomalies. +The histogram presented in the Fig. 3 shows that the +distribution of the mean squared errors MSE(σtrue, σAD) +and MSE(σtrue, σAN) is similar. +Notice that the mean +squared errors in both cases are concentrated around 10−2 +with a set of outliers with error higher than 0.1. However, +this outliers occur in the same proportion for both meth- +ods. In analysis, this shows that the inverse solver with +automatic differentiation matches that with the analytic +derivative. +In Fig. 4 the histogram presents the distribution +of +the +mean +squared +errors +between +reconstruction +Figure 4: +Histogram of the mean squared errors of +fixed conductivities, case 1, comparing the reconstructed +anomalies. +MSE(σAD, σAN) and one can see that it is highly con- +centrated around 10−3. There are some different recon- +structions between the methods, but their error is in the +order of 0.1. Again, this highlights the effectiveness of AD +compared with the analytic method. However, there are +some outliers that shows divergence in the reconstructions +between both methods. These errors seem to be related +with round-off errors when we combine this analysis with +the sanity check for the Jacobian. +To complete the discussion of this case, we allude to +the statistics in Table 1. We point to the mean and vari- +ance of the different mean squared errors. This shows that +on average the reconstruction obtained with AD is much +closer with the analytic one than with the true anomalies. +Furthermore, the variance between these reconstructions +is very small. Once again it shows the effectiveness of AD +to match the analytic derivative method and that other +inverse solver methods need to be improved in order to +obtain better reconstruction results. +6.2 +Case 2: General Conductivities +For +this +case +the +objective +is +to +determine +the +general anomaly parameterization given by σtrue += +(r, cx, cy, σin, σout). +Again, we denote σtrue as the con- +ductivity we aim to discover and mtrue for the respective +measurements. +We start from the measurements dataset for the gen- +eral conductivities with the set of 1000 voltage mea- +surements corresponding to the different anomalies. Re- +call, that in this generation we have assumed that σin is +always greater than σout. +The statistical analysis for this case is given in Table 2 +and the histogram for the different mean squared errors +are in Figs. 5 and 6. +The histogram presented in the Fig. 5 shows that the +distribution of the mean squared errors MSE(σtrue, σAD) +and MSE(σtrue, σAN) is similar. In analysis, this shows +that the inverse solver with automatic differentiation +matches that with the analytic derivative. Further, no- +8 + +250 +250 +200 +200 +150 +150 +Count +Count +100 +100 +50 +50 +10~4 +10-3 +10-2 +10-1 +100 +10~4 +10~3 +10-2 +10-1 +100 +MSE(true, gAD) +MSE(αtrue, gAN)250 +200 +150 +Count +100 +50 - +0 + +10~4 +10-3 +10~2 +10~1 +10° +MSE(αAD, αAN )Mean +S2 +Max. +Min. +MSE(σtrue, σAD) +0.2264 +0.0292 +0.9698 +0.0042 +MSE(σtrue, σAN) +0.2215 +0.0273 +0.9706 +0.0042 +MSE(σAD, σAN) +0.039 +0.0134 +0.8838 +4.4e-6 +Table 2: Statistics of mean squared errors of general con- +ductivities, case 2, that compares the reconstructed con- +ductivities obtained through the different derivative meth- +ods with the true anomalies. +Figure 5: Histogram of the mean squared errors of general +conductivities, case 2, that compares the reconstructed +anomalies obtained through the different derivative meth- +ods with the true anomalies. +tice that the mean squared errors in both cases are con- +centrated around 10−1. +In fact by setting a threshold, +we verified that there are at most 50 reconstructions for +both methods where the mean squared error with the true +anomaly is higher than 0.5, which together with the his- +tograms shows that the vast majority of reconstructions +is successful. +Figure 6: Histogram of the mean squared errors of gen- +eral conductivities, case 2, comparing the reconstructed +anomalies. +Furthermore, the histogram in Fig. 6 that presents the +histogram of MSE(σAD, σAN) shows that the errors be- +tween reconstructions are more concentrated around the +interval [10−4, 10−2]. +Again, this highlights the equiva- +lence of AD compared with the analytic method. How- +ever, there are some outliers that shows divergence in the +reconstructions between both methods. Combining this +analysis with the sanity check for the Jacobian it reveals +that this might occur due to round-off errors. +To complete the discussion of this case, we allude to the +statistics Table 2. The only aspect we would like to point +out here is the mean of the different mean squared errors. +This shows that on average the reconstruction obtained +with AD is much closer with the analytic one than with +the true anomalies. Once again it shows the effectiveness +of AD to match the analytic derivative method and that +other inverse solver methods need to be improved in order +to obtain better reconstruction results. +6.3 +Computational performance of AD +The viability of AD also depends of its scaling capabilities. +Namely, we want to understand if increasing the number +of mesh elements, and therefore the resolution and accu- +racy of the FEM turns AD unfeasible. This is relevant +because AD requires the construction of a computational +graph for the direct problem and then applies the chain- +rule throughout the nodes of the graph to compute the +derivatives. +As the number of mesh elements increases +the computational graph becomes larger and can be un- +feasible to use for it to compute the derivatives. +In order to understand this behavior, we compute for ten +different mesh sizes the Jacobian for 100 distinct general +anomalies, randomly generated as described before. For +each mesh size we measure the average GPU memory and +load usage through the Python package GPUtil. In Fig. +7 we plot the average of GPU load and memory usage +percent for each of the different mesh resolutions and in +Fig. 8 we plot the time that took to compute the Jacobian +matrices with respect to each mesh resolution. +Figure 7: Percentage of GPU load and memory usage with +respect to the number of mesh elements. +It is clear from both figures the growth in GPU memory +usage and time to execute this experiment. Moreover, for +meshes with more than 15000 elements we require more +than 8Gb of GPU memory. As of now, we cannot under- +stand the order of growth and further experiments with +finer resolution are needed. +9 + +250 +250 +200 +200 +150 +150 +Count +Count +100 +100 +50 +50 +0 +0 + +10~4 +103 +102 +10-1 +100 +10~4 +10-3 +102 +10-1 +10g +MSE(otrue, gAD) +MSE(αtrue, gAN)250 +200 +150 +Count +100 +50 +10-4 +10-3 +10-2 +10-1 +100 +MSE(gAD, GAN)95 +80 +90 +% +85 +60 +% +Load +80 +75 +40 +70 +20 +65 +. +2000 +4000 +6000 +8000 +10000 +12000 +14000 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +Number of Elements in Mesh +Number of Elements in MeshFigure 8: Time (s) elapsed to compute Jacobian matrices +for 100 random anomalies with respect to the number of +mesh elements. +7 +Conclusion +In this paper we have compared the effectiveness of AD to +solve inverse problems against classical methods with ana- +lytical formulations of the derivative. We have shown how +to adequately construct a FEM differentiable simulator in +the context of inverse problems. +We successfully intro- +duced automatic differentiation for solving inverse prob- +lems in an optimization framework, in particular, Elec- +trical Impedance Tomography. We have shown that AD +provides a simple way of computing derivatives of complex +operators, for example, arising from solutions of PDEs, +with respect to a set of parameters. +We have shown that AD is indeed effective to compute +the derivatives, since it matches the analytical computa- +tion up to minimal error. Further, it was used to solve the +Electrical Impedance Tomography inverse problem and we +shown that it is even superior to analytical methods, in +terms of time and resources. +The analytical formulation is nothing more than an ap- +plication of differentiation rules to the FEM formulation +of the direct operator. By construction AD essentially ex- +ecutes the same process, but automatically. As such, AD +and the analytical formulation can be even performing the +same operations, but the fact that AD is a plug-and-play +tool makes it advantageous to use for complex operators. +Moreover, it has proven more efficient since it takes +less time on average to solve any particular EIT prob- +lem, when compared with the analytical formulation in +our case study and scales well with the mesh resolution. +This indicates that with the right hardware AD can be +efficiently executed for large-scale problems. +With this tool, we can cast our focus into an efficient +implementation of the direct problem solvers, which is +way more understood in literature, and on the methods to +solve the inverse problem. It allows freedom to experiment +and deal with difficult equations, without much thought, +bringing focus to the practical application at hand. +Further, we expect that AD extends nicely to higher di- +mensions, while the analytic formulation will require some +re-implementation to accommodate the three dimensional +shapes of anomalies. +Future studies are interested in testing how AD easily +handles different shapes of anomalies, as well as 3D mod- +els. +Acknowledgements. +The +work +of +I. +Pombo +was +supported +by +FCT +through +CIDMA +and +projects +UIDB/04106/2020, +UIDP/04106/2020 +and +the +PhD +Scholarship SFRH/BD/143523/2019. +This work was developed during a research internship +at Inductiva Research Labs from March 2022 to Jan 2023. +The first author would like to thank the entire Inductiva +team for the continuous support and encouragement pro- +vided during the entire period of the internship and in par- +ticular thank Hugo Penedones, F´abio Cruz, David Lima +and David Carvalho for their comments and constructive +feedback given over the several versions of this manuscript. +References +[1] Adler, A., & Holder, D. (Eds.). (2021). Electrical +impedance tomography: methods, history and ap- +plications. CRC Press. +[2] Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. +J., Leary, C., Maclaurin, D., Necula, G., Paszke, +A., VanderPlas, J., Wanderman-Milne, S., & Zhang, +Q. (2018). JAX: composable transformations of +Python+ NumPy programs. Version 0.3.13, 5, 14- +24. +[3] Cheng, K. S., Isaacson, D., Newell, J. C., & Gisser, +D. G. (1989). Electrode models for electric cur- +rent computed tomography. IEEE Transactions on +Biomedical Engineering, 36(9), 918-924. +[4] Gen¸cer, N. G., & Tanzer, I. O. (1999). Forward prob- +lem solution of electromagnetic source imaging using +a new BEM formulation with high-order elements. +Physics in Medicine & Biology, 44(9), 2275. +[5] Harrach, B. (2021). An introduction to finite el- +ement methods for inverse coefficient problems +in +elliptic +PDEs. +Jahresbericht +der +Deutschen +Mathematiker-Vereinigung, 123(3), 183-210. +[6] Levenberg, K. (1944). A method for the solution of +certain non-linear problems in least squares. Quar- +terly of applied mathematics, 2(2), 164-168. +[7] Marquardt, D. W. (1963). An algorithm for least- +squares estimation of nonlinear parameters. Journal +of the society for Industrial and Applied Mathemat- +ics, 11(2), 431-441. +[8] Mueller, J. L., & Siltanen, S. (Eds.). (2012). Linear +and nonlinear inverse problems with practical appli- +cations. Society for Industrial and Applied Mathe- +matics. +10 + +12000 +10000 +8000 +Time +6000 +4000 +2000 +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +Numberof meshelements[9] Persson, P. O., & Strang, G. (2004). A simple mesh +generator in MATLAB. SIAM review, 46(2), 329- +345. +[10] Raissi, M., Perdikaris, P., & Karniadakis, G. E. +(2019). Physics-informed neural networks: A deep +learning framework for solving forward and in- +verse problems involving nonlinear partial differen- +tial equations. Journal of Computational physics, +378, 686-707. +[11] Somersalo, E., Cheney, M., & Isaacson, D. (1992). +Existence and uniqueness for electrode models for +electric current computed tomography. SIAM Jour- +nal on Applied Mathematics, 52(4), 1023-1040. +[12] Vauhkonen, P. J., Vauhkonen, M., Savolainen, T., +& Kaipio, J. P. (1999). Three-dimensional electri- +cal impedance tomography based on the complete +electrode model. IEEE Transactions on Biomedical +Engineering, 46(9), 1150-1160. +[13] Vauhkonen, P. (2004). Image Reconstruction in +Three-Dimensional Electrical Impedance Tomogra- +phy. Kuopio University Publications C. Natural and +Environmental Sciences 166. +[14] Webster, J. G. (Ed.). (1990). Electrical impedance +tomography. CRC Press. +[15] Wengert, R. E. (1964). A simple automatic deriva- +tive evaluation program. Communications of the +ACM, 7(8), 463-464. +A +FEM formulation of the Direct +Problem +In this appendix, we take a deeper dive into the Finite +Element Method applied to Electrical Impedance Tomog- +raphy. In this paper, we have used the Complete Electrode +Model (CEM) [3] to understand how current propagates +inside the body. Starting from its formulation, that we +have introduced in section 2.2. and recall here, we derive +its weak formulation and apply FEM to obtain a system +of linear equations. +The CEM takes into account the finite nature of +electrodes, current injection through them and electro- +chemical effects happening between skin and electrode sur- +face. +Let Ω describe the subject region we are evaluating. To +establish a measurement setup, we attach L electrodes at +the subject boundary ∂Ω. Through them we apply an elec- +trical current pattern I = (I1, ..., IL) into Ω. The objective +is to find the electrical potential u inside and the voltages +at electrodes V = (V1, ..., VL) that fulfill the system of +equations describing the Complete Electrode Model: +� +� +� +� +� +� +� +� +� +∇ · (σ∇u) = 0, +in Ω, +� +El σ ∂u +∂ν dS = Il, +l = 1, 2, ..., L +σ ∂u +∂ν = 0, +in ∂Ω \ ∪L +l=1El +u + zlσ ∂u +∂ν +�� +El = Vl, +l = 1, 2, ..., L +(20) +where σ is the conductivity distribution. +The first equation represents electrical current diffu- +sion. The second and third define the insertion of current +through electrodes, meaning that current spreads through +the whole electrode before being inserted into the domain +and in regions without electrode there isn’t current flow- +ing. Finally, the last equations models the electrochemical +effects at interface of skin-electrode, with zl designated as +contact impedance represent the resistance at that inter- +face. +To ensure the existence and uniqueness of a solution, +the current pattern must satisfy Kirchoff’s law and we fix +a reference voltage condition: +L +� +l=1 +Il = 0, +and +L +� +l=1 +Vl = 0. +(21) +In order to apply Finite element method, we introduce +the variational equation that describes fully (20). In [11] it +has been derived and shown that (u, V ) is a weak-solution +of (20) if for all (w, W) ∈ H1(Ω) × RL we have: +� +Ω +σ∇u · ∇vdx + +L +� +l=1 +1 +zl +� +El +(u − Vl) (w − Wl) dS += +L +� +l=1 +IlWl +(22) +This formulation joins every condition of (20) together +into one equation, which allows the simplification into a +linear system of equations. The first integral describes the +propagation of current throughout the domain, while the +second represents skin-electrode interface condition and +the right-hand side explains the insertion of current. +A.1 +FEM for Complete Electrode Model +FEM allows transforming the continuous problem, de- +scribed by the variational equation (22), into a discrete +system of equations that can be handled by linear algebra +methods. +A detailed explanation is provided in any FEM book, +and specifically for EIT [8]. Further, in Appendix we pro- +vide an explanation of each step for complete understand- +ing of those interested. Here, we only briefly describe some +of the parts required for our exposition. +In this section, we briefly describe how to apply FEM +in EIT. First, we remark that many variants can arise due +to possible different assumptions made. +11 + +First, we discretized the subject domain Ω into smaller +elements. Next, we approximate our solutions u, U by +uh(x, y) = +N +� +i=1 +αiφi(x, y) +(23) +V h = +L−1 +� +k=1 +βkηk, +(24) +where φi, ηk are basis functions. +In particular, the +ηk are defined through η1 += +(1, −1, 0, ..., 0)T , η2 += +(1, 0, −1, 0, ..., 0)T , ..., ηL−1 = (1, 0, ..., 0, −1)T ∈ RL. This +choice ensures that reference voltage condition (21) is ful- +filled. Further, N in (23) corresponds to the number nodes +forming the finite element mesh. +The approximate solutions uh and V h for the direct +problem are fully determined by the coefficients +α = [α1, ..., αN] ∈ RN, +(25) +β = [β1, ..., βL−1] ∈ RL−1. +(26) +FEM allows us to obtain a system of linear equa- +tions characterizing them. This is achieved by inserting +(uh, V h) into the variational equation (22), together with +different choices of (v, V ) = (φi, ηj). Gathering all possi- +bilities leads to a linear system of equations: +Aθ = ˜I, +(27) +where θ = [α, β] ∈ RN+L−1 and ˜I is described through +the current pattern I applied at the electrodes as follows: +˜I = +�−→0 , I1 − I2, I1 − I3, ..., I1 − IL +� +∈ RN+(L−1). +(28) +The stiffness matrix A can be computed in terms of four +blocks: +A = +� +B1 + B2 +C +CT +D +� +. +(29) +Each term is defined through integration over the do- +main and over the electrodes like: +B1 +ij = +� +Ω +σ∇φi · ∇φj dx, +i, j = 1, 2, ..., N +(30) +B2 +ij = +L +� +l=1 +1 +zl +� +El +φiφj dS, +i, j = 1, 2, ..., N +(31) +Cij = − +� +1 +z1 +� +E1 +φi dS − +1 +zj+1 +� +Ej+1 +φi dS +� +, +i = 1, 2, ..., N, j = 1, 2, ..., L − 1 +(32) +Dij = +� E1 +z1 , +i ̸= j +E1 +z1 + |Ej+1 +zj+1 , +i = j , i, j = 1, ..., L − 1, +(33) +with |Ej| being the electrode area. +The derivation of each block arises from application of +two different basis functions on the weak formulation. A +full description was done in [8]. +After solving the system for θ, the voltages V h are ob- +tained by multiplication with the basis functions matrix +M defined as: +M = +� +������� +1 +1 +1 +. . . +1 +−1 +0 +0 +. . . +0 +0 +−1 +0 +. . . +0 +... +... +... +... +... +0 +0 +0 +. . . +−1 +� +������� +(34) +through +V h = Mβ. +One detail we want to point out regarding FEM imple- +mentation concerns conductivity parameterization. +For +computational purposes, we assume that σ is piece-wise +constant, meaning that at each mesh element is constant, +and thus mathematically defined as: +σ(x, y) = +K +� +k=1 +σkχk(x, y), +(35) +where K is the total number of elements and χk is the +indicator function of the k-th element. +In this sense, matrix B1 simplifies to +B1 +ij = +� +{k: i,j∈Tk} +σk +� +Tk +∇φi · ∇φj dx +(36) +The parameterization of σ is essential to compute the +voltages variation V h with respect to a conductivity +variation, i.e., the derivative. If a parameterization was +not applied to σ, then it would be described as a function +from Ω to R. For the latter case a derivative still exists, +but it is more theoretically described, see [5]. +A.2 +Implementation Details +For implementation purposes we restrict ourselves to two- +dimensions even though the above formulation also holds +for further dimensions. +The first implementation decision is about space dis- +cretization. For simplicity sake, our choice of mesh gener- +ator is DistMesh algorithm, developed by Per-Olof Persson +and Gilbert Strang [9]. The elements are triangles and the +algorithm has been adapted to consider L equidistant elec- +trodes, with a pre-defined size, at the surface ∂Ω. +Secondly, we need to define our basis functions. +We +choose piece-wise linear functions, and therefore, for each +triangle element any basis function is linearly defined as: +φi(x, y) = ai + bix + ciy. +12 + +Moreover, the basis function are obtained in correspon- +dence to a mesh node (xj, yj) through the condition: +φi(xj, yj) = +� +1, +i = j +0, +i ̸= j . +(37) +Since, for every other node the function will be zero, +it holds that for every triangle that does not have i as a +node, φi ≡ 0 there. This simplifies the computations of +all the matrices, since most entries will be 0, due to non- +intersection of most basis functions supports. As such, A +is sparse. +Moreover, due to the equation nature being elliptic, the +stiffness matrix A is positive-definite. As such, the most +appropriate system of equations solver is the Conjugate +Gradient method (CG). +B +Derivation +of +Levenberg- +Marquardt method +A simple method for inverse problems under such an op- +timization framework is Levenberg-Marquardt method. +It is a general method since it is independent of the +simulator and the method used for differentiating it. As +such, it allows us to demonstrate the effectiveness of vari- +ous methods to compute the derivatives, in particular, of +Automatic Differentiation. +We hereby assume that σ is discretely given by a pa- +rameterization, i.e., σ ∈ Rp. +This simplifies simulation +and, more importantly, the derivatives computation pro- +cess which is now done with respect to each variable +σi, i = 1, ..., p. +An example is seen in Figure 1 where +σ = (σ1, σ2). +The minimization problem is given as +min +σ +1 +2 +��Sim(σ) − mtrue��2 +2 , +(38) +where mtrue is a set of true measured voltages with respect +to N currents applied, as already introduced. +We iteratively improve an approximate solution of the +minimization problem (2) through +σk+1 = σk + δσk +(39) +where δσk is an update step and σk is the current approx- +imate solution. This process is done until a satisfactory +solution is found. +Each method to solve the minimization problem is de- +fined by the update step δσk computation. +Levenberg-Marquardt is a second order quasi-newton +method, that approximates the Hessian through an iden- +tity regularization. In this sense, the update rule is given +as follows: +δσLM = − +� +J(σ)T J(σ) + λLMI +�−1 J(σ)T (Sim(σ) − mtrue). +(40) +Here, J(σ) denotes the Jacobian of Sim, i.e., a matrix +of voltage derivatives with respect to each parameter σi. +Further, λLM is a parameter used to approximate the Hes- +sian and that allows for improving the condition number +of J(σ)T J(σ). We determine it empirically. +The update rule derivation is given in appendix. +The Levenberg-Marquardt method is a particular type +of quasi-Newton methods. We start by deducing the gen- +eral form of quasi-Newton methods and there after funnel +on our chosen method. +We hereby assume that σ is discretely given by a pa- +rameterization, i.e., σ ∈ Rp. +This simplifies simulation +and, more importantly, the derivatives computation pro- +cess which is now done with respect to each variable +σi, i = 1, ..., p. +An example is seen in Figure 1 where +σ = (σ1, σ2). +The minimization problem is given as +min +σ +1 +2 +��Sim(σ) − mtrue��2 +2 , +(41) +where mtrue is a set of true measured voltages with respect +to N currents applied, as already introduced. +Denote by L(σ) the loss function in (41). Then, assum- +ing that we have an initial guess σ0, we can re-write (38) +as +L(σ + δσ) = 1 +2 +��Sim(σ0 + δσ) − mtrue��2 +2 +(42) +with an intent to minimize with respect to the parameter +variation δσ, which we designate by update step. There- +after, applying this iteratively we approximate our solu- +tion through +σk+1 = σk + δσk +(43) +until a satisfactory solution is found. +The Levenberg-Marquardt Algorithm is essential for to +compute the update step δσ. Taylor expansion of (42) up +to quadratic term is given by +L(σ + δσ) = L(σ) + L′(σ)δσ + 1 +2L′′(σ)(δσ)2, +(44) +where L′(σ) and L′′(σ) denotes the gradient and Hessian +of the objective function L, with respect to parameters +defining σ. +A minimum with respect to δσ has gradient zero. Thus, +we apply gradient to (44) +∂L +∂δσ (σ + δσ) = L′(σ) + L′′(σ)δσ, +and setting the gradient equal to zero yields +0 = L′(σ) + L′′(σ)δσ +⇔ δσ = − [L′′(σ)]−1 L′(σ). +13 + +Since only Sim depends on conductivity parameteriza- +tion we can compute the gradient and Hessian through: +L′(σ) = J(σ)T � +Sim(σ) − mtrue� +(45) +L′′(σ) = J(σ)T J(σ)+ +� +i +[Simi(σ)]′′ � +Simi(σ) − mtrue +i +� +, +(46) +where J is the Jacobian of simulated voltages Sim(σ) with +respect to the parameterization of σ. +Up until here the derivation is general for quasi-Newton +methods. +The Levenberg-Marquardt method distinguishes itself +from other quasi-Newton methods by avoiding computa- +tion second order derivative, substituting it by a scaled +identity matrix λLMI, λ ∈ R+, which acts as a regular- +izer by improving the condition number of the Hessian +matrix to be inverted. Now, the update can be computed +through: +δσLM = − +� +J(σ)T J(σ) + λLMI +�−1 J(σ)T (Sim(σ) − mtrue). +(47) +C +Automatic Differentiation +AD is a set of techniques to evaluate the derivative of a +function specified by a computer program. No matter how +complicated they are, any computer program is based on a +simple set of arithmetic operations and functions, like ad- +dition, multiplication, trigonometric functions, exponen- +tials, etc. +We can encode the derivative rule for all of +these simple operations and build up the full derivative of +our complex program through the chain-rule. AD evalu- +ates derivatives with exact precision. +There are two modes for AD implementation: forward- +mode and reverse-mode. In any case, they are not hard to +implement through operator overloading techniques. The +difficult part is to provide an efficient and optimal compu- +tation of these modes. However, at the present moment +there are great libraries that provide efficient implemen- +tations of AD for both modes, like JAX for Python. +The first step in AD is the creation of a computa- +tional graph of our program, that explains the decom- +position into simpler operations for which we know the +derivative. +Let’s exemplify for the following function +f(x1, x2) = sin(x1 · x2) + ex1. The first step is to break +things apart into the simpler operations: +w1 = x1, w2 = x2 +w3 = w1 · w2 +w4 = sin(w3) +w5 = ew1 +w6 = w4 + w5 =: f(w1, w2) +This decomposition is more easily visualized through +the computational graph in Fig. 9. +Figure 9: Computational Graph of f(x1, x2) = sin(x1 · +x2) + ex1 evaluated at (π/2, −3). +With the computational graph in mind, forward-mode +computes derivatives from bottom-to-top, that is from the +variables to output. As such, it allows the derivative com- +putation of all outputs with respect to a single variable. It +can evaluate the derivative simultaneously with the func- +tion, and thus it is proportional to the original code com- +plexity. +In this terms, it is more efficient for functions +f : Rn → Rm with m >> n. +Reverse-mode of AD works the other way around, that +it is, top-to-bottom. First, it requires a forward evaluation +of all the variables, and thereafter it starts computing the +derivatives from output values for the variables involved +immediately, doing that successively until the input vari- +ables. Therefore, it allows evaluation of the gradient of an +single output function. As such, it is way more efficient +for functions f : Rn → Rm with m << n. +A familiar example in these days is neural networks that +are described by way more weights that output variables, +in this particular example the reverse-mode is known as +backpropagation. +One possible limitation to take into account in AD arises +from the computational graph we described. Due to the +computer program complexity this computational graph +can be very expensive to establish and keep in memory. +In such scenarios, where the Jacobian is obtained from +a very complex graph, instead of a compact formula like +analytic formulation, it can take a long time to be evalu- +ated. As such, AD is not a tool to be inserted into play +whenever needed and considerations must be made when +implementing the Sim operator, to avoid some of these +flaws. +To bypass this problem, JAX can encode loops and con- +ditionals in primitive operations that are inherent from the +domain-specific compilers for linear algebra (XLA). Oth- +erwise, the loops are unrolled into a set of operations (may +be smaller than the general loop, but) that increases the +computational graph size. With the primitives in mind, +14 + +f(x,y) +data 5.8105 +sin(x*y) +data 1.0000 +x*y +exp(x) +data -4.7124 +data 4.8105 +y +x +data -3.0000 +data 1.5708this will be encoded on the graph with a single operation, +for which we already know the derivative. +With AD the focus is completely in an optimal imple- +mentation of the Sim operator, which is essential to ob- +tain a very efficient inverse problem solver (even with an- +alytical computation of derivatives). Thereafter, thinking +about both modes, we can apply forward-mode to com- +pute efficiently the derivatives of Sim with respect to the +parameterization (r, cx, cy, σin, σout). +Being aware of the inherent problems with both meth- +ods is essential for a proper implementation of the inverse +solver. +D +Extended Results +In this section we present some extra analysis about the +Jacobian computation with both methods in order to make +a sanity check. +The sanity check we want to verify is to check if the +Jacobian computed through automatic differentiation and +the analytic formulation match. This is what we already +expect since AD applies the chain-rule of differentiation +to FEM, which is exactly what we have done by hand to +determine the analytic formulation. The Frobenius norm +of the Jacobian difference is given as: +��JAD − Janalytic�� +F ro , +where ∥A∥F ro = +� +� +n,m +� +i,j +|aij|2 +� +� +1/2 +. +Further, we computed the Jacobian with both meth- +ods for 100 randomly generated general conductivities de- +scribed in section 3. Thereafter, we compute their differ- +ence and applied the Frobenius norm in order to obtain +an array with dimension 100. +To verify the assumption that both should evaluate to +almost the same values we make an histogram of the losses +and provide some statistics, namely, mean, variance, max- +imum and minimum. This results are provided in Fig. 10 +and Table 3. +Statistically we can infer that the Jacobian match +closely together with maximum error of 0.0552 and an +average of 0.0271. +Indeed, the histogram confirms that +most evaluations are really close together, with only some +outliers compared with the overall picture. Further, these +outliers might just be rounding off errors and are not wor- +risome since the error is still considerably small. +15 + +Mean +S2 +Max. Error +Min. Error +��JAD − Janalytic�� +F ro +0.0271 +7.94e-05 +0.0552 +0.0146 +Table 3: Statistic analysis of the error between Jacobian matrices obtained through the Frobenius norm. +Figure 10: Histogram of Jacobian error with both derivative methods evaluated with Frobenius norm. +16 + +25 +20 +15 +Count +10 +5 +2 × 10~2 +3 × 10-2 +4 × 102 +6 × 10-2 +MSE( JAD, janalytic) \ No newline at end of file diff --git a/_9FIT4oBgHgl3EQf-Suu/content/tmp_files/load_file.txt b/_9FIT4oBgHgl3EQf-Suu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..718b943efa1fb956b62fca162e3ac99c1267b06b --- /dev/null +++ b/_9FIT4oBgHgl3EQf-Suu/content/tmp_files/load_file.txt @@ -0,0 +1,784 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf,len=783 +page_content='Automatic differentiation as an effective tool in Electrical Impedance Tomography Ivan Pombo1,2, Luis Sarmento2 1 CIDMA - Center for Research and Development in Mathemathics and Applications 2 Inductiva Research Labs {ivanpombo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='ext, sarmento}@inductiva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='ai Abstract Determining physical properties inside an object without access to direct measurements of target regions can be for- mulated as a specific type of inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' One of such problems is applied in Electrical Impedance Tomography (EIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In general, EIT can be posed as a minimization prob- lem and solved by iterative methods, which require knowl- edge of derivatives of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In practice, this can be challenging because analytical closed-form so- lutions for them are hard to derive and implement effi- ciently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this paper, we study the effectiveness of automatic differentiation (AD) to solve EIT in a minimization frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We devise a case study where we compare solutions of the inverse problem obtained with AD methods and with the manually-derived formulation of the derivative against the true solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Furthermore, we study the viability of AD for large scale inverse problems by checking the memory and load re- quirements of AD as the resolution of the model increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' With powerful infrastructure, AD can pave the way for faster and simpler inverse solvers and provide better re- sults than classical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1 Introduction Electrical Impedance Tomography (EIT) is a non-invasive imaging method that produces images by determining the electrical conductivity inside a subject using only electrical measurements obtained at its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' More specifically, sinusoidal currents are applied to the subject through elec- trodes placed in certain locations at the surface of the ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The resulting voltages are then measured, making it possible to infer internal properties of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' EIT is a low-cost method and harmless for human being, since it only applies low amplitude currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Additionally, it allows for real-time monitoring of various subjects even in the most difficult conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' There are applications of this technology for medical purposes, in scenarios such as ventilation monitoring, detecting brain hemorrhages and breast cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' EIT is also used in geophysical imaging, flow analysis and other industrial purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For further insight into the applications see [14] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 1: Example of a target conductivity over the do- main Ω that represents a simple model of breast cancer where tumors have higher conductivity than the back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The domain Ω is represented by the black cir- cumference which has a conductivity of σout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In a blue circle it is represented a region with different conductivity σin from the background one σout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A particularly relevant application of EIT is in the early determination of breast cancer, specifically for young women where the risk of the ionizing X-rays of mammo- graphies outweigh the benefits of regular check-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1 describes one simplified EIT scenario where the blue region represents cancer inside the breast, denoted as a domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The assumption is that healthy and cancerous tissue have different conductivity values σ1, σ2, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The goal is to locate a potential region affected by cancer from measurements on the breast surface, which is the boundary of the domain Ω and denoted as ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='11410v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='NA] 26 Jan 2023 TargetconductivityoverdomainΩ Gout QinThe measurements are obtained by injecting into the domain Ω a fixed set of different electrical current pat- terns Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Each Ij is defined by injecting electrical current through all electrodes in a particular manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', for L electrodes we have Ij = (Ij,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', Ij,L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Simultaneously, we measure the resulting voltages Vj for each current pat- tern, obtaining a voltage measurement at each electrode, denoted as Vj = (Vj,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', Vj,L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This leads to a set of true measurements denoted by mj = (Ij, Vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Then, the corresponding inverse problem is to determine the electri- cal conductivity over Ω that leads to these measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In the particular case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1 we want to determine the conductivity outside and inside the anomaly, σout and σin, respectively, and the location of the anomaly (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This is a hard problem because in general there is no analytical expression that maps a set of electrical mea- surements back to the respective conductivity profile that generates them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To solve this inverse problem we first need to under- stand how to solve the direct problem, that is, computing electrical measurements Vj for a given set of currents Ij and conductivity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The direct problem has an easier so- lution, since the propagation of electrical current through the domain obeys the well-known Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Many methods for solving the direct problem are de- scribed in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', Finite Element Method (FEM) [12], Boundary Element Method (BEM) [4], and, more recently Deep Learning methods (DL) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Independently of the numerical method used to solve the direct problem, such a procedure is commonly desig- nated as simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Hence, for a given conductivity pro- file we can obtain through a simulation method the electri- cal measurements denoted as mSim j = (Ij, V Sim j ), for each different current pattern with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We can thus define an operator that maps conductivity into voltage measurements, here termed by direct operatorand given as: Sim : σ �→ V Sim = (V Sim 1,1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='., V Sim j,l , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', V Sim N,L) ∈ RL·N (1) where V Sim j,l represent voltages measured at the l-th elec- trode for the j-th current pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our goal is to find a conductivity profile σ that matches measurements m = (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', mN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Thus, we can formulate EIT as the following minimization problem by making use of the direct operator Sim: min σ 1 2 ��Sim(σ) − mtrue��2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (2) We use the L2-norm here for simplicity, but, in general, we could use any other norm as long as it is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Most classical methods for solving this minimization problem are based on iteratively improving the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The update requires computing the derivative of both the loss function in (2) and the Sim operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To solve the inverse problem under an optimization framework we opted for the Levenberg-Marquardt algo- rithm [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It is a simple quasi-Newton method that only requires the Jacobian computation of the Sim operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further details about the method are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In essence, the main challenges to solve the minimiza- tion problem (2) with iterative classic methods are: to ensure that the simulator is once-differentiable with respect to a conductivity parameterization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' devise a method to compute the respective derivatives of the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our study explores a simulation operator obtained through FEM, which is already well established for EIT, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 2: Circular anomaly defined over a triangular FEM mesh in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Electrodes are attached to the boundary, black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' When the Sim operator is given by FEM we can deduce an analytical closed-form of the derivative with respect to the conductivity variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It is simply obtained with re- spect to a conductivity discretization over the FEM mesh, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, it requires derivative computations with respect to conductivity values over all elements of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' If the conductivity is defined through a different pa- rameterization we can obtain the respective derivatives by the chain rule of differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For such endeavor, the analytical formulation needs to be adapted and derived for each particular parameterization of the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As a result this formula is hard to derive and implement, see [5] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Automatic differentiation (AD) is a method that au- tomatically evaluates exact derivatives for complex pro- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It exploits the simple mathematical operations the programs are built on, to automatically compute the derivative through the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' While the initial con- cept was developed in the sixties [15], only recently with advancements in hardware and efficient implementations, like JAX [2], it has gained traction for application in gen- eral problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2 Piece-wise conductivity withcircularanomaly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0 S/m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0 x ElectrodesIn this paper, we explore automatic differentiation as an alternative to manual methods for computing the Ja- cobian of differentiable simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In particular, the goal is to validate its effectiveness in solving the EIT inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' By effectiveness we mean that it is as success- ful in solving the inverse problem as previous methods, namely, through analytical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' By doing so we show its versatility compared with analytic formulation and moreover verify its viability for high resolution im- ages where .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The validation is done by comparing the absolute error between solutions obtained by solving the minimization problem with both methods to compute the derivatives and the absolute error compared with the true solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In particular, we evaluate the maximum difference between both Jacobian computations to check if they are evaluating to the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Then as a second set of checks, we explore the memory consumption of AD and show that it is still in reasonable terms as the problem scales with higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our end goal is to show feasibility and practicality of AD as a tool for lowering the entry barrier for other inverse problems in Partial Differential Equations, where AD can also be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In the following, we first introduce the EIT case study we are using for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In Section 3, we explain how the required derivatives are computed with both the analytical and AD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 4, we introduce our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Results comparing the effectiveness of both methods and viability of AD are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 5, and conclusions are drawn out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2 Establishing a case study In this section we establish a case study in order to make a clear comparison between both methods for computing derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1 EIT scenario To demonstrate our claims we focus on a two-dimensional setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We remark that this is not physically accurate since electrical current propagates in three dimensions However, it simplifies the construction of our case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' EIT is an ill-posed inverse problem [8] and thus we need to take into account the possible instability of the prob- lem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', small variations in the measurements may imply large variation on parameters solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In practice, this makes it hard to solve the inverse problem since true mea- surements, captured with real-world measuring devices, always contain noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Therefore, real solutions for noisy input data can be drastically distinct from the true solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Due to this, it becomes hard to accurately determine a very large number of parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', the value of the conductivity at all mesh elements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2), from a small number of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' An example would be a conductivity defined over a fine mesh which has a value at each mesh element, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To mitigate this problem we want to make as many measurements as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' However, the possible number of distinct measurements is constrained by the quantity of electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This occurs since for L electrodes there are only L−1 linearly independent current patterns for which the voltage measurements yield independent information of the conductivity, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The best way to mitigate this issue is to work on simpler cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' By doing so we can reduce the parameter space and have less variability on the solutions, like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Even though instability issues do not become completely fixed, the space of measurements has a lower, more tractable, dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For the sake of comparison we wish to make, it is enough to focus on conductivity profiles with a circular region of distinct conductivity value from the background, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' These anomalies are parameterized by their cen- ter (cx, cy) inside the domain Ω, radius r and conductivity value inside and outside σin, σout, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We work with this simplification because it is easier to obtain a solution to the inverse problem due to the pa- rameterization of such region being given by only a few parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, we remark that we need to make sure that the parameterization is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our choice of circular regions is based on this, since it is easy to define a smooth parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For regions with corners two smoothing procedures would be required, one to smoothen the corners and another to smooth the parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' By the reasons above, in our experiments we assume the existence of a single circular anomaly with conductivity value different from the background, like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1and denoted the parameterization variables as σ = (r, cx, cy, σin, σout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (3) We introduce now the EIT model, the conductivity pa- rameterization definition and the measurement setup we use to proceed with out comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2 Voltage measuring setup We introduce here the measuring setup that is applied for the direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this simple 2D setup, we define the Sim operator in (1) according to the case study and the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Recall that with L electrodes at the surface ∂Ω, we can at most apply L− 1 linearly independent current patterns Ij ∈ RL with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The Sim operator is ob- tained by solving the direct problem for each Ij and deter- mine the respective voltages Vj ∈ RL over the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The more measurements we can perform the better we are able to potentially reconstruct the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Therefore, we need to choose L − 1 linearly independent 3 current patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This choice is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' One possibil- ity presented in the literature [8] is obtained by injecting currents in a wave pattern through the electrodes accord- ing to Ij,l = � A cos(jθl), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L 2 , A sin � (j − L 2 )θl � , j = L 2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L − 1 (4) with θl = 2π L l and A the constant current amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' These patterns have been shown to obtain th best result on the detection of conductivities profiles with small anoma- lies in the regions furthest from the boundary, [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The experiments are performed in the following setting: Ω is a circular domain with radius rΩ10cm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Current amplitude of A = 3mA, which is a reason- able value for human subjects, and the voltages are measured in (mV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Attach L = 16 electrodes equally spaced at the boundary with each having fixed length π/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We refer to Figure 2 for a visual representation of the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Under the above setup, the simulator in equation (1) is given as Sim : R5 → RL(L−1) (5) (r, cx, cy, σin, σout) �→ (V Sim 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', V Sim j,l , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', V Sim L−1,L) with V Sim j,l ∈ R being the voltage measurement on the l-th electrode obtained by the direct problem solution for the trigonometric current pattern Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 3 Modeling EIT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1 Direct problem Currents propagating in human tissues and organs can be satisfactorily modeled by the Complete Electrode Model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It accounts for the finite nature of electrodes, for the current injection through them and for the electro- chemical effects happening between skin and electrode sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Let Ω describe the subject region we are evaluating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To establish a measurement setup, we attach L electrodes at the subject boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Through them we apply an elec- trical current pattern I = (I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', IL) into Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The objective is to find the electrical potential u inside and the voltages at electrodes V = (V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', VL) that fulfill the system of equations describing the Complete Electrode Model: � � � � � � � � � ∇ · (σ∇u) = 0, in Ω, � El σ ∂u ∂ν dS = Il, l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L σ ∂u ∂ν = 0, in ∂Ω \\ ∪L l=1El u + zlσ ∂u ∂ν �� El = Vl, l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L (6) where ν is the outward pointing normal vector at ∂Ω, dS is measuring length of the boundary and σ is the conduc- tivity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first equation represents electrical current diffu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The second and third define the insertion of cur- rent through electrodes, meaning current spreads through the whole electrode before being inserted into the domain and in regions without electrodes there isn’t current flow- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Finally, the last equations model the electrochem- ical effects at interface of skin-electrode, with zl termed as contact impedance representing the resistance at that interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To ensure the existence and uniqueness of a solution, the current pattern must satisfy Kirchoff’s law and we fix a reference voltage condition: L � l=1 Il = 0, and L � l=1 Vl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2 Modeling the circular anomaly In this section, we define the conductivity parameteriza- tion formally introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The parameterization is done through a level-set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', a function that has positive sign inside the region it de- scribes, negative on the outside and equal to zero on the region boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In particular, a circle level-set LS(x, y) can be defined through a center c = (cx, cy) and a radius r as follows LS(x, y) = r2 − � (x − cx)2 + (y − cy)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (8) The level-set function is positively valued if the point (x, y) is inside the circular anomaly, negative if it is outside and zero if its precisely at the boundary of the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, we can use the Heaviside function H(z) that equals 1 if z > 0 and 0 otherwise, to fully describe the conductivity profile of interest through σ(x, y) = σinH(LS(x, y)) + σout (1 − H(LS(x, y))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (9) Under this formulation σ is not differentiable due to the discontinuity of H at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In order to attain differen- tiability, we use a smooth approximation of the Heaviside function given as Hϵ(z) = 1 π arctan �z ϵ � + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The conductivity σ is instead established in terms of Hϵ, where ϵ > 0 works as a smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The smaller it is the closer Hϵ is to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This smoothing procedure is necessary both for the an- alytical computation as well as AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In fact, we need to take into account the mathematical differentiability for a proper implementation of derivatives through AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For ex- ample, JAX AD applies the derivative to H by follow- ing the conditional operations if else, which implies a derivative of 0 everywhere, which is not true for z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 4 4 Derivatives computation In order to solve the inverse problem in a minimization framework, we need to compute derivatives of the Sim operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this section, we deduce the analytical formula and explain how to apply AD to Sim, in order to obtain the derivatives with respect to the parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We recall that the direct solver and Sim are indepen- dent of the derivative computation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1 Analytical Computation We recall that by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (5) we have that the FEM simulator operator is given by Sim : R5 → RL(L−1) (cx, cy, r, σin, σout) �→ � V Sim 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', V Sim j,l , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', V Sim L−1,L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (10) To avoid heavy notation, we denote the vector of voltage measurements by V Sim ∈ RL(L−1) and Vn ∈ RL are the voltages measured j-th current pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The Jacobian matrix J ∈ RL(L−1)×5 is given by J = � ∂V Sim ∂cx ∂V Sim ∂cy ∂V Sim ∂r ∂V ∂σin ∂V Sim ∂σout � (11) In order to provide an analytical formulation, we specif- ically focus on the computation of derivatives for each Vn with respect to a single parameter, which if done for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L − 1 determines one column of the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Furthermore, we need to specify a method to simulate the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this paper, we have used FEM applied to the Com- plete Electrode Model described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The FEM solu- tion is θ = (α, β) ∈ RN+L−1, where α describes the elec- trical potential inside Ω and β the voltages at the elec- trodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Accordingly, we denote for each current pattern Ij the FEM solution by θj = [αj, βj] ∈ RN+L−1 with re- spect to ˜Ij on the right-hand side of the FEM system of equations (a variation of Ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' With this in mind, the voltages are computed by Vj = Mβj where M is a matrix defining the basis functions used by FEM at the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For further detail about the FEM solution we point to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Now, if we define ˜ M = [ˆ0 M] ∈ RL×(N+L−1) then we have Vn = ˜ Mθn = ˜ MA−1 ˜In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (12) As such, it holds for any parameter w of {cx, cy, r, σin, σout} that: ∂Vn ∂w = ∂ � ˜ MA−1 ˜In � ∂w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Since neither ˜ M and ˜In depend on the conductivity σ and, therefore, for any of the parameters, it holds that ∂Vn ∂w = ˜ M ∂A−1 ∂w ˜In = − ˜ MA−1 ∂A ∂w A−1 ˜In (13) with the last equality following from matrix calculus prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Thus, in essence, the computation resumes to the stiff- ness matrix derivative and noticing that A−1 ˜In = θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Set- ting γ = ˜ MA−1 the computation of the derivative in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (13) simplifies to ∂Vn ∂w = −γT ∂A ∂w θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (14) As such, the focus is on the computation of ∂A ∂w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The stiffness matrix A is composed of four blocks, like, � B1 + B2 C CT D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The block B1 is the only one depending on the conductiv- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Due to its definition there is a clear way of computing the derivatives of B1 with respect to the conductivity value σk over each mesh element (see the Appendix for further details on its definition): ∂B1 ij ∂σk = �� Tk ∇φi · ∇φj dx, if i, j ∈ Tk 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (15) Furthermore, the resulting matrix is independent of σ therefore it can be precomputed at the start and re-used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Through the chain rule we have that ∂B1 ij ∂w = K � k=0 ∂B1 ij ∂σk ∂σk ∂w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (16) We note that due to sparsity of the matrix defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (15) it can be assembled very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' However, this optimal performance is an extra layer of complexity that needs to be solved manually and AD takes care of that automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The remaining object to be computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (14) is γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Since, A is a very large sparse matrix the best way to do determine it is by solving the adjoint system equivalent to γ = ˜ MA−1 given as AT γ = ˜ M T with γ ∈ RN+(L−1)×L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (17) Since A depends on the conductivity σ this system needs to be solve once at each iteration of the inverse solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Finally, a formula for the derivatives in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (11) is ob- tained after solving the adjoint system (17) and comput- ing the derivative of B1 as in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The derivatives are compactly given through the formula ∂Vn ∂w = −γT � ∂B1 ∂w 0 0 0 � θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (18) Through this demonstration, we have seen that it can be very tedious to deduce and implement the analytical derivatives for complex problems, like ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For simple functions, an analytical derivative in compact form takes the lead in efficiency, however we want to experiment with the case of more complex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2 Automatic differentiation method In this section, we introduce how to apply JAX automatic differentiation toolbox [2] to obtain the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Fur- ther details about the inner workings of AD and JAX are explained in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Since our direct operator Sim has more output variables than input variables we note that the most-efficient AD mode is the forward-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The implementation of a differentiable simulator Sim means we can simply use JAX AD to compute the deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our Sim operator is differentiable with respect to the parameterization variables (r, cx, cy, σin, σout) that de- fine the anomaly, as introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This preparation are a requirement for both derivative methods, but now the derivative computation with AD is simply implemented through JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To do so, we implement a routine that defines the di- rect operator Sim given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The implementation is established through the solution of the direct problem through FEM, that we here hide as the simulator method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Listing 1 provides the routine with all of these in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' import jax def direct_operator(anomaly_parameters): """Simulate measurements for given input function with JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Args: anomaly_parameters: Array of shape (5,) with parametrization variables of circular anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Returns: measurements: Array of shape (nmb_electrodes(nmb_electrodes-1),) that contains the voltage measurements for all current patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' """ # Compute measurements measurements = simulator(anomaly_parameters) return measurements Listing 1: Definition of the direct operator through a gen- eral simulator method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In order to compute the Jacobian defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (11) with JAX one only needs to call jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='jacfwd(direct operator) for our direct operator as in Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To establish the inverse solver these function definitions are redundant and we can immediately call simulator and jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='jacfwd(direct operator) in the inverse solver routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This definition is just for visualization purposes in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' def jacobian(anomaly_parameters): """Compute Jacobian with JAX AD Args: anomaly_parameters: 1d array of shape (5,) with parametrization variables of circular anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Returns: Jacobian matrix of shape (nmb_electrodes(nmb_electrodes-1), 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' """ # Define the jacobian through forward-mode jacobian = jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='jacfwd(direct_operator) return jacobian(anomaly_parameters) Listing 2: Computation of the Jacobian matrix through JAX automatic differentiation toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 5 Experimental setup To compare both analytic and automatic differentiation methods, we explore their evaluation at different conduc- tivities, and how they fit in to solve the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For the latter, we consider two particular cases for the in- verse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first case, that we label as the case of fixed conductivities is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We want to deter- mine only the location parameters (r, cx, cy) and we as- sume the conductivity values inside σin and outside σout are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This scenario can represent breast cancer, for example, where we know a priori conductivity values of different tissues, and we are only concerned in determining the anomaly location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The second case, that we label as the case of gen- eral conductivities, we want to determine all param- eters (r, cx, cy, σin, σout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This is a more general scenario where we only know there is a circular anomaly and want to characterize it in terms of location, radius and conduc- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Recall that we fix a voltage measurement setup to sim- plify the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our only interest is to show that AD is as good as analytical methods in terms of solution ac- curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, we show that the memory requirements for AD scale reasonably well with the mesh resolution, to show that AD can be effectively implemented in more realistic cases involving more complex scenarios and 3D meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' All of the experiments have been run in a machine with the following hardware specifications: CPU Intel Core i5-12400F (released in Q1 2022, 12th gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='4 GHz, 6 cores, 12 threads, 64 GB RAM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' GPU NVIDIA GeForce RTX 3070 (released in Q4 2020, 6144 CUDA cores, 8 GB memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 6 We chose this machine because it has typical med-range specs and can be considered as a good example of an af- fordable solution for the numerical computation, compat- ible with the lower cost of EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We remark that besides automatic differentiation, JAX excels in optimizing the performance for a given hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Therefore, we have not performed any specific optimization, but appropriate care as been taken throughout the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1 Establish a ground truth In order to have a “lab” setup, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', one we can control the experiment from start to finish, we define a voltage measurements dataset through simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For such, we randomly initialize our conductivity parameterization un- der a certain range of parameters and determine their re- spective voltage measurements m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To test new inverse solvers we need to generate mea- surements with the highest resolution possible to avoid the so-called inverse crimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Such crimes occur by using the same resolution to obtain m and Sim operator com- putationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' By doing it, we do not account for errors arising from the approximate nature of the direct solver, which occurs when using true measurements obtained by a real-world measuring device, which adequately we can think as having infinite resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, we need to choose a higher mesh resolution for m than for Sim oper- ator, since they are obtained both through FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' With this in mind we generate our ground truth dataset of voltage measurements with the highest possible resolu- tion for our hardware specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In our work, it was established with a FEM mesh of 5815 elements that is set accordingly to have each element with a edge length of h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='035 relative to the domain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Furthermore, we generate the dataset through the fol- lowing random initialization of the anomaly parameters: Uniformly generate conductivity centers anywhere in- side the disk domain Ω = B1(0) with radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Hence, we use polar coordinates to generate the centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To start we uniformly generate an angle between [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Then, we uniformly generate a value in [0, 1] to obtain a radius sample by taking square root of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Joining both through polar coordinates gives an almost uni- formly sampled set of 2D points inside Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Uniformly generate an anomaly radius, taking into consideration the center position generated on the previous point, so that anomalies are strictly in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, for each center we select the anomaly radius uniformly from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1, 1 − |c|], where |c| is the distance from center to origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Uniformly generate conductivity values inside σin from [1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='6] S/m and outside σout from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='] S/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Such values do not encapsulate any particular medical or industrial scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our model assumes that contact impedances on each electrode are fixed and have value z = 5 × 10−6Ω·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In fact, we generate two separate datasets each with 1000 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' One for the case of fixed conductivities where we randomly generate 1000 anomalies and compute the respective measurements with fixed conductivity value inside of σin = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='4 S/m and outside of σout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='7 S/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Another for the case of general conductivities where we randomly generate 1000 anomalies and compute their measurements as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Furthermore, we provide an initial sanity check for the general dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We verified that the Jacobian computed through both methods matches with minimal error mar- gin, which may arise due to round off errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This analysis is presented in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 6 Results In order to solve the inverse problem for the two cases described above, we use a FEM mesh with 5210 elements set by h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='037 to define the Sim operator, in order to avoid inverse crimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Our chosen inverse solver is the Levenberg-Marquardt method with a line search algorithm on each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, we establish two stopping cri- teria based on a maximum number of iterations equal to 20 and a relative mean squared loss 1 2 ∥Sim(σ) − mtrue∥2 2 ∥mtrue∥2 2 < ξ (19) with a feasible threshold of ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This choice was established empirically, since after that it becomes hard to improve the anomaly reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Let σAD and σAN be the solutions obtained through the inverse solver with the different methods to compute the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In order to verify the effectiveness of AD in solving the EIT inverse problem we evaluate how σAD and σAN compare with the true solution σtrue and how they compare with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This evaluation is based on the mean squared error between the anomalies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', for two different anomaly parameterizations σ1, σ2 we evaluate MSE(σ1, σ2) := ∥σ1 − σ2∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In essence, we compute MSE(σtrue, σAD), MSE(σtrue, σAN), MSE(σAD, σAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Then, we per- form an analysis of the mean squared errors by computing simple statistics of the mean, variance, maximum and minimum error, and by plotting the histogram with a logarithmic scale in the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We remark that the following analysis is focused on a general analysis on the reconstructions obtained through the different methods and does not verifies the nature of the errors obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', we do not check if the errors are occurring for one specific parameter or for small/large values of those same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1 Case 1: Fixed Conductivities In this case our goal is to determine the anomaly param- eterized by σtrue = (r, cx, cy), since we know a priori that the conductivity inside and outside are σin = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='4 S/m and σout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='7 S/m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Here, we denote σtrue as the conductivity we aim to discover and mtrue for the respective measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We start from our measurements dataset for the fixed conductivities with the set of 1000 voltage measure- ments corresponding to different anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This num- ber of experiments was constrained by time and hardware capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The statistical analysis for this case is given in Table 1 and the histogram for the different mean squared errors are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Mean S2 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' MSE(σtrue, σAD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='4177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0020 MSE(σtrue, σAN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0455 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='4007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0020 MSE(σAD, σAN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='64e-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2702 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='51e-5 Table 1: Statistics of mean squared errors of fixed conduc- tivities, case 1, that compares the reconstructed conduc- tivities obtained through the different derivative methods with the true anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 3: Histogram of the mean squared errors of fixed conductivities, case 1, comparing the reconstructed anomalies obtained through the different derivative meth- ods with the true anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The histogram presented in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 3 shows that the distribution of the mean squared errors MSE(σtrue, σAD) and MSE(σtrue, σAN) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Notice that the mean squared errors in both cases are concentrated around 10−2 with a set of outliers with error higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' However, this outliers occur in the same proportion for both meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In analysis, this shows that the inverse solver with automatic differentiation matches that with the analytic derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 4 the histogram presents the distribution of the mean squared errors between reconstruction Figure 4: Histogram of the mean squared errors of fixed conductivities, case 1, comparing the reconstructed anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' MSE(σAD, σAN) and one can see that it is highly con- centrated around 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' There are some different recon- structions between the methods, but their error is in the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Again, this highlights the effectiveness of AD compared with the analytic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' However, there are some outliers that shows divergence in the reconstructions between both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' These errors seem to be related with round-off errors when we combine this analysis with the sanity check for the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To complete the discussion of this case, we allude to the statistics in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We point to the mean and vari- ance of the different mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This shows that on average the reconstruction obtained with AD is much closer with the analytic one than with the true anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Furthermore, the variance between these reconstructions is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Once again it shows the effectiveness of AD to match the analytic derivative method and that other inverse solver methods need to be improved in order to obtain better reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2 Case 2: General Conductivities For this case the objective is to determine the general anomaly parameterization given by σtrue = (r, cx, cy, σin, σout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Again, we denote σtrue as the con- ductivity we aim to discover and mtrue for the respective measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We start from the measurements dataset for the gen- eral conductivities with the set of 1000 voltage mea- surements corresponding to the different anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Re- call, that in this generation we have assumed that σin is always greater than σout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The statistical analysis for this case is given in Table 2 and the histogram for the different mean squared errors are in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The histogram presented in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 5 shows that the distribution of the mean squared errors MSE(σtrue, σAD) and MSE(σtrue, σAN) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In analysis, this shows that the inverse solver with automatic differentiation matches that with the analytic derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, no- 8 250 250 200 200 150 150 Count Count 100 100 50 50 10~4 10-3 10-2 10-1 100 10~4 10~3 10-2 10-1 100 MSE(true, gAD) MSE(αtrue, gAN)250 200 150 Count 100 50 - 0 + 10~4 10-3 10~2 10~1 10° MSE(αAD, αAN )Mean S2 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' MSE(σtrue, σAD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='9698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0042 MSE(σtrue, σAN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='9706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0042 MSE(σAD, σAN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='8838 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='4e-6 Table 2: Statistics of mean squared errors of general con- ductivities, case 2, that compares the reconstructed con- ductivities obtained through the different derivative meth- ods with the true anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 5: Histogram of the mean squared errors of general conductivities, case 2, that compares the reconstructed anomalies obtained through the different derivative meth- ods with the true anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' tice that the mean squared errors in both cases are con- centrated around 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In fact by setting a threshold, we verified that there are at most 50 reconstructions for both methods where the mean squared error with the true anomaly is higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='5, which together with the his- tograms shows that the vast majority of reconstructions is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 6: Histogram of the mean squared errors of gen- eral conductivities, case 2, comparing the reconstructed anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Furthermore, the histogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 6 that presents the histogram of MSE(σAD, σAN) shows that the errors be- tween reconstructions are more concentrated around the interval [10−4, 10−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Again, this highlights the equiva- lence of AD compared with the analytic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' How- ever, there are some outliers that shows divergence in the reconstructions between both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Combining this analysis with the sanity check for the Jacobian it reveals that this might occur due to round-off errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To complete the discussion of this case, we allude to the statistics Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The only aspect we would like to point out here is the mean of the different mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This shows that on average the reconstruction obtained with AD is much closer with the analytic one than with the true anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Once again it shows the effectiveness of AD to match the analytic derivative method and that other inverse solver methods need to be improved in order to obtain better reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='3 Computational performance of AD The viability of AD also depends of its scaling capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Namely, we want to understand if increasing the number of mesh elements, and therefore the resolution and accu- racy of the FEM turns AD unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This is relevant because AD requires the construction of a computational graph for the direct problem and then applies the chain- rule throughout the nodes of the graph to compute the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As the number of mesh elements increases the computational graph becomes larger and can be un- feasible to use for it to compute the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In order to understand this behavior, we compute for ten different mesh sizes the Jacobian for 100 distinct general anomalies, randomly generated as described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For each mesh size we measure the average GPU memory and load usage through the Python package GPUtil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 7 we plot the average of GPU load and memory usage percent for each of the different mesh resolutions and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 8 we plot the time that took to compute the Jacobian matrices with respect to each mesh resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 7: Percentage of GPU load and memory usage with respect to the number of mesh elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It is clear from both figures the growth in GPU memory usage and time to execute this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Moreover, for meshes with more than 15000 elements we require more than 8Gb of GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As of now, we cannot under- stand the order of growth and further experiments with finer resolution are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 9 250 250 200 200 150 150 Count Count 100 100 50 50 0 0 + 10~4 103 102 10-1 100 10~4 10-3 102 10-1 10g MSE(otrue, gAD) MSE(αtrue, gAN)250 200 150 Count 100 50 10-4 10-3 10-2 10-1 100 MSE(gAD, GAN)95 80 90 % 85 60 % Load 80 75 40 70 20 65 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 2000 4000 6000 8000 10000 12000 14000 2000 4000 6000 8000 10000 12000 14000 Number of Elements in Mesh Number of Elements in MeshFigure 8: Time (s) elapsed to compute Jacobian matrices for 100 random anomalies with respect to the number of mesh elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 7 Conclusion In this paper we have compared the effectiveness of AD to solve inverse problems against classical methods with ana- lytical formulations of the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We have shown how to adequately construct a FEM differentiable simulator in the context of inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We successfully intro- duced automatic differentiation for solving inverse prob- lems in an optimization framework, in particular, Elec- trical Impedance Tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We have shown that AD provides a simple way of computing derivatives of complex operators, for example, arising from solutions of PDEs, with respect to a set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We have shown that AD is indeed effective to compute the derivatives, since it matches the analytical computa- tion up to minimal error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, it was used to solve the Electrical Impedance Tomography inverse problem and we shown that it is even superior to analytical methods, in terms of time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The analytical formulation is nothing more than an ap- plication of differentiation rules to the FEM formulation of the direct operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' By construction AD essentially ex- ecutes the same process, but automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, AD and the analytical formulation can be even performing the same operations, but the fact that AD is a plug-and-play tool makes it advantageous to use for complex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Moreover, it has proven more efficient since it takes less time on average to solve any particular EIT prob- lem, when compared with the analytical formulation in our case study and scales well with the mesh resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This indicates that with the right hardware AD can be efficiently executed for large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' With this tool, we can cast our focus into an efficient implementation of the direct problem solvers, which is way more understood in literature, and on the methods to solve the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It allows freedom to experiment and deal with difficult equations, without much thought, bringing focus to the practical application at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, we expect that AD extends nicely to higher di- mensions, while the analytic formulation will require some re-implementation to accommodate the three dimensional shapes of anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Future studies are interested in testing how AD easily handles different shapes of anomalies, as well as 3D mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The work of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Pombo was supported by FCT through CIDMA and projects UIDB/04106/2020, UIDP/04106/2020 and the PhD Scholarship SFRH/BD/143523/2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This work was developed during a research internship at Inductiva Research Labs from March 2022 to Jan 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first author would like to thank the entire Inductiva team for the continuous support and encouragement pro- vided during the entire period of the internship and in par- ticular thank Hugo Penedones, F´abio Cruz, David Lima and David Carvalho for their comments and constructive feedback given over the several versions of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' References [1] Adler, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A simple automatic deriva- tive evaluation program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Communications of the ACM, 7(8), 463-464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A FEM formulation of the Direct Problem In this appendix, we take a deeper dive into the Finite Element Method applied to Electrical Impedance Tomog- raphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this paper, we have used the Complete Electrode Model (CEM) [3] to understand how current propagates inside the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Starting from its formulation, that we have introduced in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' and recall here, we derive its weak formulation and apply FEM to obtain a system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The CEM takes into account the finite nature of electrodes, current injection through them and electro- chemical effects happening between skin and electrode sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Let Ω describe the subject region we are evaluating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To establish a measurement setup, we attach L electrodes at the subject boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Through them we apply an elec- trical current pattern I = (I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', IL) into Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The objective is to find the electrical potential u inside and the voltages at electrodes V = (V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', VL) that fulfill the system of equations describing the Complete Electrode Model: � � � � � � � � � ∇ · (σ∇u) = 0, in Ω, � El σ ∂u ∂ν dS = Il, l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L σ ∂u ∂ν = 0, in ∂Ω \\ ∪L l=1El u + zlσ ∂u ∂ν �� El = Vl, l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L (20) where σ is the conductivity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first equation represents electrical current diffu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The second and third define the insertion of current through electrodes, meaning that current spreads through the whole electrode before being inserted into the domain and in regions without electrode there isn’t current flow- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Finally, the last equations models the electrochemical effects at interface of skin-electrode, with zl designated as contact impedance represent the resistance at that inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To ensure the existence and uniqueness of a solution, the current pattern must satisfy Kirchoff’s law and we fix a reference voltage condition: L � l=1 Il = 0, and L � l=1 Vl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (21) In order to apply Finite element method, we introduce the variational equation that describes fully (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In [11] it has been derived and shown that (u, V ) is a weak-solution of (20) if for all (w, W) ∈ H1(Ω) × RL we have: � Ω σ∇u · ∇vdx + L � l=1 1 zl � El (u − Vl) (w − Wl) dS = L � l=1 IlWl (22) This formulation joins every condition of (20) together into one equation, which allows the simplification into a linear system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first integral describes the propagation of current throughout the domain, while the second represents skin-electrode interface condition and the right-hand side explains the insertion of current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='1 FEM for Complete Electrode Model FEM allows transforming the continuous problem, de- scribed by the variational equation (22), into a discrete system of equations that can be handled by linear algebra methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A detailed explanation is provided in any FEM book, and specifically for EIT [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, in Appendix we pro- vide an explanation of each step for complete understand- ing of those interested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Here, we only briefly describe some of the parts required for our exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this section, we briefly describe how to apply FEM in EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' First, we remark that many variants can arise due to possible different assumptions made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 11 First, we discretized the subject domain Ω into smaller elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Next, we approximate our solutions u, U by uh(x, y) = N � i=1 αiφi(x, y) (23) V h = L−1 � k=1 βkηk, (24) where φi, ηk are basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In particular, the ηk are defined through η1 = (1, −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', 0)T , η2 = (1, 0, −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', 0)T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', ηL−1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', 0, −1)T ∈ RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This choice ensures that reference voltage condition (21) is ful- filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, N in (23) corresponds to the number nodes forming the finite element mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The approximate solutions uh and V h for the direct problem are fully determined by the coefficients α = [α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', αN] ∈ RN, (25) β = [β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', βL−1] ∈ RL−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (26) FEM allows us to obtain a system of linear equa- tions characterizing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This is achieved by inserting (uh, V h) into the variational equation (22), together with different choices of (v, V ) = (φi, ηj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Gathering all possi- bilities leads to a linear system of equations: Aθ = ˜I, (27) where θ = [α, β] ∈ RN+L−1 and ˜I is described through the current pattern I applied at the electrodes as follows: ˜I = �−→0 , I1 − I2, I1 − I3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', I1 − IL � ∈ RN+(L−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (28) The stiffness matrix A can be computed in terms of four blocks: A = � B1 + B2 C CT D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (29) Each term is defined through integration over the do- main and over the electrodes like: B1 ij = � Ω σ∇φi · ∇φj dx, i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', N (30) B2 ij = L � l=1 1 zl � El φiφj dS, i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', N (31) Cij = − � 1 z1 � E1 φi dS − 1 zj+1 � Ej+1 φi dS � , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', N, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L − 1 (32) Dij = � E1 z1 , i ̸= j E1 z1 + |Ej+1 zj+1 , i = j , i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', L − 1, (33) with |Ej| being the electrode area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The derivation of each block arises from application of two different basis functions on the weak formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A full description was done in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' After solving the system for θ, the voltages V h are ob- tained by multiplication with the basis functions matrix M defined as: M = � ������� 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 1 −1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 0 0 −1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' −1 � ������� (34) through V h = Mβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' One detail we want to point out regarding FEM imple- mentation concerns conductivity parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For computational purposes, we assume that σ is piece-wise constant, meaning that at each mesh element is constant, and thus mathematically defined as: σ(x, y) = K � k=1 σkχk(x, y), (35) where K is the total number of elements and χk is the indicator function of the k-th element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this sense, matrix B1 simplifies to B1 ij = � {k: i,j∈Tk} σk � Tk ∇φi · ∇φj dx (36) The parameterization of σ is essential to compute the voltages variation V h with respect to a conductivity variation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' If a parameterization was not applied to σ, then it would be described as a function from Ω to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For the latter case a derivative still exists, but it is more theoretically described, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='2 Implementation Details For implementation purposes we restrict ourselves to two- dimensions even though the above formulation also holds for further dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first implementation decision is about space dis- cretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' For simplicity sake, our choice of mesh gener- ator is DistMesh algorithm, developed by Per-Olof Persson and Gilbert Strang [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The elements are triangles and the algorithm has been adapted to consider L equidistant elec- trodes, with a pre-defined size, at the surface ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Secondly, we need to define our basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We choose piece-wise linear functions, and therefore, for each triangle element any basis function is linearly defined as: φi(x, y) = ai + bix + ciy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 12 Moreover, the basis function are obtained in correspon- dence to a mesh node (xj, yj) through the condition: φi(xj, yj) = � 1, i = j 0, i ̸= j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (37) Since, for every other node the function will be zero, it holds that for every triangle that does not have i as a node, φi ≡ 0 there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This simplifies the computations of all the matrices, since most entries will be 0, due to non- intersection of most basis functions supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, A is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Moreover, due to the equation nature being elliptic, the stiffness matrix A is positive-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, the most appropriate system of equations solver is the Conjugate Gradient method (CG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' B Derivation of Levenberg- Marquardt method A simple method for inverse problems under such an op- timization framework is Levenberg-Marquardt method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It is a general method since it is independent of the simulator and the method used for differentiating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, it allows us to demonstrate the effectiveness of vari- ous methods to compute the derivatives, in particular, of Automatic Differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We hereby assume that σ is discretely given by a pa- rameterization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', σ ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This simplifies simulation and, more importantly, the derivatives computation pro- cess which is now done with respect to each variable σi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' An example is seen in Figure 1 where σ = (σ1, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The minimization problem is given as min σ 1 2 ��Sim(σ) − mtrue��2 2 , (38) where mtrue is a set of true measured voltages with respect to N currents applied, as already introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We iteratively improve an approximate solution of the minimization problem (2) through σk+1 = σk + δσk (39) where δσk is an update step and σk is the current approx- imate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This process is done until a satisfactory solution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Each method to solve the minimization problem is de- fined by the update step δσk computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Levenberg-Marquardt is a second order quasi-newton method, that approximates the Hessian through an iden- tity regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this sense, the update rule is given as follows: δσLM = − � J(σ)T J(σ) + λLMI �−1 J(σ)T (Sim(σ) − mtrue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (40) Here, J(σ) denotes the Jacobian of Sim, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', a matrix of voltage derivatives with respect to each parameter σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, λLM is a parameter used to approximate the Hes- sian and that allows for improving the condition number of J(σ)T J(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We determine it empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The update rule derivation is given in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The Levenberg-Marquardt method is a particular type of quasi-Newton methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We start by deducing the gen- eral form of quasi-Newton methods and there after funnel on our chosen method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We hereby assume that σ is discretely given by a pa- rameterization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', σ ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This simplifies simulation and, more importantly, the derivatives computation pro- cess which is now done with respect to each variable σi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' An example is seen in Figure 1 where σ = (σ1, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The minimization problem is given as min σ 1 2 ��Sim(σ) − mtrue��2 2 , (41) where mtrue is a set of true measured voltages with respect to N currents applied, as already introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Denote by L(σ) the loss function in (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Then, assum- ing that we have an initial guess σ0, we can re-write (38) as L(σ + δσ) = 1 2 ��Sim(σ0 + δσ) − mtrue��2 2 (42) with an intent to minimize with respect to the parameter variation δσ, which we designate by update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' There- after, applying this iteratively we approximate our solu- tion through σk+1 = σk + δσk (43) until a satisfactory solution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The Levenberg-Marquardt Algorithm is essential for to compute the update step δσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Taylor expansion of (42) up to quadratic term is given by L(σ + δσ) = L(σ) + L′(σ)δσ + 1 2L′′(σ)(δσ)2, (44) where L′(σ) and L′′(σ) denotes the gradient and Hessian of the objective function L, with respect to parameters defining σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A minimum with respect to δσ has gradient zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Thus, we apply gradient to (44) ∂L ∂δσ (σ + δσ) = L′(σ) + L′′(σ)δσ, and setting the gradient equal to zero yields 0 = L′(σ) + L′′(σ)δσ ⇔ δσ = − [L′′(σ)]−1 L′(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 13 Since only Sim depends on conductivity parameteriza- tion we can compute the gradient and Hessian through: L′(σ) = J(σ)T � Sim(σ) − mtrue� (45) L′′(σ) = J(σ)T J(σ)+ � i [Simi(σ)]′′ � Simi(σ) − mtrue i � , (46) where J is the Jacobian of simulated voltages Sim(σ) with respect to the parameterization of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Up until here the derivation is general for quasi-Newton methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The Levenberg-Marquardt method distinguishes itself from other quasi-Newton methods by avoiding computa- tion second order derivative, substituting it by a scaled identity matrix λLMI, λ ∈ R+, which acts as a regular- izer by improving the condition number of the Hessian matrix to be inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Now, the update can be computed through: δσLM = − � J(σ)T J(σ) + λLMI �−1 J(σ)T (Sim(σ) − mtrue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' (47) C Automatic Differentiation AD is a set of techniques to evaluate the derivative of a function specified by a computer program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' No matter how complicated they are, any computer program is based on a simple set of arithmetic operations and functions, like ad- dition, multiplication, trigonometric functions, exponen- tials, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' We can encode the derivative rule for all of these simple operations and build up the full derivative of our complex program through the chain-rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' AD evalu- ates derivatives with exact precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' There are two modes for AD implementation: forward- mode and reverse-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In any case, they are not hard to implement through operator overloading techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The difficult part is to provide an efficient and optimal compu- tation of these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' However, at the present moment there are great libraries that provide efficient implemen- tations of AD for both modes, like JAX for Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first step in AD is the creation of a computa- tional graph of our program, that explains the decom- position into simpler operations for which we know the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Let’s exemplify for the following function f(x1, x2) = sin(x1 · x2) + ex1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The first step is to break things apart into the simpler operations: w1 = x1, w2 = x2 w3 = w1 · w2 w4 = sin(w3) w5 = ew1 w6 = w4 + w5 =: f(w1, w2) This decomposition is more easily visualized through the computational graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 9: Computational Graph of f(x1, x2) = sin(x1 · x2) + ex1 evaluated at (π/2, −3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' With the computational graph in mind, forward-mode computes derivatives from bottom-to-top, that is from the variables to output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, it allows the derivative com- putation of all outputs with respect to a single variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' It can evaluate the derivative simultaneously with the func- tion, and thus it is proportional to the original code com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In this terms, it is more efficient for functions f : Rn → Rm with m >> n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Reverse-mode of AD works the other way around, that it is, top-to-bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' First, it requires a forward evaluation of all the variables, and thereafter it starts computing the derivatives from output values for the variables involved immediately, doing that successively until the input vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Therefore, it allows evaluation of the gradient of an single output function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, it is way more efficient for functions f : Rn → Rm with m << n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' A familiar example in these days is neural networks that are described by way more weights that output variables, in this particular example the reverse-mode is known as backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' One possible limitation to take into account in AD arises from the computational graph we described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Due to the computer program complexity this computational graph can be very expensive to establish and keep in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' In such scenarios, where the Jacobian is obtained from a very complex graph, instead of a compact formula like analytic formulation, it can take a long time to be evalu- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' As such, AD is not a tool to be inserted into play whenever needed and considerations must be made when implementing the Sim operator, to avoid some of these flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To bypass this problem, JAX can encode loops and con- ditionals in primitive operations that are inherent from the domain-specific compilers for linear algebra (XLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Oth- erwise, the loops are unrolled into a set of operations (may be smaller than the general loop, but) that increases the computational graph size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' With the primitives in mind, 14 f(x,y) data 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='8105 sin(x*y) data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0000 x*y exp(x) data -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='7124 data 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='8105 y x data -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0000 data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='5708this will be encoded on the graph with a single operation, for which we already know the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' With AD the focus is completely in an optimal imple- mentation of the Sim operator, which is essential to ob- tain a very efficient inverse problem solver (even with an- alytical computation of derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Thereafter, thinking about both modes, we can apply forward-mode to com- pute efficiently the derivatives of Sim with respect to the parameterization (r, cx, cy, σin, σout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Being aware of the inherent problems with both meth- ods is essential for a proper implementation of the inverse solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' D Extended Results In this section we present some extra analysis about the Jacobian computation with both methods in order to make a sanity check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The sanity check we want to verify is to check if the Jacobian computed through automatic differentiation and the analytic formulation match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This is what we already expect since AD applies the chain-rule of differentiation to FEM, which is exactly what we have done by hand to determine the analytic formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' The Frobenius norm of the Jacobian difference is given as: ��JAD − Janalytic�� F ro , where ∥A∥F ro = � � n,m � i,j |aij|2 � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, we computed the Jacobian with both meth- ods for 100 randomly generated general conductivities de- scribed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Thereafter, we compute their differ- ence and applied the Frobenius norm in order to obtain an array with dimension 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' To verify the assumption that both should evaluate to almost the same values we make an histogram of the losses and provide some statistics, namely, mean, variance, max- imum and minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' This results are provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 10 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Statistically we can infer that the Jacobian match closely together with maximum error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0552 and an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Indeed, the histogram confirms that most evaluations are really close together, with only some outliers compared with the overall picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Further, these outliers might just be rounding off errors and are not wor- risome since the error is still considerably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 15 Mean S2 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Error Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Error ��JAD − Janalytic�� F ro 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0271 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='94e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content='0146 Table 3: Statistic analysis of the error between Jacobian matrices obtained through the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' Figure 10: Histogram of Jacobian error with both derivative methods evaluated with Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} +page_content=' 16 25 20 15 Count 10 5 2 × 10~2 3 × 10-2 4 × 102 6 × 10-2 MSE( JAD, janalytic)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FIT4oBgHgl3EQf-Suu/content/2301.11410v1.pdf'} diff --git a/aNE3T4oBgHgl3EQf2gu7/content/tmp_files/2301.04756v1.pdf.txt b/aNE3T4oBgHgl3EQf2gu7/content/tmp_files/2301.04756v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..402e673ee6770338c82fd6aee57a3b9b9b35e3a8 --- /dev/null +++ b/aNE3T4oBgHgl3EQf2gu7/content/tmp_files/2301.04756v1.pdf.txt @@ -0,0 +1,593 @@ +arXiv:2301.04756v1 [astro-ph.IM] 11 Jan 2023 +2023 January 11 +Techniques for Measuring Parallax and Proper Motion with VLBI +M. J. Reid1 +ABSTRACT +Astrometry at centimeter wavelengths using Very Long Baseline Interferom- +etry is approaching accuracies of ∼1 µas for the angle between a target and a +calibrator source separated by <∼1◦ on the sky. +The BeSSeL Survey and the +Japanese VERA project are using this to map the spiral structure of the Milky +Way by measuring trigonometric parallaxes of hundreds of maser sources associ- +ated with massive, young stars. This paper outlines how µas astrometry is done, +including details regarding the scheduling of observations, calibration of data, +and measuring positions. +Subject headings: astrometry – parallaxes – methods: data analysis – techniques: +interferometric – atmospheric effects +1. +Introduction +This paper focuses on the techniques of differential astrometry using Very Long Baseline +Interferometry (VLBI) at centimeter wavelengths. Here I use the term differential astrom- +etry to mean measuring the angular difference between a target source and one or more +background calibrator sources nearby in angle. Target sources are often molecular masers +(associated with young stars, red giants, and extragalactic sources), X-ray binaries, and +active galactic nuclei and quasars. Calibrator sources are usually quasars at sufficient dis- +tances that they have negligible proper motion and, thus, can yield “absolute” positions and +motions. +Here I will discuss the interferometric phase, φ, as the observable, in contrast to using +interferometric group delay for geodesy and “whole sky” astrometry. Phase delay, τp, is a +monochromatic measure of interferometric delay at frequency ν and is given by +τp = 1 +2π +φ +ν +, +1Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA + +– 2 – +for φ in radians. An interferometer phase shift is equivalent to a shift along a sinusoidal +interference pattern, and one cannot discriminate φ from φ ± n 2π, where n is an arbitrary +integer. In contrast, group delay is given by +τg = 1 +2π +∂φ +∂ν +, +which requires a significant bandwidth to measure. Thus, group delay is the peak of the +broadband response, often called a delay or bandwidth pattern (see, e.g., Thompson, Moran & Swenson +2017). In principle, phase delays can be more precise than group delays by a factor of ∆ν/ν, +where ∆ν is the total spanned observing bandwidth (used for group delays) and ν is the +center observing frequency. However, since absolute phase delays are not usually measur- +able (owing to 2π ambiguities), use of phase delay is generally limited to differential tech- +niques. Differencing the phase between a target (T) and a calibrator (C) nearby on the sky +(i.e., ∆φ = φT − φC) reduces most systematic sources of error by a factor of the angular +difference, ∆θ (in radians), between target and calibrator. +For detailed reviews of radio astrometry see Reid & Honma (2014) and Rioja & Dodson +(2020). Here, instead, I discuss many practical details and lessons learned over the past +several decades. Section 2 covers basic requirements for accurate VLBI astrometry, including +some useful rules-of-thumb related to limiting sources of error. In Section 3, I discuss optimal +strategies for scheduling parallax observations, as well as what calibrations are essential for +high astrometric accuracy. +Next, Section 4 walks the reader through these calibrations, +including examples of problems to watch for in real data. Some miscellaneous “things to +consider” are covered in Section 5, and I conclude with thoughts related to the future in +Section 6. +2. +Basics +Estimating a position (offset) for a point-like source will be limited by random (thermal) +noise and systematics. In an image, the central position of a Gaussian brightness distribution +can be estimated with a precision no better than about 0.5 Θ/SNR, where Θ is the Gaussian +full-width at half-maximum (FWHM) and SNR is the peak brightness divided by the image +(root-mean-square) noise level (see Eq. (1) of Reid et al. 1988). However, for moderately +strong sources, astrometric accuracy is often not limited by thermal noise. For example, +a 10 mJy source in a Very Long Baseline Array (VLBA) image with 0.1 mJy noise (i.e., +SNR=100) and a beam Θ = 1 mas would be limited to a precision of 5 µas. Even with current +calibration techniques and reference sources separated by only 1◦ on the sky, systematic errors +from uncompensated interferometric delays, usually owing to the propagation of the signal + +– 3 – +through the Earth’s atmosphere, can lead to position errors ∼ 10 µas. +For signal propagation through the troposphere, interferometric position error, ∆θ, +scales as follows: +∆θ ∼ θf × c∆τ +λ +, +(1) +where θf = λ/B is the fringe spacing, λ is the observing wavelength, B is baseline length, c +is the speed of light, and ∆τ is the propagation delay error. Essentially, Eq. (1) states that +a path delay of one wavelength shifts a position by one fringe. Note that for non-dispersive +delays wavelength cancels out, indicating that astrometric accuracy is not dependent on +fringe spacing (angular resolution); instead, accuracy is improved by decreasing delay errors +and/or increasing baseline length. While this holds for non-dispersive delays in the neutral +atmosphere, it does not hold for dispersive delays from the ionosphere. +A prerequisite for differential astromety is an accurate model for interferometric delays. +This involves knowledge of antenna locations, absolute source coordinates, Earth’s orienta- +tion parameters, atmospheric propagation delays, clock errors, etc. Of particular importance +is the accuracy of the position of the source used for phase referencing. While to first order a +position error for a source shifts all other sources referenced to it by the same amount, this is +not true to second order in the shift as the interferometer geometry changes with the position +on the sky. Indeed, a phase shift owing to a coordinate error at the position of the reference +source cannot be modeled purely as a position offset at the position of another source, leading +to an undesirable relative position error as well as degrading its image (Beasley & Conway +1995). For relative astrometry with VLBI at cm wavelengths with sources across a couple of +degrees on the sky, a reference source error of ∼ 100 mas can dramatically degrade imaging. +To avoid these problems, one should ensure that a phase-reference source has a position +accurate to better than a few mas. If the position is not known to this accuracy prior to the +observations, one can often do a preliminary pass through the correlated data and measure +it against another source with a mas-accurate absolute position. +As the angle between a calibrator and target decreases, astrometric sensitivity to delay +errors also decreases. Rough rules-of-thumb for achieving ±10 µas relative positional accu- +racy for sources separated by 1◦ on the sky, include baselines accurate to ±1 cm, source posi- +tions accurate to ±1 mas, and propagation delays accurate to ±0.03 nsec (corresponding to +path delays of ±1 cm). For many antennas and sources used in VLBI arrays, especially those +used for geodesy and absolute reference-frame astrometry, locations and source positions of +this accuracy are readily available, and the major problem involves variable atmospheric +propagation delays. +Signals delayed when passing through the atmosphere are conveniently described as + +– 4 – +occurring in tropospheric and ionospheric layers. +When looking vertically, tropospheric +delays are dominated by dry air, which contributes roughly 200 cm of path delay (path +delay is the signal delay multiplied by the speed of light), and water vapor, which often +leads to between 5 and 20 cm of path delay depending on site, elevation, and weather +(see Thompson, Moran & Swenson 2017). While the dry air component can be accurately +estimated (from ground-based pressure) and removed in the VLBI correlator, the variable +water-vapor component can only be approximated and residual vertical path delays of ±5−10 +cm are common and can typically change by ∼1 cm hr−1. Astrometric accuracy critically +depends on estimating and removing these residual delays. +Observations using geodetic +blocks (see Section 3) can greatly reduce these path-delay errors. +The ionosphere presents a more difficult calibration problem. The application of global +models of the total electron content (TEC) above a telescope, while useful to reduce delay +errors, leaves roughly ±20% of the ionospheric delay in correlated data (Walker & Chatterjee +1999). Since these delays are dispersive, scaling with observing frequency as ν−2, they are +larger at lower frequencies. At frequencies <∼10 GHz, they usually become the dominant +source of delay error. For example, at 6.7 GHz (a strong masing transition of methanol), +residual path delays after removing a TEC model are often ∼5 cm (corresponding to 5.6 +TECUs, where a TECU = 1016 e− m−2). The most successful method to deal with iono- +spheric delays is to use the MultiView technique (Rioja et al. 2017), which uses calibra- +tors surrounding the target source to solve for and remove the effects of planar ionospheric +“wedges” (see Section 3). +3. +Approaches to Observing +While one would like to avoid phase-unstable weather (e.g., early spring mornings are +generally better than late summer evenings in the southwest U.S.), one must also take into +consideration optimal dates for parallax measurements when the Earth appears furthest +from the Sun as viewed by the source. (Note that the parallax effect is maximum when the +Sun-Earth-Source alignment traces a right triangle, which implies observations near sunrise +and sunset.) Fig. 1 shows the parallax signatures for two locations in the Galactic plane: at +Galactic longitude 135◦ in the outer Milky Way and 0◦ toward the Galactic center. While +toward 135◦ longitude, the North–South parallax offsets have nearly the amplitude of the +East–West offsets, this is not the case for the vast majority of sources in the Galactic plane. +Toward the inner Galaxy, the East–West parallax amplitude greatly exceeds that of the +North–South amplitude as exemplified by the 0◦ plot. +When the parallax amplitude in the East–West direction is significantly larger than + +– 5 – +Fig. 1.— Parallax signatures for two sources at 4 kpc distance in the Galactic plane: (top panel) at +135◦ longitude and (bottom panel) at 0◦ longitude. The red solid lines trace the East–West offsets and the +blue dashed lines trace the North–South offsets vs. time of year. The plotted points indicate near-optimal +sampling. See the text for details. +in the North–South direction, one should schedule observations near the extremes of the +East–West parallax sinusoid. An optimal sampling of this sinusoid would be to place one +observation at an extremum, followed by two observations six months later, and then a +fourth one year later than the first. This scheme decorrelates all parameters in a parallax +and proper motion solution. However, an undesirable aspect of using only four observations +is that it leaves only one degree-of-freedom (after solving for offset, motion and parallax) +with which to estimate realistic parameter uncertainties from post-fit residuals (see a detailed +discussion in Reid et al. 2017). Better would be to use eight observations by doubling-up +on each of the four described above, as indicated by the additional open circles in the lower +plot of Fig. 1. +The parallax sequences just discussed assume that the sources are detectable over a full +year of observation. This is usually not an issue for methanol masers, but can be a problem +for water masers. Assuming water masers come and go on a timescale of seven months, an +optimized sequence starts at an East–West peak and includes six epochs at offset days of 0, +91, 155, 189, 241, 364 (or the reverse). This was determined by evaluating parallax accuracy +for thousands of Monte Carlo simulations, in which the dates of observation were chosen +randomly. + +– 6 – +A single astrometric observing track should include three types of observations: +3.1. +Fringe finders +Fringe-finder (a.k.a. manual phase-calibrator) sources are strong, compact sources which +are used to “find” interferometric fringes and establish clock offsets and drifts at each antenna +for use in the correlator. These sources should have mas-accurate positions, so as to avoid +residual delays changing significantly over an observing track, and, ideally, little source +structure. After correlation, one (or more) of these scans are used to estimate and remove +electronic delay and phase differences among separate intermediate frequency (IF) bands. +Always have at least 2 different fringe-finders observed briefly (e.g., 1 minute on-source) +every couple of hours. +3.2. +Geodetic blocks +Geodetic blocks typically consist of rapid observations of a dozen calibrators spread +across the sky and observed over a large range of source zenith angles (θZA). The goal is to +measure broadband (group) delays and fit for clock and residual atmospheric path-delays to +cm-level precision (Reid et al. 2009). For a given signal-to-noise ratio, group-delay precision +scales inversely with spanned bandwidth. Thus, one should maximize the spanned bandwidth +for best precision, even if this involves gaps in the frequency coverage. Since fitting delay +involves Fourier transforming frequency to delay, the frequency coverage should minimize +delay sidelobes. For example, four IF bands, centered at relative frequencies of 0, 1, 4 and 6 +×∆f, yield a “minimum-redundancy” sampling of integer multiples of 1 through 6 ∆f. All +calibrators must have positions accurate to <∼ 1 mas, in order for measured residual delays +to be dominated by atmospheric, not position, errors. +It is best to observe geodetic blocks at a high frequency (e.g., 24 GHz), so as to minimize +the contribution of the dispersive delay from the ionosphere. +For dispersive delays, the +interferometer phase at frequency ν is shifted by +φ = −c re +ν +Ne , +where c is the speed of light, re is the classical electron radius, and Ne is the electron column +density along the line of sight. Note that the phase shift is negative. The phase delay, τp, is +given by +τp = 1 +2π +φ +ν = − c re +2πν2 Ne , + +– 7 – +and the group delay, τg, is given by +τg = 1 +2π +∂φ +∂ν = + c re +2πν2 Ne . +Note that the group delay has the opposite sign of the phase delay (and the radio signal +phase shift) for a dispersive delay. Combining the phase shift and group delay formulae +yields the relation +φ = −2π ν τg . +In other words, when correcting the interferometer phase for a change in ionospheric group +delay, one must flip the sign of the correction compared to a change in tropospheric delay. +With a mix of dispersive and non-dispersive delays, corrections for group delays can be +combined, but phase-delays cannot. +If one observes geodetic blocks at frequencies below about 15 GHz, it is best to measure +multi-band delays at two well-spaced frequencies in order to estimate and remove the disper- +sive component of the delay from the total delay, yielding pure non-dispersive delay for each +source. Note that solving for a zenith delay-error with pure dispersive delays, and then using +an ionospheric mapping function to predict delays along a ray-path to a source, does not +seem to improve astrometric accuracy. For example, at the high altitude of the ionosphere, +rays from a source at, say, a zenith angle of 45◦ will pass through the ionosphere ≈ 400 km +from the antenna site, likely creating a significant azimuthal dependence. It may be possible +to solve for and remove an azimuthal dependence, but this has not yet been demonstrated. +3.3. +Phase-reference blocks +Phase-reference blocks involve cycling between sources well within the interferometer +coherence time, which is usually limited by short-term fluctuations in tropospheric water +vapor and produce phase changes proportional to observing frequency. For reasonably dry +sites, the time spanned between (the centers of the) phase-reference scans should be less than +about 40 s at 43 GHz, 80 s at 22 GHz, and 260 s at 6.7 GHz. For example, if a calibrator (C) is +the phase reference for a target (T), the sequence could be C, T, C, T, ... with scan durations +of 40 s at 22 GHz. Note that scan durations include slew and settle times, so on-source +(dwell) times will be less. For a strong target and weak calibrator, so-called inverse phase +referencing can be used: T, C, T, C, ... . At frequencies well above 10 GHz, such sequences +are adequate, and if more than one calibrator can be found within about 2◦ of the target, +they can be used to give quasi-independent relative positions (provided the calibrators are +well separated on the sky) and to provide a check on potential problems associated with +jet-like structures in some calibrators. + +– 8 – +At frequencies below ∼10 GHz, MultiView can yield excellent astrometric accuracy +(Rioja et al. 2017). Ionospheric electron densities follow the Sun, and residual path-delays +can be approximated by ionospheric “wedges” and modeled with linear gradients (a tilted +plane) on the sky. Using calibrators which surround the target, an observing sequence with +four calibrators could be as follows: C1, C2, T, C3, C4, T, ... . This approach involves fitting +the phases from the four calibrators nearest in time to the target for a given interferometer +baseline to a tilted phase-plane: +φC = φT + Sx∆x + Sy∆y , +(2) +where φT is the desired phase at the target position, Sx and Sy are phase-gradients in the x +and y directions, and ∆x and ∆y are angular offsets from the target position. +In general, three calibrators are the minimum number needed to fit a tilted phase plane. +In the example above, I mentioned using four calibrators, as this provides the opportunity +to identify and remove a “bad” calibrator, possibly owing to large phases associated with +complex source structures. Interestingly, however, only two calibrators can be used, provided +they nicely straddle the target source, such that a line between them passes very close to +the target. +For MultiView to succeed, the entire cycle of calibrators and target must be completed in +well under the interferometer coherence time (e.g. the time it takes for phases to wander by 1 +radian). For example, at 1.6 GHz, Rioja et al. (2017) successfully used a cycle time of 300 s. +It is critical that the phases on each calibrator are dominated by propagation delays and not, +for example, by position errors or source structure. So, calibrator positions must be known +to a small fraction of the interferometer fringe spacing on the longest baseline. Note, the +calibrator positions can be updated by measuring relative positions to one of the calibrators +(or possibly the target), which has a well determined absolute position. Calibrator phases +can “wrap” past ±180◦, owing to position or baseline errors or large residual tropospheric +or ionospheric delays in the correlation process, and these phase wraps must be accounted +for when fitting the phase-plane. +If the target source is sufficiently strong to use as the interferometer phase refer- +ence, inverse MultiView has some advantages. An observing sequence can be as follows: +T, C1, T, C2, T, C3, T, C4, ... , and only the time between the centers of the target (T) scans +needs to be less than the interferometer coherence time. Also, phase-wrap problems are re- +duced, compared to standard MultiView, as atmospheric phase-shifts should largely cancel +between the target and each calibrator. Hyland et al. (2022) demonstrate that calibrators +separated by up to 7◦ from the target can yield excellent results at 8 GHz. Typically this +allows for many useful calibrators. + +– 9 – +4. +Data Calibration +4.1. +Geodetic Block Analysis +The first step in calibration is to estimate clock and residual tropospheric delays at +each antenna. +This can be done with the geodetic-block data. +Start by correcting the +data for updated Earth’s orientation parameters, antenna locations, source positions, and +feed rotation. Also, ionospheric delays (usually not applied during correlation) should be +removed, based on total electron content models. Then, electronic delays and phase offsets +among different IFs and feeds should be “aligned” using a “comb” of frequencies inserted +during the observations or performing a “manual” phase-calibration using a fringe-finder +source. +Next, multi-band delays for a source on a baseline involving antennas i and j, τi,j = +τj − τi , are fitted with antenna dependent clock parameters and atmospheric zenith delay- +errors, where each antenna’s delay is given by +τ = T0 + dT +dt ∆T + τ0 sec θZA , +(3) +assuming other sources of delay error are small. The first two terms in Eq. (3) correspond +to a clock offset, T0, and a linear clock drift over time, +dT +dt , and ∆T is time relative to a +reference time, best chosen near the center of the observations; the last term is the zenith +(vertical) atmosphreric delay error scaled by the secant of the source zenith angle, sec θZA, +for the increased (slant) path-length through the troposphere.2. Note, with interferometric +data the clock parameters for one (reference) antenna must be held constant and usually are +set to zero. However, since θZA varies differently for each VLBI antenna, one can solve for +the zenith delay at all antennas. Geodetic blocks with a dozen calibrators can take about 30 +minutes to complete, depending strongly on antenna slew speeds, and should be done every +2–3 hr in order to monitor and correct for changing conditions. +An often overlooked aspect of differential astrometry is that the tropospheric delay +difference between a target and a calibrator at a given antenna is given by +∆τ ≈ τ0 +∂ sec θZA +∂θZA +∆θZA = τ0 sec θZA tan θZA ∆θZA . +2While sec θZA is appropriate for a plane-parallel atmosphere and is a reasonable approximation +for slant path delay, better θZA “mapping functions” which account for the Earth’s curvature and fi- +nite atmospheric thickness are a significant improvement for observations at large zenith angles; see +Thompson, Moran & Swenson (2017). + +– 10 – +At large zenith angles, both sec θZA and tan θZA blow up. +While this can be leveraged +to increase geodetic block sensitivity, large θZA observations should be avoided for phase- +referencing astrometry. +Fig. 2.— Geodetic block examples. Plotted are multi-band (group) delays (left panels) and residual fringe +rates (right panels) vs. time. Measurements are green dots, best-fitting model values are blue circles, and +“data minus model” residuals are red stars. The top row is for a well-behaved baseline, whereas the bottom +row shows a serious problem – a clock jump between 15:05 and 16:15 UT. +Examples of geodetic block data from two baselines of VLBA observations are shown in +Fig. 2. Data were taken near 3 and 7 GHz and the dispersive component of delay (owing to +ionospheric electrons) was calculated and subtracted from each measurement, leaving pure +non-disperive delays. The left panels plot multi-band (group) delays and the right panels +show fringe rates. Green dots are the measurements, blue circles are the fitted model and red +symbols are residuals. The upper plots (for the baseline with antennas 5 and 8) show good +data; one can see a uniform clock drift of about +0.3 nsec hr−1 and most residuals are <∼ 0.1 +nsec (or <∼ 3 cm of path delay). The lower plots (for the baseline with antennas 6 and 8) show +a serious problem. There is a large clock-drift rate of −2.5 nsec hr−1 evident between 11 and +15 UT (corroborated by an average fringe rate of about −3 mHz, corresponding to ν dT +dt for +observations at ν = 5 GHz). While this is not necessarily a problem, there is a large clock +jump between the third and fourth geodetic blocks (i.e., somewhere between 15:05 and 16:15 +UT). Note that the residuals, while not as small as for baseline 5–8, are reasonably close to +zero. This can happen as spurious large zenith-delay parameters can partially compensate +for mismodeling the clock as a linear drift. So, simply looking for small residuals is not +sufficient to spot problems. Clearly the clock jump occurred at antenna 6 (since antenna 8 +is common to both baselines and the jump does not show up in baseline 5–8). Therefore, +data involving antenna 6 from 15:05 to the end should be flagged, both in the geodetic block +and phase-referencing files, and the geodetic block data should be re-fitted. + +– 11 – +Finally, it is always a good idea to check that the clock and tropospheric delay corrections +work well by applying them to the geodetic block data, re-fitting for multi-band delays, and +verifying that the residual delays are <∼ 0.1 nsec. Alternatively, since many geodetic block +sources are strong, after applying the corrections, one can simply plot phase versus frequency +across all IFs and feeds and visually verify that the response is nearly flat. +4.2. +Phase-referencing Block Analysis +The phase-referencing data require standard calibration steps3, with the addition of tro- +pospheric delay calibration using the results of the geodetic-block fits, and for low-frequency +observations substitution of MultiView for standard phase referencing. For example, cali- +bration steps could be as follows: +1. Flag any bad data. +2. Convert interferometer visibilities (correlation coefficients) to Jansky units, using mea- +sured system temperatures and antenna gain curves, as well as any other known am- +plitude corrections (e.g., digitization loses). +3. Adjust delays and phases as done in the preliminary calibration of the geodetic block +data (using updated Earth’s orientation parameters, antenna locations and source po- +sitions, and applying ionospheric TEC models.). +4. Correct phases for antenna feed rotations; for example, for circularly polarized feeds +shift phases by the feed parallactic angles with opposite signs for right and left cir- +cularly polarized feeds (and be sure to verify the proper sign convention). Note, this +holds for linearly polarized feeds correlated against circularly polarized feeds, but for +linear against linear feeds there is no phase shift induced by feed rotation. The con- +version of VLBI data with some or all antennas having linearly polarized feeds to a +common circularly polarized basis is complicated and deserves its own paper (see e.g., +Mart´ı-Vidal et al. 2016). +5. Apply clock and tropospheric delay corrections based on fitting geodetic block data. +Then remove electronic delays and phases among IFs and feeds using either a cali- +bration frequency comb (inserted at the receiver) or using a manual phase-calibration +3see http://www.aips.nrao.edu/vlbarun.shtml for examples + +– 12 – +on a strong source. For spectral-line sources (e.g., masers) a subtle but serious prob- +lem can occur in this step. If a second-pass correlation is done which retains only a +portion of the total IF bandwidth (so-called “zoom mode” used to limit the number +of spectral channels when high spectral-resolution is desired), then one should also +correlate the manual phase-calibration source(s) in the same manner and apply them +to the spectral-line data. One cannot directly transfer the manual phase-calibration +from broadband to narrowband data without considering the frequency location of the +narrowband data relative to the edge frequency of its associated broadband, and some +software analysis packages (i.e., AIPS and CASA) do not have this capability. +Fig. 3.— Interferometer reference phase simulations. Each dot represents a phase measured on a calibra- +tor, to be interpolated between two adjacent (closely spaced) measurements and removed from the target +source. A strong target can be used as the phase reference for a weak calibrator, so-called “inverse phase +referencing,” if desired. Top panel: high signal-to-noise phases showing small variations over ∼ 5 minutes +and larger variations over hours. For high-quality differential astrometry, reference phases should resemble +these. Middle panel: the same phases but generated with low signal-to-noise and these are far from optimal +for phase referencing. Bottom panel: high signal-to-noise phases, but with a high residual fringe-rate as +evidenced by rapid “phase wrapping.” These indicate a large position error for the phase-reference source, +or other geometric or clock errors, and are not suitable for accurate differential astrometry. +6. If the local oscillator was held constant during the observing track, then spectral-line + +– 13 – +sources will shift in frequency owing to the Earth’s rotation and orbit about the Sun. +Then interferometer visibilities should be corrected to hold a spectral line steady in +frequency. If the visibilities are in the delay-lag domain, V (τ), this can be accomplished +by an appropriate linear shift of visibility phase as a function of delay-lag, which +corresponds to a shift in spectral frequency (by the Fourier transform shift theorem). +If, instead, the visibilities are in the frequency domain, V (ν), one can Fourier transform +to delay-lag, perform the linear phase shift, and then inverse Fourier transform back +to frequency. +7. Phase reference the data. This can use a single source, either a calibrator (standard +phase-referencing) or the target (inverse phase-referencing), or it can involve multiple +calibrators (MultiView). Fig. 3 shows three simulated examples of reference phases +calculated from a single source. The top panel shows high SNR phases which might +be obtained at ∼1 cm wavelength. The slow phase wander of hundreds of degrees of +phase over hours is typical of path delays of ∼1 wavelength owing to variable water +vapor. Within groups spanning about 15 minutes, the phases look “worm-like,” which +comes from small-scale fluctuations in water vapor, and one can reliably interpolate +between individual points where the source to be referenced was observed. +The middle panel of Fig. 3 shows the same simulated data, except a low SNR of unity +was used for each point. Interpolation between adjacent points would be very inaccu- +rate and, when applied to another source, imaging would be considerably degraded. +The bottom panel shows the effects of a large error in the position of the reference +source, as evidenced by the “patterned” appearance made by rapid phase wrapping. +Other errors, such as incorrect baselines or uncompensated clock drifts, can also do +this. In this case, differential astrometric accuracy would be degraded by second-order +effects of phase referencing as discussed earlier. +As outlined in Section 3, at frequencies below ∼ 10 GHz, path-delays from ionospheric +“wedges” can be removed from data with the MultiView calibration technique. Fig. +4 shows simulated phases on one baseline at an instant in time for four calibrators +surrounding a target. A tilted plane fitted to the calibrator phases as a function of +(X, Y )-offset yields a phase of 50◦ at the position of the target source, which could +then be removed from the target source data prior to astrometric measurements. +As discussed in Section 3, phase wraps can be a serious problem. Fig. 5 gives a simple +example of a phase-wrap issue. +The phase for the C3 calibrator, coming from the +arctan(Vimag, Vreal) of an interferometer visibility, is generally defined between −180◦ +and +180◦ and, in this example, is −175◦. However, this phase should be +185◦ (i.e., +−175◦ + 360◦) and must be set to that value when fitting a MultiView tilted-plane + +– 14 – +Fig. 4.— MultiView phase-calibration simulation of an ionospheric wedge with gradients of 10◦ and −5◦ +of phase per degree of angular offset in the X− and Y −directions. Calibration sources are offset from the +target at (X, Y ) = (0◦, 0◦) by (−4◦, +3◦) for C1, (−2◦, −1◦) for C2, (+3◦, −3◦) for C3, and (+2◦, +3◦) for +C4. Fitting a tilted plane to the phases of the four calibrators (ψC1, ψC2, ψC3, and ψC4) returns the 50◦ +phase at the origin to be removed from the target source. In general, the minimum number of calibrators +needed to define the phase plane is three, but it can be useful to use four to test for a “bad” calibrator (e.g., +with source-structure phase). In cases where the line between two calibrators comes very close to the target +source, one can use only two calibrators. +to all calibrator phases. Otherwise, MultiView would return a spurious phase to the +target. +If the target source is sufficiently strong and compact so as to yield a high SNR in +a single scan, it can be used as a “preliminary” phase reference for the surrounding +calibrators (so-called inverse MultiView). This has the important advantage of pre- +calibrating the data on a shorter time-scale than standard MultiView, which requires +cycling around all calibrators and the target well within the interferometer coherence- +time limitation. This pre-calibration also results in “differenced phases,” which can +diminish phase-wrap problems. The next step in inverse MultiView is the same as stan- +dard MultiView, i.e. fitting a tilted plane to the differenced phases of the calibrators +and subtracting the phase offset at the target position from the target phases (which +were zero after the pre-calibration step). This returns “clean” phases, which give the +desired position information for the target. Hyland et al. (2022) have demonstrated +that inverse MultiView can work well for 8 GHz observations and achieve ≈ 20 µas +(single-epoch) astrometry. + +– 15 – +Fig. 5.— MultiView phase-calibration simulation, similar to that in Fig. 4, but with larger gradients of +30◦ and −15◦ of phase per degree of angular offset in the X− and Y −directions. Fitting a tilted plane to +the phases of the four calibrators (ψC1, ψC2, ψC3, and ψC4) would also yield the 50◦ phase at the origin to +be removed from the target source. However, with the larger phase slope, the phase for C3 is now 185◦, +which would generally be reported as −175◦. Fitting a tilted plane using that phase would result in a very +large error in the slope and in the fitted phase at the target position. Such phase-wraps must be corrected +for MultiView to work. +It is essential that the calibrators used for MultiView have position errors small enough +so as not to contribute significantly to the measured phases. If a priori positions are +only known to, say, ±0.5θf, then measured phases could have 180◦ unwanted contri- +bution from their position errors. For a fringe spacing θf ∼ 1 mas, this corresponds to +a position error of only 0.5 mas, and not all calibrators have this positional accuracy. +However, what is more important is relative position accuracy among the MultiView +calibrators. Relative position accuracy better than ±0.1θf can usually be achieved +through a first-pass of phase referencing and imaging of all sources relative to a single +source with a very accurate position. After correcting the interferometer visibilities +for the measured position offsets, one can then repeat the calibration sequence, now +with greatly reduced contributions to phases from poor positions, and then do the +MultiView step. + +– 16 – +5. +Things to Consider +A cardinal rule for astrometry used to estimate parallax and/or proper motion is to +avoid changing any observational parameters over the entire program. This offers the best +chance for canceling systematic errors. For example, if a source has structure within the +resolution of the array and either the source or the interferometer (u, v)-coverage changes +between observations, then one can get centroid shifts that are unrelated to parallax or +proper motion and can degrade their measurement. +For sources with modest structure, such as a double with separation comparable to +the interferometer resolution, accurate astrometry can usually benefit from some degree of +“super-resolution.” Often when imaging, one can use a round CLEAN restoring beam even +if the dirty beam FWHM is elliptical. I have found it useful to separate compact emission +components by setting the CLEAN restoring beam to +θF W HM = Sr +� +θmaj × θmin , +where the θmaj and θmin are the “dirty” beam’s major and minor FWHM sizes, and using a +moderate super-resolution factor, Sr, between 0.5 and 1.0. +While one can consider directly fitting interferometer phases for position offsets, it +seems advisable to always make images in order to assess whether or not a source has a +complicated structure. If a source (either the target, which might be an X-ray binary, or +a quasar calibrator) has a jet-like structure, which changes among observing epochs, one +should consider rotating the measured positions and the parallax/proper motion model to +be along and perpendicular to the jet direction (e.g., see Miller-Jones et al. 2021). This +allows one to down-weight the “corrupted” data in the jet direction and rely more on the +“clean” data in the perpendicular direction. +When measuring group delays, e.g. for geodetic block observations, one should spread +the spanned bandwidth in order to get more accurate delays. On the flip side, for phase- +reference blocks, it is better to pack the bands together in order to minimize the frequency +spanned and have less sensitivity to delay errors. Note that modern digital base-band con- +verters (DBBCs) may introduce delay jumps when changing between different setups, and +this can significantly degrade (or destroy) astrometry. If one’s DBBCs have such problems, +it might be best to observe geodetic and phase-reference blocks with the same electronic +setup. +Calibrators offset from a target in decl. are often not as good for astrometry as those +offset in R.A., since phases associated with atmospheric (zenith-angle dependent) delay errors +can more closely mimic decl. than R.A. offsets. + +– 17 – +6. +Closing Thoughts +The most accurate parallax measurements to date have uncertainties near ±5 µas (see +Table 1 of Reid et al. 2019, and references therein). In many cases, these come from mea- +surements at 22 GHz and are limited by uncompensated tropospheric delays. Since these are +likely to be quasi-random, parallax measurement can be improved with a greater number of +observations (N), with precision scaling as 1/N1/2 (if optimally sampled). Currently, it is +not clear if the non-dispersive phase-delays associated with fluctuations in water vapor are +uncorrelated over several degrees on the sky or, instead, have a “planar” structure which +could be addressed by MultiView calibration. +Three-dimensional (3D) kinematic distances offer a promising method of estimating +distances to sources well past the Galactic center, where observed motions can approach 470 +km s−1, i.e., twice the rotation speed of the Galaxy (Reid 2022). Thus, even at a distance +of 20 kpc, this leads to a proper motion of about 5 mas y−1, or over one year about 100 +times the magnitude of the parallax amplitude. For young stars hosting masers, non-circular +motions are ∼10 km s−1, and hence contribute less than a couple of percent to the distance +error budget. Usually, proper motion can be measured much more easily than parallax, in +part because motion precision scales as 1/N3/2 (for uniform sampling in time). Note, that if +one observes a source at integer year intervals, the parallax effect vanishes, simplifying the +motion measurement. Measurements with future VLBI arrays, including the SKA and the +ngVLA, may be able to obtain hundreds-to-thousands of 3D kinematic distances for very +distant Galactic sources. +REFERENCES +Beasley, A. J. & Conway, J. E. 1995, ASPC, 82, 327 +Hyland, L. J., Reid, M. J., Ellingsen, S. P. et al. 2022, ApJ, 932, 52 +Mart´ı-Vidal, I., Roy, A., Conway, J. & Zensus, A. J. 2016, A&A, 587, A143 +Miller-Jones, J. C. A., Bahramian, A., Orosz, J. A. et al. 2021, Sci., 371, 1046 +Reid, M. J., Schneps, M. H., Moran, J. M. et al. 1988, ApJ, 330, 809 +Reid, M. J., Menten, K. M., Brunthaler, A. et al. 2009, ApJ, 693, 397 +Reid, M. J. & Honma, M. 2014, ARA&A, 52, 39 +Reid, M. J., Brunthaler, A., Menten, K. M. et al. 2017, AJ, 154, 63 + +– 18 – +Reid, M. J., Menten, K. M. Brunthaler, A., et al. 2019, ApJ, 885, 131 +Reid, M. J. 2022, AJ, 164, 133 +Rioja, M. J., Dodson, R., Orosz, G., Imai, H. & Frey, S. 2017, AJ, 153, 105 +Rioja, M. J. & Dodson, R. 2020, Astron. Astrophys. Rev., 28:6 +Walker & Chatterjee 1999, VLBA Scientific Memo 23, +https://library.nrao.edu/vlbas.shtml +Thompson, A. R., Moran, J. M. & Swenson, G. W. Jr. 2017, Interferometry and Synthesis +in Radio Astronomy, 3rd ed., (Springer Open, Switzerland) +This preprint was prepared with the AAS LATEX macros v5.0. + diff --git a/aNE3T4oBgHgl3EQf2gu7/content/tmp_files/load_file.txt b/aNE3T4oBgHgl3EQf2gu7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e74e255d43dbcc95c5d018c809b7b09e21d8e77 --- /dev/null +++ b/aNE3T4oBgHgl3EQf2gu7/content/tmp_files/load_file.txt @@ -0,0 +1,460 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf,len=459 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='04756v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='IM] 11 Jan 2023 2023 January 11 Techniques for Measuring Parallax and Proper Motion with VLBI M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Reid1 ABSTRACT Astrometry at centimeter wavelengths using Very Long Baseline Interferom- etry is approaching accuracies of ∼1 µas for the angle between a target and a calibrator source separated by <∼1◦ on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The BeSSeL Survey and the Japanese VERA project are using this to map the spiral structure of the Milky Way by measuring trigonometric parallaxes of hundreds of maser sources associ- ated with massive, young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This paper outlines how µas astrometry is done, including details regarding the scheduling of observations, calibration of data, and measuring positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Subject headings: astrometry – parallaxes – methods: data analysis – techniques: interferometric – atmospheric effects 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Introduction This paper focuses on the techniques of differential astrometry using Very Long Baseline Interferometry (VLBI) at centimeter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Here I use the term differential astrom- etry to mean measuring the angular difference between a target source and one or more background calibrator sources nearby in angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Target sources are often molecular masers (associated with young stars, red giants, and extragalactic sources), X-ray binaries, and active galactic nuclei and quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Calibrator sources are usually quasars at sufficient dis- tances that they have negligible proper motion and, thus, can yield “absolute” positions and motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Here I will discuss the interferometric phase, φ, as the observable, in contrast to using interferometric group delay for geodesy and “whole sky” astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Phase delay, τp, is a monochromatic measure of interferometric delay at frequency ν and is given by τp = 1 2π φ ν , 1Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA – 2 – for φ in radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' An interferometer phase shift is equivalent to a shift along a sinusoidal interference pattern, and one cannot discriminate φ from φ ± n 2π, where n is an arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In contrast, group delay is given by τg = 1 2π ∂φ ∂ν , which requires a significant bandwidth to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Thus, group delay is the peak of the broadband response, often called a delay or bandwidth pattern (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', Thompson, Moran & Swenson 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In principle, phase delays can be more precise than group delays by a factor of ∆ν/ν, where ∆ν is the total spanned observing bandwidth (used for group delays) and ν is the center observing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' However, since absolute phase delays are not usually measur- able (owing to 2π ambiguities), use of phase delay is generally limited to differential tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Differencing the phase between a target (T) and a calibrator (C) nearby on the sky (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', ∆φ = φT − φC) reduces most systematic sources of error by a factor of the angular difference, ∆θ (in radians), between target and calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For detailed reviews of radio astrometry see Reid & Honma (2014) and Rioja & Dodson (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Here, instead, I discuss many practical details and lessons learned over the past several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Section 2 covers basic requirements for accurate VLBI astrometry, including some useful rules-of-thumb related to limiting sources of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In Section 3, I discuss optimal strategies for scheduling parallax observations, as well as what calibrations are essential for high astrometric accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Next, Section 4 walks the reader through these calibrations, including examples of problems to watch for in real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Some miscellaneous “things to consider” are covered in Section 5, and I conclude with thoughts related to the future in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Basics Estimating a position (offset) for a point-like source will be limited by random (thermal) noise and systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In an image, the central position of a Gaussian brightness distribution can be estimated with a precision no better than about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='5 Θ/SNR, where Θ is the Gaussian full-width at half-maximum (FWHM) and SNR is the peak brightness divided by the image (root-mean-square) noise level (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' (1) of Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' However, for moderately strong sources, astrometric accuracy is often not limited by thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, a 10 mJy source in a Very Long Baseline Array (VLBA) image with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='1 mJy noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', SNR=100) and a beam Θ = 1 mas would be limited to a precision of 5 µas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Even with current calibration techniques and reference sources separated by only 1◦ on the sky, systematic errors from uncompensated interferometric delays, usually owing to the propagation of the signal – 3 – through the Earth’s atmosphere, can lead to position errors ∼ 10 µas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For signal propagation through the troposphere, interferometric position error, ∆θ, scales as follows: ∆θ ∼ θf × c∆τ λ , (1) where θf = λ/B is the fringe spacing, λ is the observing wavelength, B is baseline length, c is the speed of light, and ∆τ is the propagation delay error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Essentially, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' (1) states that a path delay of one wavelength shifts a position by one fringe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note that for non-dispersive delays wavelength cancels out, indicating that astrometric accuracy is not dependent on fringe spacing (angular resolution);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' instead, accuracy is improved by decreasing delay errors and/or increasing baseline length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' While this holds for non-dispersive delays in the neutral atmosphere, it does not hold for dispersive delays from the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' A prerequisite for differential astromety is an accurate model for interferometric delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This involves knowledge of antenna locations, absolute source coordinates, Earth’s orienta- tion parameters, atmospheric propagation delays, clock errors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Of particular importance is the accuracy of the position of the source used for phase referencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' While to first order a position error for a source shifts all other sources referenced to it by the same amount, this is not true to second order in the shift as the interferometer geometry changes with the position on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Indeed, a phase shift owing to a coordinate error at the position of the reference source cannot be modeled purely as a position offset at the position of another source, leading to an undesirable relative position error as well as degrading its image (Beasley & Conway 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For relative astrometry with VLBI at cm wavelengths with sources across a couple of degrees on the sky, a reference source error of ∼ 100 mas can dramatically degrade imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' To avoid these problems, one should ensure that a phase-reference source has a position accurate to better than a few mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If the position is not known to this accuracy prior to the observations, one can often do a preliminary pass through the correlated data and measure it against another source with a mas-accurate absolute position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' As the angle between a calibrator and target decreases, astrometric sensitivity to delay errors also decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Rough rules-of-thumb for achieving ±10 µas relative positional accu- racy for sources separated by 1◦ on the sky, include baselines accurate to ±1 cm, source posi- tions accurate to ±1 mas, and propagation delays accurate to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='03 nsec (corresponding to path delays of ±1 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For many antennas and sources used in VLBI arrays, especially those used for geodesy and absolute reference-frame astrometry, locations and source positions of this accuracy are readily available, and the major problem involves variable atmospheric propagation delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Signals delayed when passing through the atmosphere are conveniently described as – 4 – occurring in tropospheric and ionospheric layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' When looking vertically, tropospheric delays are dominated by dry air, which contributes roughly 200 cm of path delay (path delay is the signal delay multiplied by the speed of light), and water vapor, which often leads to between 5 and 20 cm of path delay depending on site, elevation, and weather (see Thompson, Moran & Swenson 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' While the dry air component can be accurately estimated (from ground-based pressure) and removed in the VLBI correlator, the variable water-vapor component can only be approximated and residual vertical path delays of ±5−10 cm are common and can typically change by ∼1 cm hr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Astrometric accuracy critically depends on estimating and removing these residual delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Observations using geodetic blocks (see Section 3) can greatly reduce these path-delay errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The ionosphere presents a more difficult calibration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The application of global models of the total electron content (TEC) above a telescope, while useful to reduce delay errors, leaves roughly ±20% of the ionospheric delay in correlated data (Walker & Chatterjee 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Since these delays are dispersive, scaling with observing frequency as ν−2, they are larger at lower frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' At frequencies <∼10 GHz, they usually become the dominant source of delay error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='7 GHz (a strong masing transition of methanol), residual path delays after removing a TEC model are often ∼5 cm (corresponding to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='6 TECUs, where a TECU = 1016 e− m−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The most successful method to deal with iono- spheric delays is to use the MultiView technique (Rioja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2017), which uses calibra- tors surrounding the target source to solve for and remove the effects of planar ionospheric “wedges” (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Approaches to Observing While one would like to avoid phase-unstable weather (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', early spring mornings are generally better than late summer evenings in the southwest U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='), one must also take into consideration optimal dates for parallax measurements when the Earth appears furthest from the Sun as viewed by the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' (Note that the parallax effect is maximum when the Sun-Earth-Source alignment traces a right triangle, which implies observations near sunrise and sunset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 1 shows the parallax signatures for two locations in the Galactic plane: at Galactic longitude 135◦ in the outer Milky Way and 0◦ toward the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' While toward 135◦ longitude, the North–South parallax offsets have nearly the amplitude of the East–West offsets, this is not the case for the vast majority of sources in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Toward the inner Galaxy, the East–West parallax amplitude greatly exceeds that of the North–South amplitude as exemplified by the 0◦ plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' When the parallax amplitude in the East–West direction is significantly larger than – 5 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='— Parallax signatures for two sources at 4 kpc distance in the Galactic plane: (top panel) at 135◦ longitude and (bottom panel) at 0◦ longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The red solid lines trace the East–West offsets and the blue dashed lines trace the North–South offsets vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' time of year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The plotted points indicate near-optimal sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' See the text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' in the North–South direction, one should schedule observations near the extremes of the East–West parallax sinusoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' An optimal sampling of this sinusoid would be to place one observation at an extremum, followed by two observations six months later, and then a fourth one year later than the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This scheme decorrelates all parameters in a parallax and proper motion solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' However, an undesirable aspect of using only four observations is that it leaves only one degree-of-freedom (after solving for offset, motion and parallax) with which to estimate realistic parameter uncertainties from post-fit residuals (see a detailed discussion in Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Better would be to use eight observations by doubling-up on each of the four described above, as indicated by the additional open circles in the lower plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The parallax sequences just discussed assume that the sources are detectable over a full year of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This is usually not an issue for methanol masers, but can be a problem for water masers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Assuming water masers come and go on a timescale of seven months, an optimized sequence starts at an East–West peak and includes six epochs at offset days of 0, 91, 155, 189, 241, 364 (or the reverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This was determined by evaluating parallax accuracy for thousands of Monte Carlo simulations, in which the dates of observation were chosen randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 6 – A single astrometric observing track should include three types of observations: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fringe finders Fringe-finder (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' manual phase-calibrator) sources are strong, compact sources which are used to “find” interferometric fringes and establish clock offsets and drifts at each antenna for use in the correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' These sources should have mas-accurate positions, so as to avoid residual delays changing significantly over an observing track, and, ideally, little source structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' After correlation, one (or more) of these scans are used to estimate and remove electronic delay and phase differences among separate intermediate frequency (IF) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Always have at least 2 different fringe-finders observed briefly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', 1 minute on-source) every couple of hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Geodetic blocks Geodetic blocks typically consist of rapid observations of a dozen calibrators spread across the sky and observed over a large range of source zenith angles (θZA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The goal is to measure broadband (group) delays and fit for clock and residual atmospheric path-delays to cm-level precision (Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For a given signal-to-noise ratio, group-delay precision scales inversely with spanned bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Thus, one should maximize the spanned bandwidth for best precision, even if this involves gaps in the frequency coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Since fitting delay involves Fourier transforming frequency to delay, the frequency coverage should minimize delay sidelobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, four IF bands, centered at relative frequencies of 0, 1, 4 and 6 ×∆f, yield a “minimum-redundancy” sampling of integer multiples of 1 through 6 ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' All calibrators must have positions accurate to <∼ 1 mas, in order for measured residual delays to be dominated by atmospheric, not position, errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' It is best to observe geodetic blocks at a high frequency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', 24 GHz), so as to minimize the contribution of the dispersive delay from the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For dispersive delays, the interferometer phase at frequency ν is shifted by φ = −c re ν Ne , where c is the speed of light, re is the classical electron radius, and Ne is the electron column density along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note that the phase shift is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The phase delay, τp, is given by τp = 1 2π φ ν = − c re 2πν2 Ne , – 7 – and the group delay, τg, is given by τg = 1 2π ∂φ ∂ν = + c re 2πν2 Ne .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note that the group delay has the opposite sign of the phase delay (and the radio signal phase shift) for a dispersive delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Combining the phase shift and group delay formulae yields the relation φ = −2π ν τg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In other words, when correcting the interferometer phase for a change in ionospheric group delay, one must flip the sign of the correction compared to a change in tropospheric delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' With a mix of dispersive and non-dispersive delays, corrections for group delays can be combined, but phase-delays cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If one observes geodetic blocks at frequencies below about 15 GHz, it is best to measure multi-band delays at two well-spaced frequencies in order to estimate and remove the disper- sive component of the delay from the total delay, yielding pure non-dispersive delay for each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note that solving for a zenith delay-error with pure dispersive delays, and then using an ionospheric mapping function to predict delays along a ray-path to a source, does not seem to improve astrometric accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, at the high altitude of the ionosphere, rays from a source at, say, a zenith angle of 45◦ will pass through the ionosphere ≈ 400 km from the antenna site, likely creating a significant azimuthal dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' It may be possible to solve for and remove an azimuthal dependence, but this has not yet been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Phase-reference blocks Phase-reference blocks involve cycling between sources well within the interferometer coherence time, which is usually limited by short-term fluctuations in tropospheric water vapor and produce phase changes proportional to observing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For reasonably dry sites, the time spanned between (the centers of the) phase-reference scans should be less than about 40 s at 43 GHz, 80 s at 22 GHz, and 260 s at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='7 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, if a calibrator (C) is the phase reference for a target (T), the sequence could be C, T, C, T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' with scan durations of 40 s at 22 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note that scan durations include slew and settle times, so on-source (dwell) times will be less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For a strong target and weak calibrator, so-called inverse phase referencing can be used: T, C, T, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' At frequencies well above 10 GHz, such sequences are adequate, and if more than one calibrator can be found within about 2◦ of the target, they can be used to give quasi-independent relative positions (provided the calibrators are well separated on the sky) and to provide a check on potential problems associated with jet-like structures in some calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 8 – At frequencies below ∼10 GHz, MultiView can yield excellent astrometric accuracy (Rioja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Ionospheric electron densities follow the Sun, and residual path-delays can be approximated by ionospheric “wedges” and modeled with linear gradients (a tilted plane) on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Using calibrators which surround the target, an observing sequence with four calibrators could be as follows: C1, C2, T, C3, C4, T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This approach involves fitting the phases from the four calibrators nearest in time to the target for a given interferometer baseline to a tilted phase-plane: φC = φT + Sx∆x + Sy∆y , (2) where φT is the desired phase at the target position, Sx and Sy are phase-gradients in the x and y directions, and ∆x and ∆y are angular offsets from the target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In general, three calibrators are the minimum number needed to fit a tilted phase plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In the example above, I mentioned using four calibrators, as this provides the opportunity to identify and remove a “bad” calibrator, possibly owing to large phases associated with complex source structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Interestingly, however, only two calibrators can be used, provided they nicely straddle the target source, such that a line between them passes very close to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For MultiView to succeed, the entire cycle of calibrators and target must be completed in well under the interferometer coherence time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' the time it takes for phases to wander by 1 radian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='6 GHz, Rioja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' (2017) successfully used a cycle time of 300 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' It is critical that the phases on each calibrator are dominated by propagation delays and not, for example, by position errors or source structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' So, calibrator positions must be known to a small fraction of the interferometer fringe spacing on the longest baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note, the calibrator positions can be updated by measuring relative positions to one of the calibrators (or possibly the target), which has a well determined absolute position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Calibrator phases can “wrap” past ±180◦, owing to position or baseline errors or large residual tropospheric or ionospheric delays in the correlation process, and these phase wraps must be accounted for when fitting the phase-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If the target source is sufficiently strong to use as the interferometer phase refer- ence, inverse MultiView has some advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' An observing sequence can be as follows: T, C1, T, C2, T, C3, T, C4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' , and only the time between the centers of the target (T) scans needs to be less than the interferometer coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Also, phase-wrap problems are re- duced, compared to standard MultiView, as atmospheric phase-shifts should largely cancel between the target and each calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Hyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' (2022) demonstrate that calibrators separated by up to 7◦ from the target can yield excellent results at 8 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Typically this allows for many useful calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 9 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Data Calibration 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Geodetic Block Analysis The first step in calibration is to estimate clock and residual tropospheric delays at each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This can be done with the geodetic-block data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Start by correcting the data for updated Earth’s orientation parameters, antenna locations, source positions, and feed rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Also, ionospheric delays (usually not applied during correlation) should be removed, based on total electron content models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Then, electronic delays and phase offsets among different IFs and feeds should be “aligned” using a “comb” of frequencies inserted during the observations or performing a “manual” phase-calibration using a fringe-finder source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Next, multi-band delays for a source on a baseline involving antennas i and j, τi,j = τj − τi , are fitted with antenna dependent clock parameters and atmospheric zenith delay- errors, where each antenna’s delay is given by τ = T0 + dT dt ∆T + τ0 sec θZA , (3) assuming other sources of delay error are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The first two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' (3) correspond to a clock offset, T0, and a linear clock drift over time, dT dt , and ∆T is time relative to a reference time, best chosen near the center of the observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' the last term is the zenith (vertical) atmosphreric delay error scaled by the secant of the source zenith angle, sec θZA, for the increased (slant) path-length through the troposphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note, with interferometric data the clock parameters for one (reference) antenna must be held constant and usually are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' However, since θZA varies differently for each VLBI antenna, one can solve for the zenith delay at all antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Geodetic blocks with a dozen calibrators can take about 30 minutes to complete, depending strongly on antenna slew speeds, and should be done every 2–3 hr in order to monitor and correct for changing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' An often overlooked aspect of differential astrometry is that the tropospheric delay difference between a target and a calibrator at a given antenna is given by ∆τ ≈ τ0 ∂ sec θZA ∂θZA ∆θZA = τ0 sec θZA tan θZA ∆θZA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2While sec θZA is appropriate for a plane-parallel atmosphere and is a reasonable approximation for slant path delay, better θZA “mapping functions” which account for the Earth’s curvature and fi- nite atmospheric thickness are a significant improvement for observations at large zenith angles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' see Thompson, Moran & Swenson (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 10 – At large zenith angles, both sec θZA and tan θZA blow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' While this can be leveraged to increase geodetic block sensitivity, large θZA observations should be avoided for phase- referencing astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='— Geodetic block examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Plotted are multi-band (group) delays (left panels) and residual fringe rates (right panels) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Measurements are green dots, best-fitting model values are blue circles, and “data minus model” residuals are red stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The top row is for a well-behaved baseline, whereas the bottom row shows a serious problem – a clock jump between 15:05 and 16:15 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Examples of geodetic block data from two baselines of VLBA observations are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Data were taken near 3 and 7 GHz and the dispersive component of delay (owing to ionospheric electrons) was calculated and subtracted from each measurement, leaving pure non-disperive delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The left panels plot multi-band (group) delays and the right panels show fringe rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Green dots are the measurements, blue circles are the fitted model and red symbols are residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The upper plots (for the baseline with antennas 5 and 8) show good data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' one can see a uniform clock drift of about +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='3 nsec hr−1 and most residuals are <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='1 nsec (or <∼ 3 cm of path delay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The lower plots (for the baseline with antennas 6 and 8) show a serious problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' There is a large clock-drift rate of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='5 nsec hr−1 evident between 11 and 15 UT (corroborated by an average fringe rate of about −3 mHz, corresponding to ν dT dt for observations at ν = 5 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' While this is not necessarily a problem, there is a large clock jump between the third and fourth geodetic blocks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', somewhere between 15:05 and 16:15 UT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note that the residuals, while not as small as for baseline 5–8, are reasonably close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This can happen as spurious large zenith-delay parameters can partially compensate for mismodeling the clock as a linear drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' So, simply looking for small residuals is not sufficient to spot problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Clearly the clock jump occurred at antenna 6 (since antenna 8 is common to both baselines and the jump does not show up in baseline 5–8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Therefore, data involving antenna 6 from 15:05 to the end should be flagged, both in the geodetic block and phase-referencing files, and the geodetic block data should be re-fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 11 – Finally, it is always a good idea to check that the clock and tropospheric delay corrections work well by applying them to the geodetic block data, re-fitting for multi-band delays, and verifying that the residual delays are <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='1 nsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Alternatively, since many geodetic block sources are strong, after applying the corrections, one can simply plot phase versus frequency across all IFs and feeds and visually verify that the response is nearly flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Phase-referencing Block Analysis The phase-referencing data require standard calibration steps3, with the addition of tro- pospheric delay calibration using the results of the geodetic-block fits, and for low-frequency observations substitution of MultiView for standard phase referencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, cali- bration steps could be as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Flag any bad data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Convert interferometer visibilities (correlation coefficients) to Jansky units, using mea- sured system temperatures and antenna gain curves, as well as any other known am- plitude corrections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', digitization loses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Adjust delays and phases as done in the preliminary calibration of the geodetic block data (using updated Earth’s orientation parameters, antenna locations and source po- sitions, and applying ionospheric TEC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Correct phases for antenna feed rotations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' for example, for circularly polarized feeds shift phases by the feed parallactic angles with opposite signs for right and left cir- cularly polarized feeds (and be sure to verify the proper sign convention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note, this holds for linearly polarized feeds correlated against circularly polarized feeds, but for linear against linear feeds there is no phase shift induced by feed rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The con- version of VLBI data with some or all antennas having linearly polarized feeds to a common circularly polarized basis is complicated and deserves its own paper (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', Mart´ı-Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Apply clock and tropospheric delay corrections based on fitting geodetic block data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Then remove electronic delays and phases among IFs and feeds using either a cali- bration frequency comb (inserted at the receiver) or using a manual phase-calibration 3see http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='aips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='edu/vlbarun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='shtml for examples – 12 – on a strong source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For spectral-line sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', masers) a subtle but serious prob- lem can occur in this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If a second-pass correlation is done which retains only a portion of the total IF bandwidth (so-called “zoom mode” used to limit the number of spectral channels when high spectral-resolution is desired), then one should also correlate the manual phase-calibration source(s) in the same manner and apply them to the spectral-line data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' One cannot directly transfer the manual phase-calibration from broadband to narrowband data without considering the frequency location of the narrowband data relative to the edge frequency of its associated broadband, and some software analysis packages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', AIPS and CASA) do not have this capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='— Interferometer reference phase simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Each dot represents a phase measured on a calibra- tor, to be interpolated between two adjacent (closely spaced) measurements and removed from the target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' A strong target can be used as the phase reference for a weak calibrator, so-called “inverse phase referencing,” if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Top panel: high signal-to-noise phases showing small variations over ∼ 5 minutes and larger variations over hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For high-quality differential astrometry, reference phases should resemble these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Middle panel: the same phases but generated with low signal-to-noise and these are far from optimal for phase referencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Bottom panel: high signal-to-noise phases, but with a high residual fringe-rate as evidenced by rapid “phase wrapping.” These indicate a large position error for the phase-reference source, or other geometric or clock errors, and are not suitable for accurate differential astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If the local oscillator was held constant during the observing track, then spectral-line – 13 – sources will shift in frequency owing to the Earth’s rotation and orbit about the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Then interferometer visibilities should be corrected to hold a spectral line steady in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If the visibilities are in the delay-lag domain, V (τ), this can be accomplished by an appropriate linear shift of visibility phase as a function of delay-lag, which corresponds to a shift in spectral frequency (by the Fourier transform shift theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If, instead, the visibilities are in the frequency domain, V (ν), one can Fourier transform to delay-lag, perform the linear phase shift, and then inverse Fourier transform back to frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Phase reference the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This can use a single source, either a calibrator (standard phase-referencing) or the target (inverse phase-referencing), or it can involve multiple calibrators (MultiView).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 3 shows three simulated examples of reference phases calculated from a single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The top panel shows high SNR phases which might be obtained at ∼1 cm wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The slow phase wander of hundreds of degrees of phase over hours is typical of path delays of ∼1 wavelength owing to variable water vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Within groups spanning about 15 minutes, the phases look “worm-like,” which comes from small-scale fluctuations in water vapor, and one can reliably interpolate between individual points where the source to be referenced was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 3 shows the same simulated data, except a low SNR of unity was used for each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Interpolation between adjacent points would be very inaccu- rate and, when applied to another source, imaging would be considerably degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The bottom panel shows the effects of a large error in the position of the reference source, as evidenced by the “patterned” appearance made by rapid phase wrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Other errors, such as incorrect baselines or uncompensated clock drifts, can also do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In this case, differential astrometric accuracy would be degraded by second-order effects of phase referencing as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' As outlined in Section 3, at frequencies below ∼ 10 GHz, path-delays from ionospheric “wedges” can be removed from data with the MultiView calibration technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 4 shows simulated phases on one baseline at an instant in time for four calibrators surrounding a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' A tilted plane fitted to the calibrator phases as a function of (X, Y )-offset yields a phase of 50◦ at the position of the target source, which could then be removed from the target source data prior to astrometric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' As discussed in Section 3, phase wraps can be a serious problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 5 gives a simple example of a phase-wrap issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The phase for the C3 calibrator, coming from the arctan(Vimag, Vreal) of an interferometer visibility, is generally defined between −180◦ and +180◦ and, in this example, is −175◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' However, this phase should be +185◦ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', −175◦ + 360◦) and must be set to that value when fitting a MultiView tilted-plane – 14 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='— MultiView phase-calibration simulation of an ionospheric wedge with gradients of 10◦ and −5◦ of phase per degree of angular offset in the X− and Y −directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Calibration sources are offset from the target at (X, Y ) = (0◦, 0◦) by (−4◦, +3◦) for C1, (−2◦, −1◦) for C2, (+3◦, −3◦) for C3, and (+2◦, +3◦) for C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fitting a tilted plane to the phases of the four calibrators (ψC1, ψC2, ψC3, and ψC4) returns the 50◦ phase at the origin to be removed from the target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In general, the minimum number of calibrators needed to define the phase plane is three, but it can be useful to use four to test for a “bad” calibrator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', with source-structure phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In cases where the line between two calibrators comes very close to the target source, one can use only two calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' to all calibrator phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Otherwise, MultiView would return a spurious phase to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If the target source is sufficiently strong and compact so as to yield a high SNR in a single scan, it can be used as a “preliminary” phase reference for the surrounding calibrators (so-called inverse MultiView).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This has the important advantage of pre- calibrating the data on a shorter time-scale than standard MultiView, which requires cycling around all calibrators and the target well within the interferometer coherence- time limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This pre-calibration also results in “differenced phases,” which can diminish phase-wrap problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' The next step in inverse MultiView is the same as stan- dard MultiView, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' fitting a tilted plane to the differenced phases of the calibrators and subtracting the phase offset at the target position from the target phases (which were zero after the pre-calibration step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This returns “clean” phases, which give the desired position information for the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Hyland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' (2022) have demonstrated that inverse MultiView can work well for 8 GHz observations and achieve ≈ 20 µas (single-epoch) astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 15 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='— MultiView phase-calibration simulation, similar to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 4, but with larger gradients of 30◦ and −15◦ of phase per degree of angular offset in the X− and Y −directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fitting a tilted plane to the phases of the four calibrators (ψC1, ψC2, ψC3, and ψC4) would also yield the 50◦ phase at the origin to be removed from the target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' However, with the larger phase slope, the phase for C3 is now 185◦, which would generally be reported as −175◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Fitting a tilted plane using that phase would result in a very large error in the slope and in the fitted phase at the target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Such phase-wraps must be corrected for MultiView to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' It is essential that the calibrators used for MultiView have position errors small enough so as not to contribute significantly to the measured phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If a priori positions are only known to, say, ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='5θf, then measured phases could have 180◦ unwanted contri- bution from their position errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For a fringe spacing θf ∼ 1 mas, this corresponds to a position error of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='5 mas, and not all calibrators have this positional accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' However, what is more important is relative position accuracy among the MultiView calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Relative position accuracy better than ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='1θf can usually be achieved through a first-pass of phase referencing and imaging of all sources relative to a single source with a very accurate position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' After correcting the interferometer visibilities for the measured position offsets, one can then repeat the calibration sequence, now with greatly reduced contributions to phases from poor positions, and then do the MultiView step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 16 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Things to Consider A cardinal rule for astrometry used to estimate parallax and/or proper motion is to avoid changing any observational parameters over the entire program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This offers the best chance for canceling systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For example, if a source has structure within the resolution of the array and either the source or the interferometer (u, v)-coverage changes between observations, then one can get centroid shifts that are unrelated to parallax or proper motion and can degrade their measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For sources with modest structure, such as a double with separation comparable to the interferometer resolution, accurate astrometry can usually benefit from some degree of “super-resolution.” Often when imaging, one can use a round CLEAN restoring beam even if the dirty beam FWHM is elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' I have found it useful to separate compact emission components by setting the CLEAN restoring beam to θF W HM = Sr � θmaj × θmin , where the θmaj and θmin are the “dirty” beam’s major and minor FWHM sizes, and using a moderate super-resolution factor, Sr, between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' While one can consider directly fitting interferometer phases for position offsets, it seems advisable to always make images in order to assess whether or not a source has a complicated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If a source (either the target, which might be an X-ray binary, or a quasar calibrator) has a jet-like structure, which changes among observing epochs, one should consider rotating the measured positions and the parallax/proper motion model to be along and perpendicular to the jet direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', see Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' This allows one to down-weight the “corrupted” data in the jet direction and rely more on the “clean” data in the perpendicular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' When measuring group delays, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' for geodetic block observations, one should spread the spanned bandwidth in order to get more accurate delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' On the flip side, for phase- reference blocks, it is better to pack the bands together in order to minimize the frequency spanned and have less sensitivity to delay errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note that modern digital base-band con- verters (DBBCs) may introduce delay jumps when changing between different setups, and this can significantly degrade (or destroy) astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' If one’s DBBCs have such problems, it might be best to observe geodetic and phase-reference blocks with the same electronic setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Calibrators offset from a target in decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' are often not as good for astrometry as those offset in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', since phases associated with atmospheric (zenith-angle dependent) delay errors can more closely mimic decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' – 17 – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Closing Thoughts The most accurate parallax measurements to date have uncertainties near ±5 µas (see Table 1 of Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 2019, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' In many cases, these come from mea- surements at 22 GHz and are limited by uncompensated tropospheric delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Since these are likely to be quasi-random, parallax measurement can be improved with a greater number of observations (N), with precision scaling as 1/N1/2 (if optimally sampled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Currently, it is not clear if the non-dispersive phase-delays associated with fluctuations in water vapor are uncorrelated over several degrees on the sky or, instead, have a “planar” structure which could be addressed by MultiView calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Three-dimensional (3D) kinematic distances offer a promising method of estimating distances to sources well past the Galactic center, where observed motions can approach 470 km s−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=', twice the rotation speed of the Galaxy (Reid 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Thus, even at a distance of 20 kpc, this leads to a proper motion of about 5 mas y−1, or over one year about 100 times the magnitude of the parallax amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' For young stars hosting masers, non-circular motions are ∼10 km s−1, and hence contribute less than a couple of percent to the distance error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Usually, proper motion can be measured much more easily than parallax, in part because motion precision scales as 1/N3/2 (for uniform sampling in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Note, that if one observes a source at integer year intervals, the parallax effect vanishes, simplifying the motion measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' Measurements with future VLBI arrays, including the SKA and the ngVLA, may be able to obtain hundreds-to-thousands of 3D kinematic distances for very distant Galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' REFERENCES Beasley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' & Conway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content=' 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+page_content=', (Springer Open, Switzerland) This preprint was prepared with the AAS LATEX macros v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQf2gu7/content/2301.04756v1.pdf'} diff --git a/adAyT4oBgHgl3EQf9_ol/content/tmp_files/2301.00883v1.pdf.txt b/adAyT4oBgHgl3EQf9_ol/content/tmp_files/2301.00883v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..be4f7281d2d65e0500a216984c8f71351e27d189 --- /dev/null +++ b/adAyT4oBgHgl3EQf9_ol/content/tmp_files/2301.00883v1.pdf.txt @@ -0,0 +1,1781 @@ +THE MINIMAL PROJECTIVE BUNDLE DIMENSION +AND TORIC 2-FANO MANIFOLDS +CAROLINA ARAUJO, ROYA BEHESHTI, ANA-MARIA CASTRAVET, KELLY JABBUSCH, +SVETLANA MAKAROVA, ENRICA MAZZON, AND NIVEDITA VISWANATHAN, +WITH AN APPENDIX BY WILL REYNOLDS +Abstract. Motivated by the problem of classifying toric 2-Fano manifolds, we introduce +a new invariant for smooth projective toric varieties, the minimal projective bundle dimen- +sion. This invariant m(X) ∈ {1, . . . , dim(X)} captures the minimal degree of a dominating +family of rational curves on X or, equivalently, the minimal length of a centrally symmet- +ric primitive relation for the fan of X. We classify smooth projective toric varieties with +m(X) ≥ dim(X)−2, and show that projective spaces are the only 2-Fano manifolds among +smooth projective toric varieties with m(X) ∈ {1, dim(X) − 2, dim(X) − 1, dim(X)}. +Contents +1. +Introduction +1 +2. +Primitive collections +5 +2.1. +Notation and background +5 +2.2. +The minimal P-dimension +7 +2.3. +Some properties of primitive collections +10 +2.4. +Primitive collections on toric Fano manifolds +11 +3. +Toric Fano manifolds with m(X) = 1 +12 +4. +Proof of Theorem 1.3 +16 +4.1. +First case: {x, y, z} is a primitive collection +17 +4.2. +Second case: none of {x, y, z} form a primitive collection +23 +5. +Toric Fano manifolds with m(X) = n − 2 +24 +Appendix A. +Code for computing primitive collections +26 +References +30 +1. Introduction +Fano varieties are projective varieties with positive first Chern class. Over the complex +numbers, this condition is equivalent to the existence of a metric with positive Ricci cur- +vature. Basic examples of Fano varieties include projective spaces and Grassmannians. +The positivity condition has further geometric implications, e.g., Fano varieties over the +complex numbers are simply connected. This has an analogue on the algebro-geometric +side: any Fano variety is covered by rational curves [Mor79], and is in fact rationally con- +nected [KMM92; Cam92], i.e., there are rational curves connecting any two of its points. +In a series of papers, de Jong and Starr introduce and investigate possible candidates for +the notion of higher rational connectedness [dHS11; dS07; dS06b; dS06c; Sta06], inspired +1 +arXiv:2301.00883v1 [math.AG] 2 Jan 2023 + +2 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +by the natural analogue in topology. In particular, in [dS06b] they define 2-Fano mani- +folds. A smooth projective variety X is 2-Fano if it is Fano and its second Chern character +ch2(TX) = 1 +2c1(TX)2 − c2(TX) is positive, i.e., ch2(TX) · S > 0 for every surface S in X. +In a similar way, one can define k-Fano varieties for any k ≥ 2, and aim at their classifi- +cation. For instance, Pn is n-Fano, and it is conjectured that it is the only n-dimensional +n-Fano manifold. The geometry of higher Fano manifolds has been fairly investigated, and +in several special cases they are shown to enjoy the expected nice properties. For instance, +2-Fano manifolds satisfying some mild assumptions are covered by rational surfaces [dS07], +and similar results hold for higher Fano manifolds [Suz21], [Nag19]. There is a classification +of 2-Fano manifolds of high index [AC13] and, more recently, a classification of homoge- +neous 2-Fano manifolds [Ara+22]. On the other hand, very few examples of higher Fano +manifolds are known. Quite strikingly, all known examples of 2-Fano manifolds have Picard +rank 1 and relatively large index. +It is natural for algebraic geometers to turn to the pool of toric varieties when looking +for intuition or examples. It is well known that projective spaces are the only projective +toric manifolds with Picard rank 1. Thus, a classification of toric 2-Fano manifolds could +either provide the first examples of 2-Fano manifolds with higher Picard rank, or it could +be an evidence that every 2-Fano manifold has Picard rank 1. Geometric properties of a +toric variety can often be checked in the combinatorics of the associated fan. This bridge +has been exploited in search of new examples of toric 2-Fano manifolds [Nob11], [Nob12], +[Sat12], [Sat16], [SS20], [SSS21], [Shr20]. Despite the efforts, a complete (computer aided) +classification is only known up to dimension 8 [Nob11], [SSS21], and projective spaces +remain the only known examples of toric 2-Fano manifolds. The sparsity of higher Fano +manifolds leads to the following conjecture. +Conjecture 1.1. ([SSS21, Conjecture 4.3]) The only toric 2-Fano manifolds are projective +spaces. +In this paper, we propose a new strategy to approach Conjecture 1.1. We follow the +philosophy introduced in [AC12], namely, to investigate 2-Fano manifolds by studying their +minimal dominating families of rational curves. By [CFH14], minimal dominating families +of rational curves on a smooth projective toric variety X correspond to primitive relations +of the form +(1) +x0 + · · · + xm = 0, +satisfied by some of the primitive integral generators xi of the corresponding fan. These +primitive relations are called centrally symmetric of order m + 1. By [CFH14], a centrally +symmetric primitive relation of order m + 1 yields a Pm-bundle structure X◦ → T on a +dense open subset X◦ of X. If dim(T) ≥ 1, and the complement X \X◦ has codimension at +least 2 in X, then one can construct a complete surface S ⊂ X◦ such that ch2(TX) · S ≤ 0, +showing that X is not 2-Fano. So our basic strategy consists of trying to describe, in a +rather explicit way, a suitable birational map ϕ : X ��� Y transforming X into a projective +toric variety Y admitting a Pm-bundle structure on a big open subset. We then hope to be +able to compare the second Chern characters ch2(TX) and ch2(TY ) to show that X is not +2-Fano, except if X = Pm and ϕ is the identity. +To follow this strategy, we introduce a new invariant of a smooth projective toric variety +X, the minimal projective bundle dimension of X, minimal P-dimension in short, which is + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +3 +of independent interest (Definition 2.10): +m(X) = min +� +m ∈ Z>0 +�� there is a relation as in (1) +� +∈ {1, . . . , dim X}. +By going through the database of toric Fano manifolds of low dimension and computing +their primitive collections, one obtains Table 1, indicating the number of Fano manifolds +for each value of m. Appendix A contains the code used to compute primitive collections. +dim(X) +# Fanos +#(m=1) +#(m=2) +#(m=3) +#(m=4) +#(m=5) +#(m=6) +4 +124 +107 +15 +1 +1 +5 +866 +744 +112 +8 +1 +1 +6 +7622 +6333 +1174 +105 +8 +1 +1 +Table 1. The minimal P-dimension of toric Fano manifolds of low dimension. +When m(X) = 1, Casagrande constructs in [Cas03b] a sequence of blowdowns and flips +from X to a toric variety admitting a global P1-bundle structure. This allows us to make +the basic strategy work, yielding the following result. +Theorem 1.2. Let X be a smooth toric Fano variety with m(X) = 1. Then X is not +2-Fano. +Table 1 suggests that this result covers “most” toric varieties, and not just the fringe cases. +Next we turn our attention to toric Fano manifolds X with large values of m(X). Projec- +tive spaces are the only smooth projective toric varieties admitting a centrally symmetric +primitive relation of order dim(X) + 1. In [CFH14, Proposition 3.8], Chen, Fu and Hwang +classify toric Fano manifolds admitting a centrally symmetric primitive relation of order +dim(X). There are three such varieties, and two of them also admit a centrally symmetric +primitive relation of order 2. As a consequence, the only n-dimensional toric Fano manifold +X with m(X) = n − 1 is the blowup of Pn along a linear Pn−2. In [BW22], Beheshti and +Wormleighton investigate smooth projective toric varieties admitting a centrally symmetric +primitive relation of order dim(X)−1, showing that they have Picard rank ρ(X) ≤ 5. Most +of these varieties also admit centrally symmetric primitive relations of order 2 or 3, and we +prove the following bound for the remaining ones. Theorem 1.4 shows that this bound is +sharp. +Theorem 1.3. Let X be a smooth toric Fano variety with dim(X) = n ≥ 6 and m(X) ≥ 3. +If X has a centrally symmetric primitive relation of order n − 1, +x0 + x1 + · · · + xn−2 = 0, +then ρ(X) ≤ 3. Moreover, m(X) = n − 2 and the above relation is the only centrally +symmetric primitive relation of X. +Using Theorem 1.3 and Batyrev’s description of smooth projective toric varieties with +Picard rank 3, we are able to classify n-dimensional smooth toric Fano varieties with +m(X) = n − 2. There are eight distinct isomorphism classes when n ≥ 6, which can be +explicitly described. The following statement summarizes the classification of toric Fano +manifolds with m(X) ≥ dim(X) − 2. + +4 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +Theorem 1.4. We have the following classification of smooth toric Fano varieties with +m(X) ≥ dim(X) − 2. +(1) The only n-dimensional smooth toric Fano variety X with m(X) = n is Pn. +(2) For n ≥ 3, the only n-dimensional smooth toric Fano variety X with m(X) = n − 1 +is the blowup of Pn along a linear Pn−2. +(3) For n ≥ 6, there are eight distinct isomorphism classes of n-dimensional smooth +toric Fano varieties X with m(X) = n − 2. Namely: +(a) X = PS(E) is a Pn−2-bundle over a toric surface S, where (S, E) is one of the +following: +• S = P2 and E = OP2(1) ⊕ O⊕n−2 +P2 +, +• S = P2 and E = OP2(1) ⊕ OP2(1) ⊕ O⊕n−3 +P2 +, +• S = P2 and E = OP2(2) ⊕ O⊕n−2 +P2 +, +• S = P1 × P1 and E = OP1×P1(1, 1) ⊕ O⊕n−2 +P1×P1, +• S = P1 × P1 and E = OP1×P1(1, 0) ⊕ OP1×P1(0, 1) ⊕ O⊕n−3 +P1×P1, +• S = F1 and E = OF1(e + f) ⊕ O⊕n−2 +F1 +, where e ⊂ F1 is the −1-curve, and +f ⊂ F1 is a fiber of F1 → P1. +In the first three cases, ρ(X) = 2, while in the latter three cases, ρ(X) = 3. +(b) Let Y ≃ PP2� +OP2(1) ⊕ O⊕n−2 +P2 +� +be the blowup of Pn along a linear subspace +L = Pn−3, and denote by E ⊂ Y the exceptional divisor. Then X is the blowup +of Y along a codimension 2 center Z ⊂ Y , where: +• Z is the intersection of E with the strict transform of a hyperplane of Pn +containing the linear subspace L, or +• Z is the intersection of the strict transforms of two hyperplanes of Pn, +one containing the linear subspace L, and the other one not containing +it. +In both cases, ρ(X) = 3. +Corollary 1.5. The projective space Pn is the only smooth n-dimensional toric 2-Fano +variety with m(X) ∈ {1, n − 2, n − 1, n}. +This paper is organized as follows. +In Section 2, we review some results from toric +geometry and fix notation. In particular, we discuss centrally symmetric primitive relations +on smooth projective toric varieties, describing explicitly their open subsets admitting a +projective space bundle structure (Proposition 2.13). In Section 3, we study smooth toric +Fano varieties with m(X) = 1, and prove Theorem 1.2. In Section 4, we investigate smooth +projective n-dimensional toric varieties admitting a centrally symmetric primitive relation +of order n − 1, and prove Theorem 1.3. In Section 5, we use this result, together with +Batyrev’s description of smooth projective toric varieties with Picard rank 3, to prove +Theorem 1.4. +Acknowledgements. +This collaboration started as a working group at the ICERM +“Women in Algebraic Geometry Collaborative Research Workshop” in July 2020. A great +part of this work was developed during the follow-up “Collaborate@ICERM” meeting in +May 2022. We thank ICERM for the financial support and the great working conditions +provided to us during our visit. We are very grateful to Cinzia Casagrande and Milena +Hering for many enlightening discussions and explanations about toric varieties, and to + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +5 +Will Reynolds for running his code and providing us with precious data about primitive +relations of toric Fano manifolds. +Carolina Araujo was partially supported by CAPES/COFECUB, CNPq and FAPERJ +Research Fellowships. Roya Beheshti was supported by NSF grant DMS-2101935. Ana- +Maria Castravet was partially supported by the ANR 20-CE40-0023 grant FanoHK. Enrica +Mazzon was supported by the collaborative research center SFB 1085 Higher Invariants +- Interactions between Arithmetic Geometry and Global Analysis funded by the Deutsche +Forschungsgemeinschaft. Nivedita Viswanathan was supported by the EPSRC New Hori- +zons Grant No.EP/V048619/1. +2. Primitive collections +2.1. Notation and background. A toric variety is a normal complex variety X that +contains a torus T = (C∗)n as a dense open subset, together with an action of T on X that +extends the natural action of T on itself. There is a one-to-one correspondence between +n-dimensional toric varieties and fans in Qn. Let N be a free abelian group of rank n, and +consider the vector space NQ = N ⊗Z Q. A fan in NQ is a nonempty finite collection Σ of +strongly convex polyhedral cones in NQ such that every face of a cone in Σ is also a cone in +Σ, and the intersection of two cones in Σ is a face of each. We write δ ≺ τ to express that +δ is a face of τ. One-dimensional cones in Σ are called rays, and each ray is generated by +a primitive vector in N. The set of primitive vectors of N generating rays of Σ is denoted +by G(Σ). We will write a cone τ ∈ Σ in terms of its primitive generators, τ = ⟨v1, . . . , vl⟩, +saying that the vi’s generate τ, and setting G(τ) := {v1, . . . , vl} ⊆ G(Σ). +We denote by XΣ the toric variety corresponding to a fan Σ. Conversely, given a toric +variety X, we denote by ΣX the fan associated to X. There is a one-to-one inclusion- +reversing correspondence between cones in Σ and T-orbit closures in XΣ. Given a cone +τ ∈ Σ, we write V (τ) ⊂ XΣ for the corresponding T-orbit closure, or V (v1, . . . , vl) when +G(τ) = {v1, . . . , vl}. Note that dim(τ) = codimXΣ V (τ). We refer to [Ful93] and [CLS11] +for the background on toric varieties. +In this paper, we are mostly interested in smooth and proper toric varieties. The smooth- +ness conditions translates into the fan Σ being regular, i.e., for each cone τ ∈ Σ, the set +of generators G(τ) is part of a basis of N ([CLS11, Definition 1.2.16]). The properness +condition translates into the fan Σ being complete, i.e., its support being the whole NQ. In +what follows, smooth and proper toric varieties will be simply called toric manifolds. We +would like to classify toric 2-Fano manifolds. Given a toric manifold X, there is an exact +sequence ([CLS11, Theorem 8.1.1]): +0 → Ω1 +X → N ∨ ⊗Z OX → +� +v∈G(ΣX) +OV (v) → 0 , +from which one easily computes: +c1(X) = +� +v∈G(ΣX) +V (v) +and +ch2(X) = 1 +2 +� +v∈G(ΣX) +V (v)2. +Definition 2.1. ([Bat91, Definition 2.6]) Let Σ be a regular complete fan in NQ. +A +primitive collection P ⊆ G(Σ) is a nonempty set of primitive vectors of N that does not +generate a cone of Σ, but such that any proper subset of P does. +Equivalently, P = + +6 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +{v1, . . . , vr} ⊆ G(Σ) is a primitive collection if and only if +⟨v1, . . . , vr⟩ /∈ Σ +and +⟨v1, . . . , ˇvi, . . . , vr⟩ ∈ Σ +for any i = 1, . . . , r. We denote by PC(Σ) the set of primitive collections of Σ. For a +toric manifold X, we will talk about primitive collections of X and write PC(X), meaning +PC(ΣX). +Definition 2.2. Let Σ be a regular complete fan in NQ. +Given a primitive collection +P = {v1, . . . , vr} ∈ PC(Σ), let σ(P) = ⟨w1, . . . , ws⟩ be the minimal cone of Σ such that +v1 + · · · + vr ∈ σ(P). Then there is a relation +r(P): v1 + · · · + vr = µ1w1 + · · · + µsws, +where µj ∈ Z≥0 for j = 1, . . . , s. We call r(P) the primitive relation associated to P. We +define the order of P as ord(P) = |P| = r, while the degree of P as deg(P) = r − �s +j=1 µj. +By [Bat91, Proposition 3.1], for any primitive collection P, we have P ∩ σ(P) = ∅. In +particular, {v1, . . . , vr} ∩ {w1, . . . , ws} = ∅. +Definition 2.3. Let Σ be a regular complete fan in NQ. +A primitive collection P = +{x0, . . . , xk} of Σ is called centrally symmetric if σ(P) = {0}, i.e. +r(P): x0 + · · · + xk = 0. +Lemma 2.4. Let Σ be a regular complete fan in NQ, and let P, Q be two distinct centrally +symmetric primitive collections. Then P ∩ Q = ∅. +Proof. Write r(P): x0 + · · · + xk = 0 and r(Q): y0 + · · · + yl = 0. Assume that P ∩ Q ̸= ∅, +then without loss of generality we may assume that x0 = y0. But then subtracting this +vector from both relations, we get +x1 + · · · + xk = y1 + · · · + yl, +which shows that interiors of two distinct cones intersect. This is a contradiction. +□ +Lemma 2.5. Let Σ be a regular complete fan in NQ, and let P, Q be two distinct centrally +symmetric primitive collections. Then Span P ∩Span Q = {0}, in particular |P|+|Q|−2 ≤ +dim NQ. +Proof. Write r(P): x0 + · · · + xk = 0 and r(Q): y0 + · · · + yl = 0. +Take any vector +v ∈ Span P ∩ Span Q, so we can write it as +v = +� +aixi = +� +bjyj. +By possibly adding r(P) and r(Q) to the sums, we can get that all ai, bj ≥ 0, and up +to relabelling the ai, bj, we can assume a0 = b0 = 0. But this shows that v is in the +intersection of two cones, ⟨x1, . . . , xk⟩ ∩ ⟨y1, . . . , yl⟩, and the sets of generators are disjoint +by Lemma 2.4, so ⟨x1, . . . , xk⟩ ∩ ⟨y1, . . . , yl⟩ = {0} and v = 0. +The last claim follows from considering the dimensions of Span P and Span Q. +□ +Let A1(XΣ) be the group of algebraic 1-cycles on XΣ modulo numerical equivalence, and +set N1(XΣ) = A1(XΣ) ⊗Z Q. The Mori cone NE(XΣ) ⊂ N1(XΣ) is the cone generated + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +7 +by the classes of effective curves. A primitive integral class generating an extremal ray of +NE(XΣ) is called an extremal class. There is an exact sequence: +0 +> A1(XΣ) +> ZG(Σ) +> N +> 0, +[C] +> +� +C · V (v) +� +v∈G(Σ) +� +νv +� +v∈G(Σ) +> +� +v∈G(Σ) +νvv. +Thus the elements of A1(XΣ) are identified with integral relations between the elements of +G(Σ). If the class [C] corresponds to the relation � +v νvv = 0, then we have −KXΣ · C = +� +v νv. +Proposition 2.6. ([Cas03a, Lemma 1.4]) +Let Σ be a regular complete fan in NQ. A +relation +α1x1 + · · · + αlxl − β1y1 − · · · − βmym = 0, +with αi, βj ∈ Z>0, defines an effective class in N1(XΣ) provided that ⟨y1, . . . , ym⟩ is a cone +of Σ. +We will usually write the above relation as +α1x1 + · · · + αlxl = β1y1 + · · · + βmym. +It follows that primitive relations correspond to effective curve classes. By abuse of notation, +we will identify a primitive relation r(P) with the corresponding curve class. Note that +deg(P) = −KXΣ ·r(P). In the projective case we have the following description of NE(XΣ). +Proposition 2.7. ([Bat91, Theorem 2.15]) Let Σ be a regular complete fan in NQ, and +assume that XΣ is projective. Then the Mori cone is generated by primitive relations: +NE(XΣ) = +� +P∈PC(XΣ) +Q≥0 r(P). +Proposition 2.8. ([Rei83, Theorem 2.4]) +Let Σ be a regular complete fan in NQ, and +assume that XΣ is projective. Let γ be an extremal class in NE(XΣ) whose corresponding +primitive relation is +r(P): v1 + · · · + vr = µ1w1 + · · · + µsws. +Let τ = ⟨z1, . . . , zl⟩ be a cone of Σ such that G(τ) ∩ P = G(τ) ∩ G(σ(P)) = ∅, and such +that ⟨σ(P), τ⟩ = ⟨w1, . . . , ws, z1, . . . , zl⟩ is a cone of Σ. Then, for each i = 1, . . . , r, +⟨P \ {vi}, σ(P), τ⟩ = ⟨v1, . . . , ˇvi, . . . , vr, w1, . . . , ws, z1, . . . , zl⟩ +is also a cone of Σ. +2.2. The minimal P-dimension. Let X be a toric manifold with regular complete fan +ΣX in NQ. In this section, we discuss centrally symmetric primitive collections, introduced +in Definition 2.3. +Proposition 2.9. ([Bat91, Proposition 3.2]) If X is projective, then ΣX has a centrally +symmetric primitive collection of order k + 1 +(2) +r(P): x0 + · · · + xk = 0 +for some k ∈ {1, . . . , dim(X)}. + +8 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +Definition 2.10. For a projective toric manifold X, we define the minimal P-dimension +as +m(X) := min +� +m ∈ Z>0 +��� +ΣX has a centrally symmetric +primitive collection of order m + 1 +� +. +The next remark explains the terminology of Definition 2.10 and highlights the signifi- +cance of studying centrally symmetric primitive collections. +Remark 2.11. In [CFH14], Chen, Fu and Hwang provide a new geometric proof of Propo- +sition 2.9 by relating centrally symmetric primitive collections to minimal dominating fam- +ilies of rational curves. We review some aspects of the theory of rational curves on varieties +and refer to [Kol96] for details. Given a smooth and proper uniruled variety X, there is +a scheme RatCurvesn(X) parametrizing rational curves on X. A dominating family of ra- +tional curves on X is an irreducible component of RatCurvesn(X) parametrizing rational +curves that sweep out a dense open subset of X. A dominating family of rational curves +H is said to be minimal if, for a general point x ∈ X, the subvariety of H parametrizing +curves through x is proper. When X is projective, there always exists a minimal domi- +nating family of rational curves on X. For instance, one can take H to be a dominating +family of rational curves on X having minimal degree with respect to some fixed ample +line bundle on X. +When X = XΣ is a toric variety, there is a one-to-one correspondence between minimal +dominating families of rational curves H on X and centrally symmetric primitive collections +of Σ ([CFH14, Proposition 3.2]). Moreover, if the centrally symmetric primitive collection +has order k + 1 as in Equation (2) above, then there is a dense T-invariant open subset U +of X and a Pk-bundle π: U → W such that the general curve parametrized by H is a line +on a general fiber π ([CFH14, Corollary 2.6]). +It follows from this discussion that the minimal P-dimension m(X) is the smallest integer +k such that X admits a generic Pk-bundle structure. We have +m(X) ∈ {1, . . . , n = dim(X)}, +and m(X) = n if and only if X ≃ Pn. By [CFH14, Proposition 3.8], there are three toric +Fano manifolds admitting a centrally symmetric primitive relation of order n = dim(X), +namely: Pn−1 × P1, the blowup of Pn−1 × P1 along a linear Pn−2, and the blowup of Pn +along a linear Pn−2. The first two varieties also admit a generic P1-bundle structure. As +a consequence, the only n-dimensional toric Fano manifold X with m(X) = n − 1 is the +blowup of Pn along a linear Pn−2. In Section 5, we shall classify n-dimensional toric Fano +manifolds X with m(X) = n − 2. +Let P ∈ PC(X) be a centrally symmetric primitive collection of order k+1. As explained +in Remark 2.11, P induces a Pk-bundle structure on a dense T-invariant open subset U of +X. In [CFH14, Corollary 2.6], the T-invariant open subset U was taken as small as possible, +namely, U ∼= Pk × (C∗)n−k. For our purposes, we want to take U as big as possible. So +our next goal is to describe explicitly the biggest T-invariant open subset of X on which P +induces a Pk-bundle structure. +Notation 2.12. Let P = {x0, . . . , xk} ∈ PC(X) be a centrally symmetric primitive collec- +tion. Denote by EP the set of cones σ = ⟨v1, . . . , vr⟩ ∈ ΣX such that P ∩ G(σ) = ∅, and +{v1, . . . , vr, xj1, . . . , xjs} ∈ PC(X) for some s ≥ 1, i.e., +EP := {σ ∈ ΣX | P ∩ G(σ) = ∅ and ∃P ′ ⊊ P such that P ′ ∪ G(σ) ∈ PC(X)} . + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +9 +We write +V (EP) := +� +σ∈EP +V (σ) ⊂ X. +Proposition 2.13. Let P = {x0, . . . , xk} ∈ PC(X) be a centrally symmetric primitive +collection, and let V (EP) be as in Notation 2.12. Then the open subset U = X \ V (EP) +admits a Pk-bundle structure over a smooth toric variety. +In order to prove Proposition 2.13, we first prove two auxiliary lemmas. +Lemma 2.14. Let P = {x0, . . . , xk} ∈ PC(X) be a centrally symmetric primitive collec- +tion, let V (EP) be as in Notation 2.12, and set U = X \ V (EP). Then the fan ΣU of U +consists of all cones of ΣX of the form +(3) +τ ′ = ⟨τ, xj1, . . . , xjm⟩, +where 0 ≤ m ≤ k, and τ ∈ ΣX is such that ⟨τ, P \ {xi}⟩ ∈ Σ for every i ∈ {0, . . . , k}. +(When m = 0, Equation (3) means that τ ′ = τ.) +Proof. Recall that a cone σ ∈ ΣX corresponds to a T-orbit, which is dense and open in +V (σ). Hence, a cone σ ∈ ΣX is in ΣU if and only if the corresponding orbit does not +intersect V (EP), which is equivalent to saying that V (σ) ̸⊆ V (EP). It is immediate that +the cones of the form (3) define a fan Σ′ ⊂ ΣX in NQ, and XΣ′ is a dense open subset of X. +We now prove that the toric variety XΣ′ coincides with U by showing that a cone σ ∈ ΣX +is of the form (3) if and only if V (σ) ̸⊆ V (EP). +Consider σ ∈ ΣX \ Σ′, which means that ⟨G(σ) ∪ P \ {xi}⟩ /∈ ΣX for some i. Then +the set G(σ) ∪ P \ {xi} contains a primitive collection S, so the cone τ := ⟨S \ P⟩ is +in EP. +But notice that τ ≺ σ, so V (σ) ⊆ V (τ) ⊆ V (EP) and hence σ is not in ΣU. +Conversely, if σ ∈ ΣX \ ΣU, then V (σ) ⊆ V (EP), hence there exists τ ∈ EP such that +V (σ) ⊆ V (τ) and G(τ) ∪ P ′ ∈ PC(X) for some P ′ ⊂ P. Since G(τ) ⊆ G(σ), we conclude +that ⟨G(σ) ∪ P ′⟩ /∈ ΣX, i.e., σ ̸∈ Σ′. +□ +Consider the sequences +0 +NP := ker(φ) +N +N = N/Z⟨x0, . . . , xk⟩ +0, +0 +(NP)Q +NQ +N Q +0, +Σ0 +ΣU +ΣU, +φ +φQ +where φ is the quotient map, the fan Σ0 of (NP)Q ≃ Qk+1 is the subfan of ΣU of cones of +the form (3) with τ = {0} (in particular note that XΣ0 ≃ Pk), and ΣU = {φQ(σ) | σ ∈ ΣU}. +Lemma 2.15. Let the notation be as above. Then ΣU is a toric fan, and the linear map +φQ is compatible with the fans ΣU and ΣU. +Proof. The cones of ΣU are exactly φQ(τ) for τ ∈ ΣU such that G(τ) ∩ P = ∅, so for +simplicity we only consider these τ. +- It is immediate that the cones of ΣU are rational polyhedral, and that the faces of +φQ(τ) are φQ(δ), for all subcones δ ≺ τ. + +10 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +- We need to show that the cone φQ(τ) is strongly convex, i.e., if y ∈ φQ(τ) and +−y ∈ φQ(τ), then y = 0. This follows automatically from the fact that the images +of the generators φQ(G(τ)) = {v1, . . . , vr} are linearly independent, which we prove +by contradiction. If they are linearly dependent, then there exist a1, . . . , ar ∈ Q, +not all 0, such that Σr +i=1aivi = 0 in N Q. This implies that there exist bj ∈ Q +for j = 1, . . . , k such that Σr +i=1aivi = �k +j=1 bjxj, which is a contradiction since +⟨G(τ) ∪ P \ {x0}⟩ ∈ Σ and hence its primitive generators are linearly independent +in NQ. +- ΣU and ΣU are compatible with φQ as we have φQ(τ ′) ∈ ΣU for any τ ′ ∈ ΣU. +□ +Proof of Proposition 2.13. Let the notation be as above. Let ˆΣU be the collection of fans +of the form (3) above with m = 0. It follows from the description of the cones of ΣU in +Lemma 2.14 that +(1) φQ maps each cone ˆτ ∈ ˆΣU bijectively to a cone τ ∈ ΣU such that φ(ˆτ ∩N) = τ ∩N. +Furthermore, the map ˆτ �→ τ defines a bijection ˆΣU → ΣU; +(2) given cones ˆτ ∈ ˆΣU and τ0 ∈ Σ0, the sum ˆτ + τ0 lies in ΣU and every cone of ΣU +arises in this way. +In the notation of [CLS11, Definition 3.3.18], we say that ΣU is split by ΣU and Σ0. We +conclude by [CLS11, Theorem 3.3.19] that U = X \ V (EP) is a locally trivial fiber bundle +over XΣU with fiber XΣ0 ≃ Pk. It follows automatically that XΣU is smooth, since it is the +base of a locally trivial fibration with a smooth total space. +□ +2.3. Some properties of primitive collections. Before focusing on toric Fano mani- +folds, we collect here two useful properties of primitive collections of arbitrary toric man- +ifolds. +The first one, by Sato, describes the behaviour of primitive collections under a +smooth toric blowdown. The second one, by Batyrev, describes primitive collections on +toric manifolds of Picard rank 3. +Proposition 2.16. ([Sat00, Corollary 4.9]) Let X be a toric manifold, and let f : X → Y +be the contraction associated to an extremal class in NE(X), corresponding to a primitive +relation of the form +r(Q): t1 + · · · + ts = z. +Then the fan ΣY is obtained from ΣX by removing the ray generated by z, and X is the +blowup of Y along V (t1, . . . , ts). Furthermore, the primitive collections of Y are precisely +the following PY ∈ PC(Y ): +• PY = PX for some PX ∈ PC(X) such that z /∈ PX and PX ̸= Q = {t1, . . . , ts}; +• PY = (PX \ {z}) ∪ {t1, . . . , tr} for some PX ∈ PC(X) such that z ∈ PX and +(PX \ {z}) ∪ S /∈ PC(X) for any subset S ⊊ {t1, . . . , tr}. +Proposition 2.17. ([Bat91, Theorem 5.7, Theorem 6.6]) +Let X be a projective toric +manifold with ρ(X) = 3. Then the number of primitive collections of ΣX is either l = 3 +or l = 5. Moreover, the set of generators G(ΣX) can be written as a disjoint union of l +nonempty subsets +G(ΣX) = X0 ⊔ · · · ⊔ Xl−1 +that define primitive collections and relations as follows: + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +11 +• Case l = 3. +Each X0, X1, X2 is a primitive collection, and the corresponding +primitive relations are extremal. +• Case l = 5. There are five primitive collections of the form Xi ⊔ Xi+1, 0 ≤ i ≤ 4, +where X5 := X0. To describe the primitive relations of X, we use the following +notation. We fix a labelling (v1, . . . , vk) for the elements of Xi. If c = (c1, . . . , ck) ∈ +Zk, then c · Xi stands for c1v1 + · · · + ckvk. Moreover, we set 1 = (1, . . . , 1). Then +there are vectors c and b of nonnegative integers such that at least one entry in c is +zero (up to relabelling, we may assume that c1 = 0), and the primitive relations of +X are the following: +r0 : +1 · X0 + 1 · X1 = c · X2 + (b + 1) · X3 +r1 : +1 · X1 + 1 · X2 = 1 · X4, +r2 : +1 · X2 + 1 · X3 = 0, +r3 : +1 · X3 + 1 · X4 = 1 · X1, +r4 : +1 · X4 + 1 · X0 = c · X2 + b · X3. +The relations r0, r1 and r3 are extremal, while r2 = r1 + r3 and r4 = r0 + r3. +Remark 2.18. In [SS20, Corollary 1.2], Sato and Suyama use Proposition 2.17 to show +that projective spaces are the only toric 2-Fano manifolds with Picard rank ρ ≤ 3. +2.4. Primitive collections on toric Fano manifolds. Let X be a projective toric man- +ifold with regular complete fan ΣX in NQ. +Proposition 2.19. ([Bat99, Proposition 2.3.6]) The toric variety X is Fano if and only if +all primitive collections of ΣX have strictly positive degree. +Proposition 2.20. ([Cas03a, Corollary 4.4]) Assume that X is Fano, and let P ∈ PC(X). +If deg(P) = 1, then the corresponding curve class is extremal. +Proposition 2.21. Assume that X is Fano, and let x ∈ G(ΣX). +(1) There is at most one primitive collection of order 2 and degree 2 containing x. If it +exists, then it is of the form x + (−x) = 0, and m(X) = 1. +(2) ([Cas03b, Lemma 3.3]) There are at most two primitive collections of order 2 and +degree 1 containing x. If there are exactly two of them, then they are of the form +x + y = (−w) and x + w = (−y), and m(X) = 1. +Corollary 2.22. Assume that X is Fano and m(X) > 1. Then any x ∈ G(ΣX) is contained +in at most one primitive collection of order 2. If there is such a primitive collection, then +it is of the form x + y = z. +Definition 2.23. Let x ∈ G(ΣX). We say that y ∈ G(ΣX) is an opponent of x if ⟨x, y⟩ /∈ +ΣX. +Notation 2.24. Assume that X is Fano and m(X) > 1. By Corollary 2.22, each vector +x ∈ G(ΣX) has at most one opponent. If such an opponent exists, we denote it by x′. +Lemma 2.25. Assume that X is Fano and m(X) > 1. Consider a pair of opponents +x, x′ ∈ G(ΣX). If there exist y, z ∈ G(ΣX) such that x + x′ = y + z, then z = y′. + +12 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +Proof. If y = x or y = x′, the claim follows automatically. So we assume otherwise. Note +that {x, x′} ∈ PC(X), and y and z do not form a cone, as otherwise x + x′ = y + z would +give us a primitive relation of degree 0, which is impossible for a toric Fano manifold. +□ +Lemma 2.26. Assume that X is Fano and m(X) > 1. Assume that there exist x, y, z, u, v +in G(ΣX) such that (∗) x + y + z = u + v and such that ⟨u, v⟩ ∈ ΣX. Then exactly one of +the following must happen: +a. +The vectors x, y, z are pairwise distinct, and {x, y, z} is a primitive collection with +primitive relation (∗). In particular, the corresponding curve class is extremal. +b. +Up to relabeling, v = z, y = x′ and x + x′ = u. +Proof. Assume two of {x, y, z} do not form a cone. For example, assume x, y do not form a +cone. Then y = x′, the opponent of x. Let x + x′ = α, for some α ∈ G(ΣX). We have that +α+z = u+v. As ⟨u, v⟩ ∈ ΣX, Proposition 2.6 implies that α+z = u+v corresponds to an +effective class of degree 0, which therefore implies that {α, z} = {u, v}. Up to relabeling, +we may assume v = z, and hence, x + x′ = u and we are in the situation b. +Assume now that any two vectors in {x, y, z} form a cone. Then x, y, z are mutually +disjoint and ⟨x, y, z⟩ /∈ ΣX, as otherwise by Proposition 2.6 we obtain an effective curve +class of degree −1, contradicting the fact that X is Fano. It follows that {x, y, z} is a +primitive collection. +Since ⟨u, v⟩ ∈ ΣX, it follows that (∗) is the associated primitive +relation. Proposition 2.20 now implies that the corresponding curve class is extremal. +□ +3. Toric Fano manifolds with m(X) = 1 +In this section, we study toric Fano manifolds with m(X) = 1, and follow the strategy +outlined in the introduction to show that they cannot be 2-Fano. +For any x ∈ G(ΣX), the set of primitive collections containing x is denoted by +PCx(X) = {P ∈ PC(X) | x ∈ P}. +Proposition 3.1. ([Cas03b, Lemma 3.1]) Assume that X is a toric Fano manifold and +that P = {x, −x} ∈ PC(X). +(1) Any Q ∈ PCx(X) \ {P} has degree 1 (hence is extremal by Proposition 2.20), and +r(Q) is of the form +r(Q): x + y1 + · · · + yh +� +�� +� +∈ ⟨Q\{x}⟩ += z1 + · · · + zh +� +�� +� +∈σ(Q) +, +where we denote ⟨Q \ {x}⟩ := ⟨y1, . . . , yh⟩ and σ(Q) := ⟨z1, . . . , zh⟩. +(2) For any R ∈ PCx(X) \ {P, Q}, we have +V (R \ {x}) ∩ V (Q \ {x}) = ∅ +and +V (R \ {x}) ∩ V (σ(Q)) = ∅. +(3) For any Q ∈ PCx(X) \ {P} with r(Q): x + y1 + · · · + yh = z1 + · · · + zh we have +Q′ = {−x, z1, . . . , zh} ∈ PC−x(X), Q′ has degree 1 (hence is extremal) and +r(Q′): − x + z1 + · · · + zh +� +�� +� +∈⟨Q′\{−x}⟩=σ(Q) += y1 + · · · + yh +� +�� +� +∈σ(Q′)=⟨Q\{x}⟩ +. + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +13 +Corollary 3.2. Let X be a toric manifold, and P = {x, −x} ∈ PC(X). +With Nota- +tion 2.12, +(4) +EP = {⟨Q \ {v}⟩ | Q ∈ PCv(X) \ {P}, +v = ±x} . +If moreover X is Fano, then V (EP) has 0, 2 or 4 components of codimension 1 in X. +Proof. Let X be a toric manifold, and P = {x, −x} ∈ PC(X). The description of V (EP) +in Equation (4) follows from Notation 2.12. In the Fano case, the number of components +of codimension 1 of V (EP) equals the number of primitive collections of order 2 and degree +1 containing x or −x, which is 0, 2 or 4 by Proposition 2.21 and Proposition 3.1. +□ +Proposition 3.3. Let X be a toric Fano manifold, and P = {x, −x} ∈ PC(X). Then +there exists a birational morphism f : X → Y such that +• PY := {x, −x} ∈ PC(Y ), +• V (EPY ) has codimension ≥ 2 in Y , +• f is a composition of at most two blow-downs with disjoint centers and smooth +target: +Exc(f) = +� +Q∈PCx(X)\{P} : +ord(Q)=2 +V (σ(Q)) ⊂ X, +f(Exc(f)) = +� +Q∈PCx(X)\{P} : +ord(Q)=2 +V (Q) ⊂ Y. +(5) +Proof. If V (EP) has codimension ≥ 2 in X, i.e., if P is the unique primitive collection of +order 2 containing x, then the statement holds with f = Id. +Assume now that V (EP) has 2 components of codimension 1, i.e. we have primitive +relations +r(P): x + (−x) = 0, +r(Q1): x + y = z, +r(Q′ +1): − x + z = y, +and, for any other R ∈ PCx(X) ∪ PC−x(X), one has ord(R \ {±x}) ≥ 2. Let f1 : X → Y +be the smooth blow-down induced by the extremal ray of NE(X) corresponding to r(Q1). +By Proposition 2.16 and Proposition 3.1, PCx(Y ) ∪ PC−x(Y ) consists of +- PY = {x, −x}; +- RY = RX for some RX ∈ PCx(X)∪PC−x(X)\{Q1} such that z /∈ RX. In particular +we have ord(RY \ {±x}) ≥ 2; +- RY = (RX\{z})∪{x, y} for some RX ∈ PCz(X) such that (RX\{z})∪{x} /∈ PC(X) +and (RX \ {z}) ∪ {y} /∈ PC(X). In particular, we have ⟨RY \ {x}⟩ = ⟨RX \ {z}, y⟩, +so ord(RY \ {x}) ≥ 2. +It follows that V (EPY ) has codimension ≥ 2 in Y , so the proposition holds with f = f1. +Assume now that V (EP) has 4 components of codimension 1, i.e., we have +r(P): x + (−x) = 0 +r(Q1): x + y = −w +r(Q′ +1): − x + (−w) = y +y + (−y) = 0 +r(Q2): x + w = −y +r(Q′ +2): − x + (−y) = w +w + (−w) = 0 +w + y = −x +−y + (−w) = x + +14 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +by Proposition 2.21. By [Cas03b, p.1487, Case (3)], any other primitive collection R ∈ +PC(X) is disjoint from {x, −x, y, −y, w, −w}. Let f1 : X → X1 be the smooth blow-down +induced by the extremal ray of NE(X) corresponding to r(Q1). By Proposition 2.16 the +primitive collections of X1 containing x or −x are only +PX1 = P +(Q2)X1 = Q2 +(Q′ +2)X1 = Q′ +2. +It follows that r(Q2) corresponds to an extremal curve class in NE(X1). Let f2 : X1 → X2 +be the smooth blow-down induced by the extremal ray of NE(X) corresponding to r(Q2). +By Proposition 2.16 PX2 = {x, −x} is the only primitive collection in PCx(X2)∪PC−x(X2). +This implies that V (EPX2) = ∅, so the proposition holds with Y = X2 and f = f2 ◦ f1. +□ +Construction 3.4. Let Y be a projective toric manifold of dimension ≥ 3, and P = +{x, −x} ∈ PC(Y ) a centrally symmetric primitive collection of order 2. Let V (EP) ⊂ Y be +the closed subset defined in Notation 2.12, set U := Y \ V (EP), and let π : U → W be the +P1-bundle given by Proposition 2.13. +Assume that V (EP) has codimension ≥ 2 in Y , and let Z ⊂ Y be any given closed subset +of codimension ≥ 2 in Y . Note that a toric manifold is rational, hence rationally connected, +so by [Kol96, Proposition II.3.7], there is a smooth (very free) rational curve C ⊂ U \ Z. +Consider the surface S := π−1(π(C)) ⊂ U, and let n: ˜S → S be its normalization. Then +˜S is a Hirzebruch surface with P1-bundle structure ˜π : ˜S → P1 induced by π. The curve +class of the image of a fiber of ˜π on Y corresponds to the centrally symmetric relation +x + (−x) = 0. By taking C general, we may assume that π(C) and π(Z) meet transversely +in at most finitely many general points. Hence S and Z meet transversely in at most finitely +many points. +Proposition 3.5. Let Y be a projective toric manifold of dimension ≥ 3, and PY = +{x, −x} ∈ PC(Y ) a centrally symmetric primitive collection of order 2. Assume that V (EP) +has codimension ≥ 2 in Y , and let S ⊂ Y be as in Construction 3.4. Then S · ch2(Y ) = 0. +Proof. Let S = π−1(π(C)) be the surface from Construction 3.4, n: ˜S → S its normaliza- +tion, ˜π: ˜S → P1 the P1-bundle structure induced by π, and F a fiber of ˜π. Our goal is to + +Z. +V(Ep) +S +C +(Z) +J(C)THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +15 +compute +S · ch2(Y ) = 1 +2 +� +v∈G(ΣY ) +S · V (v)2 = 1 +2 +� +v∈G(ΣY ) +� +n∗V (v) +�2. +Recall that the curve class of the image of F in Y is associated to the relation x+(−x) = 0. +By restricting the divisors n∗V (v) to F, we have: +� +� +� +� +� +n∗V (v) · F = 0 +if v ̸= x, −x, +n∗V (x) · F = 1, +n∗V (−x) · F = 1. +Hence there are sections σ, σ′ of ˜π: ˜S → P1, and α, β, γ ∈ Z such that, on ˜S, +� +� +� +� +� +n∗V (v) = αF +if v ̸= x, −x, +n∗V (x) = σ + βF, +n∗V (−x) = σ′ + γF. +Therefore +� +v∈G(ΣY ) +� +n∗V (v) +�2 = +� +v̸=x,−x + +� +n∗V (v) +�2 + +� +n∗V (x) +�2 + +� +n∗V (−x) +�2 += σ2 + 2β + σ′2 + 2γ += σ2 − 2(σ · σ′) + σ′2 +as σ · σ′ + β + γ = n∗V (x) · n∗V (−x) = 0 += (σ − σ′)2 = 0 +as σ − σ′ is a multiple of F, +and this concludes the proof. +□ +Lemma 3.6. ([dS06a, Lemma 5.1]) Consider the blowup diagram +E ⊂ +j> X := BlZ Y +Z +π:=f|E∨ +⊂ +> Y +f +∨ +where both Y and Z are smooth projective varieties and codimY Z = c ≥ 2. Then we have +the following relation between the 2nd Chern characters of X and Y : +ch2(X) = f ∗ ch2(Y ) + c + 1 +2 +E2 − j∗π∗ c1(NZ/Y ). +Corollary 3.7. In the setting of Lemma 3.6, assume that c = 2. Let S ⊂ Y be a surface +that intersects Z at most transversely at k ≥ 0 points, and let SX ⊂ X be its strict +transform. Then +ch2(X) · SX = ch2(Y ) · S − 3 +2 · k. +Proof. Write S ∩Z = {p1, . . . , pk}. Then SX is isomorphic to the blowup of S at p1, . . . , pk, +SX ∩ E = ∪k +i=1ei, where ei ≃ P1 is the exceptional curve over pi, and (e2 +i )SX = −1. By + +16 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +Lemma 3.6, +ch2(X) · SX = f ∗ ch2(Y ) · SX + c + 1 +2 +E2 · SX − j∗π∗ c1(NZ/Y ) · SX += ch2(Y ) · S + 3 +2(E|SX)2 − π∗ c1(NZ/Y ) · SX|E += ch2(Y ) · S + 3 +2 +k +� +i=1 +(ei)2 − π∗ c1(NZ/Y ) · +k +� +i=1 +ei += ch2(Y ) · S − 3 +2k − +k +� +i=1 +((((((( +c1(NZ/Y ) · pi. +□ +We are ready to prove Theorem 1.2: a toric Fano manifold X with m(X) = 1 is not +2-Fano. +Proof of Theorem 1.2. Let X be a toric Fano manifold with m(X) = 1, and fix a centrally +symmetric primitive relation r(P): x + (−x) = 0. Let f : X → Y be as in Proposition 3.3, +and π: U = Y \ V (EPY ) → W the P1-bundle structure induced by r(PY ): x + (−x) = 0 +(see Proposition 2.13). By Equation (5), Z := f(Exc(f)) has codimension ≥ 2 in Y . Let +S ⊂ Y be the surface given by Construction 3.4. Then S and Z meet transversely in at +most finitely many points. It follows from Proposition 3.5 that ch2(Y ) · S = 0. Let SX be +the strict transform of S in X. By Corollary 3.7, ch2(X) · SX ≤ ch2(Y ) · S = 0, and so X +is not 2-Fano. +□ +4. Proof of Theorem 1.3 +In this section, we work in the setting of Theorem 1.3: +(I) X is a Fano manifold of dimension n ≥ 6 with fan Σ = ΣX in NQ; +(II) m(X) ≥ 3; +(III) r(P): x0 + · · · + xn−2 = 0 is a centrally symmetric primitive relation in Σ. +The equality m(X) = n − 2, as well as uniqueness of the centrally symmetric primitive +collection, follows immediately from Lemma 2.5. Now the goal is to prove that ρ(X) ≤ 3, +and this will follow from Proposition 4.11, Proposition 4.16 and Proposition 4.17. +By +Remark 2.18, we conclude that the projective space Pn is the only smooth n-dimensional +toric 2-Fano variety with m(X) ≥ n − 2 (Corollary 1.5). +Let P := {x0, . . . , xn−2} and set Γ = Span P ⊂ NQ. Consider the quotient +π: NQ ≃ Qn → Qn/Γ ≃ Q2. +Since X is complete, the support of Σ is equal to NQ, and hence we can find generators +(IV) x, y, z ∈ G(Σ) \ P, for which +(V) 0 ∈ Conv(π(x), π(y), π(z)). +Lemma 4.1. The nonnegative span ⟨x, y, z⟩ is not a cone in Σ. + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +17 +Proof. We will argue by contradiction and assume that ⟨x, y, z⟩ ∈ Σ. By (V), we can find +a nonnegative triple of constants (c1, c2, c3) such that c1π(x) + c2π(y) + c3π(z) = 0, or in +other words +v := c1x + c2y + c3z = a0x0 + · · · + an−2xn−2 +for some constants ai. Note that by adding some multiple of r(P) (III) to the right hand +side, we can assume the ai’s are nonnegative and such that at least one of them, say aj, is +0. It follows that v lies in two cones of Σ, namely +v ∈ ⟨x, y, z⟩ ∩ ⟨x0, . . . , ˇxj, . . . , xn−2⟩, +which is impossible since {x, y, z} ∩ S = ∅. +□ +Since {x, y, z} does not span a cone, we conclude that +(1) either it is a primitive collection, +(2) or two of these vectors do not form a cone. +The former case is the more technical one, and we start with it in Section 4.1. After we +are done analyzing it, we can assume that none of the triples {x, y, z} as in (IV) and (V) +form a primitive collection, and this case will be treated in Section 4.2. +4.1. First case: {x, y, z} is a primitive collection. We recall that if X is a projective +toric Fano manifold and m(X) > 1, then any x ∈ G(Σ) has at most one opponent by +Corollary 2.22, where the opponent of x is an element x′ ∈ G(Σ) such that {x, x′} ∈ PC(X). +When we write {x, x′}, we mean either the set of two elements if x′ exists, or the singleton +{x} if x′ does not exist. +Remark 4.2. In the setting (I)—(V), pick a generator u ∈ G(Σ). Then (V) and plane +geometry imply that the convex hull of π(u) together with two of the vectors π(x), π(y), +π(z) contains 0. +Lemma 4.3. In the setting (I)—(V), assume in addition that Q := {x, y, z} is a primitive +collection. Then the corresponding primitive relation is +• either r(Q): x + y + z = xi + xj for possibly equal xi, xj ∈ P, +• or r(Q): x + y + z = v for some v ∈ G(Σ). +Proof. Since X is Fano (I), the degree of the primitive relation x + y + z = A is positive, +so we can have only three possibilities for A. The first one with A = 0 is actually not +possible by our assumption (II). The second is A = v, and we cannot say much about v at +the moment. The last possibility is +r(Q): x + y + z = u + v +for some, possibly equal, u, v ∈ G(Σ). By Proposition 2.20, this is an extremal primi- +tive relation, so by Proposition 2.8 applied to r(Q) and τ = {0}, we get that ⟨u, x, y⟩, +⟨u, y, z⟩, ⟨u, x, z⟩ ∈ Σ. By Remark 4.2, we may assume without loss of generality that +0 ∈ Conv(π(u), π(x), π(y)), hence by Lemma 4.1, we get u ∈ P. +The same argument +applies to conclude v ∈ P. +□ +Hence we have two cases to consider: when deg(Q) is 1 and when it is 2. + +18 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +4.1.1. Degree one. In this case, by Lemma 4.3, we have +(VI) a primitive relation r(Q): x + y + z = xi + xj for possibly equal xi, xj ∈ P. +Lemma 4.4. In the setting (I)—(VI), assume in addition that G(Σ) is contained in P ∪ +{x, y, z, x′, y′, z′}. Then x′, y′, z′ do not exist. Consequently, ρ(X) ≤ 2. +Proof. By contradiction, assume that x′ ∈ G(Σ) exists, so {x, x′} is a primitive collection, +and let x + x′ = α be the corresponding primitive relation. Clearly α ̸= x, x′. The relation +r(Q) (VI) gives +α + y + z = xi + xj + x′, +which shows α ̸∈ P ∪ {y, z} since otherwise the left hand side (LHS) forms a cone and we +get an effective class of degree zero. Indeed, if α ∈ {y, z} it is clear that the LHS would +form a cone; if α ∈ P applying Proposition 2.8 to r(Q) and τ = ⟨α⟩ would follow that the +LHS is a cone. +So without loss of generality, we can assume x + x′ = y′. Applying the same argument +to y′, we get that y +y′ = x′ or y +y′ = z′. The former would imply x+y = 0, which is not +possible by (II), so y +y′ = z′. Again, applying the same argument to z′, we get z +z′ = x′. +Summing the three primitive relations, we obtain x+y+z = 0, which contradicts (VI). +□ +This leaves us with the case when +(VII) there exists u ∈ G(Σ) \ (P ∪ {x, y, z, x′, y′, z′}). +By Remark 4.2, we can assume without loss of generality that +(VIII) 0 ∈ Conv(π(x), π(y), π(u)). +Lemma 4.5. Assume (I)—(VIII), then R := {x, y, u} is a primitive collection with primi- +tive relation r(R): x + y + u = v for some v ∈ G(Σ). +Proof. By Lemma 4.1, we have ⟨x, y, u⟩ /∈ Σ, and since u ̸= x′, y′, we conclude that {x, y, u} +is a primitive collection. By Lemma 4.3, we can have either r(R): x + y + u = xk + xl or +r(R): x + y + u = v. In the former case, combining r(R) with r(Q) (VI) provides us with +a relation +u + xi + xj = z + xk + xl. +By applying Proposition 2.8 to r(Q) (VI) and τ = ⟨xk, xl⟩, we get ⟨z, xk, xl⟩ ∈ Σ (here +we are using the assumption dim(X) = n ≥ 6 (I)). Hence, by Proposition 2.6, we get an +effective curve class of degree 0, contradicting the Fano assumption (I). +□ +We will write down the result of Lemma 4.5 as an additional assumption, remembering +that it is implied by the previous assumptions: +(IX) We have a primitive relation r(R): x + y + u = v for some v ∈ G(Σ). +Lemma 4.6. Assume (I)—(IX), then +G(Σ) ⊂ P ∪ {x, y, z, u, x′, y′, u′, v′}. +In particular, z′ ∈ P ∪ {x, y, z, u, x′, y′, u′, v′}, and since v ̸= x, y, u, v′, we have that v ∈ P +or v ∈ {z, x′, y′, u′}. +Proof. Take any w ∈ G(Σ) \ (P ∪ {x, y, z, u, x′, y′, u′}). By Remark 4.2, the convex hull +of π(w) together with two of π(x), π(y), π(u) contains 0, yielding an analog of (VIII). By + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +19 +Lemma 4.1 and from w ̸= x′, y′, u′, it follows that one of {w, x, u}, {w, y, u}, {w, x, y} is a +primitive collection. +We will prove that the corresponding primitive relation has the form w + x + u = b or +w+y+u = b or w+x+y = b for some b ∈ G(Σ). Assume for a contradiction that this is not +the case. By Lemma 4.3, we have that one of w+x+u, w+y+u, or w+x+y equals xk+xl, +for possibly equal xk, xl ∈ P. Hence there exist a, b ∈ G(Σ) (one of which is w ̸= z) such +that either x+a+b or y +a+b equals xk +xl. Assume x+a+b = xk +xl. Combining this +with r(Q) (VI), it follows that y +z +xk +xl = a+b+xi +xj. By applying Proposition 2.8 +to r(Q) (VI) and τ = ⟨xk, xl⟩, we get ⟨y, z, xk, xl⟩ ∈ Σ (here we are using the assumption +dim(X) = n ≥ 6 (I)). Hence, by Proposition 2.6, we get an effective curve class of degree +0. The class is non-trivial since w ̸= y, z, xk, xl. This contradicts the assumption that X is +Fano. The case y + a + b = xk + xl is similar. +So we have a primitive relation of the form w+x+u = b or w+y+u = b or w+x+y = b +for some b ∈ G(Σ). Combining this with r(R) (IX), we get b + y = w + v or b + x = w + v +or b + u = w + v. All possibilities imply that w = v′, as otherwise, by Proposition 2.6, we +obtain an effective class of degree 0 (non-trivial since w, v ̸= y, u), which contradicts the +Fano assumption (I). +□ +Lemma 4.7. Assume (I)—(IX). Then x, y don’t have opponents, +G(Σ) ⊂ P ∪ {x, y, z, u, u′, v′}, +and we have a trichotomy: v ∈ P or v = z or v = u′. +Proof. We will only show that x′ does not exist, and the argument for y′ is symmetric. +Suppose to the contrary that x′ ∈ G(Σ), and let x + x′ = α be the corresponding primitive +relation, so α ̸= x, x′. Then we can substitute x = α − x′ into r(Q) (VI) to get +α + y + z = x′ + xi + xj, +which shows α /∈ P ∪ {y, z}, as otherwise we would get an effective curve class of degree 0. +Substituting x = α − x′ into r(R) (IX) gives the relation +(6) +α + y + u = v + x′. +Note that ⟨v, x′⟩ ∈ Σ by Corollary 2.22. We claim that Equation (6) is an extremal +primitive relation by Lemma 2.26. +Assume not, then Lemma 2.26 implies that one of +{α, y, u} is in {v, x′} and the remaining two vectors are opponents. Since u ̸= y′, then +either α, y are opponents and u ∈ {v, x′}, or α, u are opponents and y ∈ {v, x′}. From +(IX), we have u ̸= v and y ̸= v; (VII) means u ̸= x′; and (VI) implies y ̸= x′. So both cases +are impossible. +So Equation (6) is an extremal primitive relation. +In particular, α ̸= y′, u, u′. +By +Proposition 2.8, v forms a cone with α, hence α ̸= v′, which contradicts Lemma 4.6. +□ +Lemma 4.8. Assume (I)—(IX). Then we have +G(Σ) ⊂ P ∪ {x, y, z, u, v′} +and a dichotomy: v ∈ P or v = z. If moreover u′ exists, we have u + u′ = z, v = xi and +u = x′ +j. + +20 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +Proof. Assume that u′ exists and let u + u′ = α be the corresponding primitive relation. +Then substituting it into r(R) (IX) gives a relation +(7) +x + y + α = u′ + v. +By r(R) (IX), v ̸= u, so ⟨u′, v⟩ ∈ Σ. Since there are no x′ and y′, it follows from Lemma 2.26 +that Equation (7) is an extremal primitive relation. It follows that α /∈ {u, u′, x, y, v′}. +Moreover, α /∈ P, as otherwise applying Proposition 2.8 to r(Q) and τ = ⟨α⟩ would +imply ⟨x, y, α⟩ ∈ Σ. +So by Lemma 4.7, the only possibility is that α = z. +Therefore +u′ + v = xi + xj by r(Q) (VI), so we have, after possibly relabeling, that v = xi, u′ = xj, +which by Lemma 4.7 proves the statement. +If instead u′ does not exist, the statement follows directly from Lemma 4.7. +□ +Lemma 4.9. Assume (I)—(IX). Then z doesn’t have an opponent. +Proof. We argue by contradiction and assume that z′ exists. Clearly z′ ̸= x, y, z by r(Q) +(VI) and z′ ̸= u by (VII). Furthermore, z′ ̸= u′, otherwise we have z = u by Corollary 2.22, +which contradicts (VII). Finally, z′ /∈ P, otherwise z′ = xk implies z = x′ +k, and r(Q) (VI) +becomes +x + y + x′ +k = xi + xj, +but applying Proposition 2.8 to r(Q) and τ = ⟨xk⟩, we get ⟨xk, x′ +k⟩ ∈ Σ, a contradiction. +Thus by Lemma 4.8, the only possibility is z′ = v′, so z = v by Corollary 2.22. +By +Lemma 4.8, this implies G(Σ) ⊂ P ∪ {x, y, z, u, z′}. +Consider the primitive relation z + z′ = β ∈ G(Σ). We will show that β = u. Indeed, it +is clear that β ̸= z, z′. Combining z + z′ = β with r(Q) (VI), we have +x + y + β = xi + xj + z′. +If β ∈ {x, y}, the left hand side is a cone, and we get a non-trivial effective relation of degree +0, which is impossible by (I). If β ∈ P, applying Proposition 2.8 to r(Q) and τ = ⟨β⟩ implies +that ⟨x, y, β⟩ ∈ Σ, and we obtain a contradiction as before. +But now, substituting z +z′ = u into r(R) (IX) yields x+y +z′ = 0, hence contradicting +m(X) ≥ 3 (II). +□ +Lemma 4.10. Assume (I)—(VII). Then ρ(X) ≤ 3. +Proof. Let us summarize the consequences of (I)—(VII): +(VIII) 0 ∈ Conv(π(x), π(y), π(u)) after possibly relabeling x, y, z; +(IX) we have a primitive relation r(R): x + y + u = v (Lemma 4.5); +• x, y, z don’t have opponents (Lemma 4.7, Lemma 4.9); +• G(Σ) ⊂ P ∪ {x, y, z, u, v′} (Lemma 4.8), so ρ(X) ≤ 4; +• we have v ∈ P or v = z (Lemma 4.8), so we can consider two cases. +Case v = z. By Lemma 4.9, v′ = z′ does not exist, hence it follows from Lemma 4.8 that +G(Σ) ⊂ P ∪ {x, y, z, u}, so ρ(X) ≤ 3. +Case v ∈ P, say v = xl. If v = xl doesn’t have an opponent, then G(Σ) ⊂ P ∪ {x, y, z, u} +and ρ(X) ≤ 3. So the tricky case is when v = xl has an opponent x′ +l /∈ P. +Let v + v′ = β ∈ G(Σ). By r(R), we have that x + y + u + v′ = v + v′ = β. Since +m(X) ≥ 3, β ̸= x, y, u, v′. Hence β ∈ P ∪ {z}. If β ∈ P, let β = xk. Then v′ = xk − xl and +k ̸= l. Using r(P), we obtain that v′ = 2xk + � +t̸=k,l xt. As the vectors on the right hand +side form a cone, we get an effective curve class of degree 2 − n, which is impossible by (I). + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +21 +So β = z. By r(Q), we obtain that x + y + v′ = xi + xj − xl. If l ̸= i, j, then again, +by using r(P) (III), we may write xi + xj − xl as xi + xj + � +t̸=l xt. As the vectors on the +right hand side form a cone, we obtain an effective curve class of degree 3 − n, which is +impossible by (I). +So l = i or l = j. Up to symmetry, we may assume l = i, so v = xi, xi +x′ +i = z. By r(Q), +we have that x + y + x′ +i = xj. Combining this with r(R), we obtain that x′ +i + xi = u + xj. +Furthermore, u ̸= v′ = x′ +i and u ̸= xi. If u, xj form a cone, we obtain an effective non-trivial +curve class of degree 0, which is impossible by (I). So u = x′ +j, v = xi and we have two +primitive relations +xi + x′ +i = z +and +xj + x′ +j = z. +Then either i = j, in which case ρ(X) ≤ 3, or i ̸= j, and we notice that they are both +degree 1 and hence extremal, so we can perform the contraction associated to one of them, +say we contract the curve class xi + x′ +i = z: +X +Y, +V (z) +V (xi, x′ +i). +By Proposition 2.16, +r(PY ): x0 + · · · + xn−2 = 0, +r(RY ): x + y + x′ +j = xi, +r(Q′): x + y + x′ +i = xj, +r(Q′′): xj + x′ +j = xi + x′ +i +are primitive relations in Y . Since ρ(Y ) = 3, we apply Proposition 2.17. We adopt the +same notation as in Proposition 2.17 and observe that we are in the case l = 5, hence +G(Σ) = ⊔4 +h=0Xh = {x0, . . . , ˇxj, . . . , xn−2} ⊔ {xj} ⊔ {x, y} ⊔ {x′ +j} ⊔ {x′ +i}, +and either X2 ⊔X3 = {x0, . . . , xn−2}, or c = b = 0 and X4 ⊔X0 = {x0, . . . , xn−2}. However, +all possibilities for {xj} lead to a contradiction. Indeed: +- if X3 = {xj}, then we have r3 : 1 · X3 + 1 · X4 = xj + x′ +j = xi + x′ +i ̸= 1 · Xh for all +h = 0, . . . , 4; +- if X2 = {xj}, then we have r1 : 1 · X1 + 1 · X2 = x′ +j + xj = xi + x′ +i ̸= 1 · Xh for all +h = 0, . . . , 4; +- if X4 = {xj}, then we have r3 : 1 · X3 + 1 · X4 = x′ +j + xj = xi + x′ +i ̸= 1 · Xh for all +h = 0, . . . , 4; +- if X0 = {xj}, then we have r0 : 1·X0+1·X1 = xj+x′ +j = xi+x′ +i ̸=  + +c · X2+(��b+1)X3 = +1 · X3. +This concludes the last case, and we get that ρ(X) ≤ 3. +□ +Proposition 4.11. Let X be a projective toric Fano manifold of dim(X) = n ≥ 5, m(X) ≥ +3, which admits a primitive relation r(P): x0 + x1 + · · · + xn−2 = 0 (I)—(III). Let x, y, z ∈ +G(Σ) \ P be such that 0 ∈ Conv(π(x), π(y), π(z)) (IV)—(V). Assume in addition that +{x, y, z} is a primitive collection of degree 1 (VI). Then ρ(X) ≤ 3. + +22 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +Proof. We recall that assumptions (I)—(III) imply (IV)—(V). Then either G(Σ) ⊂ P ∪ +{x, y, z, x′, y′, z′}, in which case ρ(X) ≤ 2 by Lemma 4.4, or there exists a vector u ∈ +G(Σ) \ (P ∪ {x, y, z, x′, y′, z′}) (VII), in which case Lemma 4.10 implies ρ(X) ≤ 3, and we +are done. +□ +4.1.2. Degree two. Still working in the setting (I)—(V), we assume that {x, y, z} is a prim- +itive collection whose primitive relation has degree 2, and by Proposition 4.11, we can +exclude the case when degree one primitive collections as in (IV)—(VI) exist. In other +words, these are the additional assumptions for this part of the proof: +(X) we have a primitive relation r(Q): x + y + z = v for some v ∈ G(Σ); +(XI) there is no primitive collection {a, b, c} ⊂ G(Σ) \ P with 0 ∈ Conv(π(a), π(b), π(c)) +whose primitive relation has degree 1. +Lemma 4.12. In the setting (I)—(V) and (X)—(XI), we have +G(Σ) ⊂ P ∪ {x, y, z, x′, y′, z′, v′} +and +v ∈ P ∪ {x′, y′, z′}. +Proof. If there is a generator u ∈ G(Σ) \ (P ∪ {x, y, z, x′, y′, z′}), then by Remark 4.2 and +(XI) we can assume that we have a primitive relation of the form +x + y + u = w. +Combining it with (X), we get u + v = w + z, so u = v′, otherwise ⟨u, v⟩ ∈ Σ and we get +an effective non-trivial curve class of degree zero. +In particular, since v ̸= v′, we have v ∈ P ∪ {x′, y′, z′}. +□ +Lemma 4.13. In the setting (I)—(V) and (X)—(XI), assume that v = xm ∈ P. Then x′ +m +does not exist. +Proof. The primitive relation r(Q) (X) becomes x + y + z = xm, and by Lemma 4.12, we +have G(Σ) ⊂ P ∪ {x, y, z, x′, y′, z′, x′ +m}. +Assume to the contrary that we have x′ +m ∈ G(Σ). Clearly, x′ +m /∈ P because xm forms +a 2-dimensional cone in Σ with any other generator xi ∈ P. By Remark 4.2, x′ +m makes +a primitive collection with two of x, y, z. Without loss of generality, assume we have a +primitive relation of the form +x + y + x′ +m = w. +Combining it with r(Q) (X), we get x′ +m + xm = z + w, so w = z′ by Lemma 2.25. Let +α = xm+x′ +m = z+z′. Then α ̸∈ P because x′ +m = α−xm is not in Span P. Now substituting +z = α − z′ into r(Q) (X) yields +(8) +x + y + α = xm + z′. +Since xm ̸= z by r(Q) (X), we have ⟨xm, z′⟩ ∈ Σ, so Equation (8) is an extremal primitive +relation. This implies that α ̸= x, y, z, x′, y′, z′, x′ +m, a contradiction with Lemma 4.12. +□ +Lemma 4.14. In the setting (I)—(V) and (X)—(XI), assume that v = xm ∈ P. Then +ρ(X) ≤ 3. +Proof. By Lemma 4.12 and Lemma 4.13, we have G(Σ) ⊂ P ∪ {x, y, z, x′, y′, z′}, and, as +before, the primitive relation (X) is r(Q): x + y + z = xm. + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +23 +We show that at most one of x′, y′, z′ exists. Suppose to the contrary that for example +x′, y′ ∈ G(Σ), and let α = x + x′, β = y + y′ ∈ G(Σ). Since +α + y + z = xm + x′ +and +⟨xm, x′⟩ ∈ Σ, +applying Lemma 2.26 shows that this is an extremal primitive relation: indeed, either +α and y are opponents and z ∈ {xm, x′}, or α and z are opponents and y ∈ {xm, x′}. +But y, z /∈ P ∪ {x′}, so this is a contradiction. It follows that α ̸∈ {x, x′, y, y′, z, z′}, so +α = xl ∈ P; similarly, β = xk ∈ P. But we have +α + β + z = x′ + y′ + xm, +which is not possible since we show that the right hand side forms a cone. Indeed, applying +Proposition 2.8 to x + x′ = xl and τ = ⟨xm, xk⟩ (we use here that n ≥ 5), we obtain +⟨xm, xk, x′⟩ ∈ Σ, and applying then Proposition 2.8 to y +y′ = xk and τ = ⟨xm, x′⟩ we have +that ⟨xm, x′, y′⟩ ∈ Σ. So, after possibly relabelling x, y, z, we have G(Σ) ⊂ P ∪{x, y, z, x′}, +hence ρ(X) ≤ 3. +□ +Lemma 4.15. In the setting (I)—(V) and (X)—(XI), assume that v = x′. Then ρ(X) ≤ 3. +Proof. We have r(Q): x+y+z = x′. We know G(Σ) ⊂ P∪{x, y, z, x′, y′, z′} by Lemma 4.12. +So it is enough to show y′ and z′ do not exist. Suppose to the contrary that, for example, +y′ ∈ G(Σ), and let β = y + y′. Then +β + x + z = x′ + y′ +and +⟨x′, y′⟩ ∈ Σ, +so again, by Lemma 2.26, this is an extremal primitive relation. Indeed, assume the relation +is not primitive. By Lemma 2.26, either β and x are opponents and z ∈ {x′, y′}, or β and z +are opponents and x ∈ {x′, y′}. But z, x /∈ {x′, y′}, since {x, y, z} is a primitive collection. +Hence β + x + z = y′ + x′ is a primitive relation. It follows from Proposition 2.8 that +⟨x, x′⟩ ∈ Σ, which is a contradiction. +□ +Proposition 4.16. Let X be a projective toric Fano manifold of dim(X) = n ≥ 5, m(X) ≥ +3, which admits a primitive relation r(P): x0 + x1 + · · · + xn−2 = 0 (I)—(III). Assume that +any primitive collection {x, y, z} such that x, y, z ∈ G(Σ)\P and 0 ∈ Conv(π(x), π(y), π(z)) +(IV) —(V) has degree 2 (XI), and that there exists such a triple x, y, z (X). Then ρ(X) ≤ 3. +Proof. The statement follows from Lemma 4.12, Lemma 4.14 and Lemma 4.15. +□ +4.2. Second case: none of {x, y, z} form a primitive collection. +Proposition 4.17. Let X be a toric Fano manifold of dim(X) = n ≥ 5, m(X) ≥ 3, which +admits a primitive relation r(P): x0 + x1 + · · · + xn−2 = 0 (I)—(III). Assume in addition +that none of the triples {x, y, z} ⊆ G(Σ)\P such that 0 ∈ Conv(π(x), π(y), π(z)) (IV)—(V) +form a primitive collection. Then ρ(X) ≤ 3. +Before proving Proposition 4.17, we will formulate the following useful lemma. +Lemma 4.18. In the setting of Proposition 4.17, take any triple {x, y, z} ⊆ G(Σ) \ +P with ⟨x, y⟩ ∈ Σ. +Assume that 0 ∈ Conv(π(x), π(y), π(z)), or equivalently, π(z) ∈ +⟨−π(x), −π(y)⟩. Then z = x′ or z = y′. +Proof. By Lemma 4.1, x, y, z do not span a cone, and by assumption, they do not form a +primitive collection. So two of the vectors must not form a cone. Since we assumed that +⟨x, y⟩ ∈ Σ, we must have z = x′ or z = y′. +□ + +24 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +Proof of Proposition 4.17. We select x, y ∈ G(Σ) with ⟨x, y⟩ ∈ Σ and such that the cone +generated by π(x) and π(y) is maximal among cones in Q2 ≃ Qn/Γ coming from such +pairs. If there is z ∈ G(Σ) such that π(z) is outside the cone ⟨π(x), π(y)⟩, then we show +that z = x′ or z = y′. Indeed, the case π(z) ∈ ⟨−π(x), −π(y)⟩ is covered by Lemma 4.18; +π(z) ∈ ⟨π(x)⟩ or π(z) ∈ ⟨π(y)⟩ cannot happen by an argument similar to Lemma 4.1; and +in the remaining case, π(z) is in ⟨π(x), −π(y)⟩ or ⟨−π(x), π(y)⟩, then we use maximality of +the cone generated by π(x) and π(y). +Let v be such that π(v) is in the open half plane determined by Span π(x) not containing +π(y). +Up to relabelling of x, y, we can assume that such a v exists. +If v = x′, then +x + x′ = y′, since π(x + x′) is non-zero and is outside of ⟨π(y), π(x)⟩. So +⟨π(x), π(y)⟩ ⊂ ⟨−π(y), −π(x′)⟩ ∪ ⟨−π(x′), −π(y′)⟩ ∪ ⟨−π(y′), −π(x)⟩. +It follows from Lemma 4.18 that G(Σ) = P ∪ {x, y, x′, y′}, which concludes this case. +If now v = y′, in case π(v) ∈ ⟨−π(x), −π(y)⟩, we notice that y + y′ = x′ as above, and in +case π(v) ∈ ⟨π(x), −π(y)⟩, by completeness, we find w such that π(w) ∈ ⟨−π(x), π(y)⟩ ∪ +⟨−π(x), −π(y)⟩ and notice w = x′. In either case, we get +Q2 = ⟨−π(x), −π(y)⟩ ∪ ⟨−π(y), −π(x′)⟩ ∪ ⟨−π(x′), −π(y′)⟩ ∪ ⟨−π(y′), −π(x)⟩, +and now Lemma 4.18 gives G(Σ) = P ∪ {x, y, x′, y′}. +□ +5. Toric Fano manifolds with m(X) = n − 2 +In this section, we classify all n-dimensional toric Fano manifolds X with m(X) = n − 2 +and n ≥ 6 (Theorem 1.4). By Theorem 1.3, we know that ρ(X) ≤ 3. Theorem 5.1 and +Theorem 5.2 classify n-dimensional toric Fano manifolds X with m(X) = n − 2, n ≥ 5, +and Picard rank ρ(X) = 2 and 3, respectively. Together, these results yield a classification +of toric Fano manifolds with m(X) = n − 2 and n ≥ 6. +Theorem 5.1. Let X be a toric Fano manifold of dimension n ≥ 5, m(X) = n − 2 and +ρ(X) = 2. Then X ≃ PP2(E), where E is one of the following vector bundles on P2: +• E = OP2(1) ⊕ O⊕n−2 +P2 +, +• E = OP2(1) ⊕ OP2(1) ⊕ O⊕n−3 +P2 +, +• E = OP2(2) ⊕ O⊕n−2 +P2 +. +Proof. Let X be an n-dimensional toric Fano manifold with m(X) = n − 2 and ρ(X) = +2. We recall that toric manifolds with Picard rank 2 are classified by [Kle88]: they are +projective space bundles over projective spaces. The assumption m(X) = n − 2 implies +that X is a Pn−2-bundle over P2. +We can write X = P(E), where E = OP2(a0) ⊕ OP2(a1) ⊕ · · · ⊕ OP2(an−2) and a0 ≥ a1 ≥ +· · · ≥ an−2 = 0. The Fano assumption on X is equivalent to saying that �n−2 +i=0 ai ≤ 2. Thus +we have the following cases: +(1) E = O⊕n−1 +P2 +and X ≃ Pn−2 × P2, +(2) E = OP2(1) ⊕ O⊕n−2 +P2 +, +(3) E = OP2(1) ⊕ OP2(1) ⊕ O⊕n−3 +P2 +, +(4) E = OP2(2) ⊕ O⊕n−2 +P2 +. +In Case (1), we have m(X) = 2. Cases (2)—(4) provide the complete list of toric Fano +manifolds of Picard rank 2 and m(X) = n − 2. +□ + +THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS +25 +Theorem 5.2. Let X be a toric Fano manifold of dimension n ≥ 5, m(X) = n − 2 and +ρ(X) = 3. Then one of the following holds: +(1) X = PS(E) is a Pn−2-bundle over a toric surface S, where (S, E) is one of the +following: +• S = P1 × P1 and E = OP1×P1(1, 1) ⊕ O⊕n−2 +P1×P1, +• S = P1 × P1 and E = OP1×P1(1, 0) ⊕ OP1×P1(0, 1) ⊕ O⊕n−3 +P1×P1, +• S = F1 and E = OF1(e+f)⊕O⊕n−2 +F1 +, where e ⊂ F1 is the −1-curve, and f ⊂ F1 +is a fiber of F1 → P1. +(2) Let Y ≃ PP2� +OP2(1)⊕O⊕n−2 +P2 +� +be the blowup of Pn along a linear subspace L = Pn−3, +and denote by E ⊂ Y the exceptional divisor. Then X is the blowup of Y along a +codimension 2 center Z ⊂ Y , where: +• Z is the intersection of E with the strict transform of a hyperplane of Pn +containing the linear subspace L, or +• Z is the intersection of the strict transforms of two hyperplanes of Pn, one +containing the linear subspace L, and the other one not containing it. +Proof. We apply Batyrev’s classification of toric manifolds with ρ(X) = 3, stated in Propo- +sition 2.17. We adopt the same notation as in Proposition 2.17, and treat separately the +cases when the number of primitive collections is l = 3 and l = 5. +Case l = 3. As G(Σ) = X0 ⊔ X1 ⊔ X2 and m(X) = n − 2, we have X0 = {x0, . . . , xn−2}, +X1 = {v1, v2} and X2 = {z1, z2}. The corresponding primitive relations are all extremal +by Proposition 2.17. +By Proposition 2.13, X is a Pn−2-bundle over a surface S. +Up +to relabelling in X0, X1 and X2, three possible choices for the remaining two primitive +relations are (noting that m(X) = n − 2 > 1): +� +v1 + v2 = x0, +z1 + z2 = v1; +� +v1 + v2 = x0, +z1 + z2 = x0; +� +v1 + v2 = x0, +z1 + z2 = x1. +It follows that S is isomorphic to F1 when r(X1) and r(X2) are as in the first column +above, or to P1 × P1 in two other cases. +Case l = 5. We denote by li = |Xi| ≥ 1 the cardinality of Xi. +If (c, b) = (0, 0), then both r2 : 1 · X2 + 1 · X3 = 0 and r4 : 1 · X4 + 1 · X0 = 0 are centrally +symmetric primitive relations. +By the assumption m(X) = n − 2, we have 2n − 2 ≤ +l2 + l3 + l4 + l0 ≤ |G(Σ)| − 1 = n + 2, which implies n ≤ 4, a contradiction. +So we have (c, b) ̸= (0, 0), and P := X2 ⊔ X3 is the only centrally symmetric primitive +collection, so l2 + l3 = n − 1 and l0 + l1 + l4 = 4, with l0, l1, l4 ∈ {1, 2}. As deg(r0) > 0, we +get the inequality +3 ≥ l0 + l1 > +� +ci + +� +bj + l3, +which is only satisfied when l3 = 1, l0 + l1 = 3 and exactly one entry in +(c1, c2, . . . , cl2, b1) +equals one, while the others are all zero. Up to relabelling, there are two cases: c1 = 1 or +b1 = 1. +We then have l4 = 4 − (l0 + l1) = 1. From deg(r3) > 0, we get +l3 + l4 > l1, + +26 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +which means 2 > l1, and this ensures l1 = 1, hence (l0, l1, l4) = (2, 1, 1). +To sum up, we denote X0 = {v1, v2}, X1 = {y}, X2 = {x1, . . . , xn−2}, X3 = {x0} and +X4 = {z}, and get the following two possibilities for X: +• b1 = 1 and cj = 0 for every j: +r0 : +v1 + v2 + y = 2x0, +r1 : +y + x1 + · · · + xn−2 = z, +r2 : +x0 + · · · + xn−2 = 0, +r3 : +x0 + z = y, +r4 : +z + v1 + v2 = x0. +• c1 = 1, b1 = 0 and cj = 0 for every j > 1: +r0 : +v1 + v2 + y = x1 + x0, +r1 : +y + x1 + · · · + xn−2 = z, +r2 : +x0 + · · · + xn−2 = 0, +r3 : +x0 + z = y, +r4 : +z + v1 + v2 = x1. +By Proposition 2.13, the open U = X \ +� +V (y) ∪ V (z) +� +has a Pn−2-bundle structure. The +relation r3 corresponds to an extremal curve class by Proposition 2.20, which induces a +smooth blow-down h: X → Y . By Proposition 2.16, the primitive relations in Y in cases +b1 = 1 and c1 = 1 are, respectively: +� +x0 + · · · + xn−2 = 0, +z + v1 + v2 = x0; +� +x0 + · · · + xn−2 = 0, +z + v1 + v2 = x1. +It follows that Y ≃ P +� +OP2(1) ⊕ O⊕n−2 +P2 +� +, i.e., Y is the blowup of Pn along a linear subspace +L = V (z, v1, v2) ≃ Pn−3 ⊂ Pn, with exceptional divisor E = V (x0) ⊂ Y in the first case, and +E = V (x1) ⊂ Y in the second case. The center of the blowup h: X → Y is V (x0, z) ⊂ Y , +yielding the two varieties described in (2). +□ +Appendix A. Code for computing primitive collections +by Will Reynolds +Let Σ be the fan of a toric manifold of dimension n. For each integer k ∈ {1, . . . , n}, +we denote by Σ(k) the subset of Σ consisting of the k-dimensional cones. To determine +whether a given subset P ⊆ G(Σ) is a primitive collection, we proceed in two steps. First +make sure there does not exist σ ∈ Σ(n) with P ⊆ G(σ), and if there does, stop; otherwise, +for each v ∈ P, make sure there exist σ ∈ Σ(n) with P \ {v} ⊆ G(σ). In the special case +|P| = 2 it suffices to check that there does not exist σ ∈ Σ(n) with P ⊆ G(σ). +Note that, by definition, if P is a primitive collection and P ⊆ Q, then Q is not a primitive +collection. Therefore, when looking for primitive collections, we go through subsets of G(Σ) +in increasing cardinality. +Assuming an implementation of the above basic algorithm, a reasonably efficient way to +list all of the primitive collections of a fan is to arrive at such a list by eliminating P ⊆ G(Σ) + +which are not primitive collections. The first step is to remove any P with |P| = 1. Then +for each P we check whether it is a primitive collection. If it is, we keep it and remove +all sets containing it. If it is not, we remove it. One way of implementing this method of +listing primitive collections is implemented in pseudocode in Algorithm 1. +This algorithm is impractical if G(Σ) is too large. In practice, on a modern laptop, it +works reasonably well up to about |G(Σ)| = 17, partly because of the combination of the +following factors: first, eliminating all of the supersets of any P with |P| = 2 cuts down +the remaining search space significantly, and second, the relative abundance of primitive +collections of size 2, at least among toric Fano varieties. For example, the 124 toric Fano +4-folds altogether have 785 primitive collections, of which 566 have cardinality 2. +This last factor makes the computation of the value of m(X) for a given toric Fano +variety X easier as well. Of the toric Fano varieties of a given dimension n (for n ≤ 6) +those X with m(X) = 1 make up an overwhelming majority. This means that computing +m(X) is usually extremely fast, even in the most straightforward way. The following table +summarizes the data for dim(X) ∈ {4, 5, 6}. +dim(X) +# Fanos +#(m=1) +#(m=2) +#(m=3) +#(m=4) +#(m=5) +#(m=6) +4 +124 +107 +15 +1 +1 +5 +866 +744 +112 +8 +1 +1 +6 +7622 +6333 +1174 +105 +8 +1 +1 +27 + +Algorithm 1: List primitive collections of a given fan +Input: fan Σ; +n ← dim Σ; +PC ← {P ⊆ G(Σ) : |P| > 1}; +for P ∈ PC satisfying |P| = 2 do +if there exists σ ∈ Σ(n) such that P ⊆ G(σ) then +PC ← PC \ {P}; +else +PC ← PC \ {Q ∈ G(Σ) : P ⊊ Q}; +end +end +for i ∈ {3, . . . , n} do +for P ∈ PC satisfying |P| = i do +if there exists σ ∈ Σ(n) such that P ⊆ G(σ) then +PC ← PC \ {P}; +else +b ← True; +for v ∈ P do +if there does not exist σ ∈ Σ(n) with P \ {v} ⊆ G(σ) then +b ← False; +end +end +if b then +PC ← PC \ {Q ⊆ G(Σ) : P ⊊ Q}; +else +PC ← PC \ {P}; +end +end +end +end +Output: PC; +For convenience, we also provide Macaulay2 code implementing the algorithm for com- +puting primitive collections. +coneExistenceCheck = (S, fan) -> ( +for cone in fan do ( +if isSubset(S, cone) then ( +return true; +); +); +return false; +); +28 + +properSubsetCheck = (S, fan) -> ( +for ray in S do ( +if coneExistenceCheck(S-set{ray}, fan) == false then ( +return false; +); +); +return true; +); +isPrimitiveCollection = (P, Var) -> ( +if coneExistenceCheck(P, orbits(Var, 0)) then ( +return false) +else ( +return properSubsetCheck(P, orbits(Var, 0) +); +); +supsetsOfPrimColl = (E, B) -> ( +return set{for P in E-set{B} when isSubset(B, P) list P}; +); +primitiveCollections = (Var) -> ( +n = length rays Var; +primColls = select(subsets(toList(0..n-1)), x -> length x > 1); +for P in subsets(toList(0..n-1), 2) do ( +if coneExistenceCheck(P, orbits(Var, 0)) == false then ( +primColls = primColls - supsetsOfPrimColl(primColls, P);) +else ( +primColls = primColls - set{P}; +); +); +for i in toList(3..n) do ( +for P in subsets(toList(0..n-1), i) do ( +if member(P, primColls) == false then continue; +if isPrimitiveCollection(P, Var) then ( +primColls = primColls - supsetsOfPrimColl(primColls, P); +) else ( +primColls = primColls - set{P}; +); +); +); +return sort primColls; +); +29 + +30 +ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN +References +[AC12] +C. 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Japan 73.3 (2021), pp. 949–964. +issn: 0025-5645. + +32 +REFERENCES +Carolina Araujo, IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil +Email address: caraujo@impa.br +Roya Beheshti, Department of Mathematics & Statistics, Washington University in St. +Louis, St. Louis, MO, 63130 +Email address: beheshti@wustl.edu +Ana-Maria Castravet, Universit´e Paris-Saclay, UVSQ, Laboratoire de Math´ematiques +de Versailles, 45 Avenue des ´Etats Unis, 78035 Versailles, France +Email address: ana-maria.castravet@uvsq.fr +Kelly Jabbusch, Department of Mathematics & Statistics, University of Michigan– +Dearborn, 4901 Evergreen Rd, Dearborn, Michigan, 48128, USA +Email address: jabbusch@umich.edu +Svetlana Makarova, Department of Mathematics, University of Pennsylvania, 209 S +33rd St, Philadelphia, PA 19104, USA +Email address: murmuno@sas.upenn.edu +Enrica Mazzon, Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, Universit¨atsstrasse +31, 93040 Regensburg, Germany, and Department of Mathematics, University of Michigan, +Ann Arbor, Michigan, 48109, USA +Email address: e.mazzon15@alumni.imperial.ac.uk +Nivedita Viswanathan, Department of Mathematical Sciences, Loughborough Univer- +sity, Loughborough, LE11 3TU, UK and School of Mathematical Sciences, University +Park Campus, The University of Nottingham, Nottingham, NG7 2RD, UK +Email address: N.Viswanathan@lboro.ac.uk +Will Reynolds, School of Mathematics, The University of Edinburgh, Edinburgh, EH9 +3FD, UK +Email address: W.R.N.Reynolds@sms.ed.ac.uk + diff --git a/adAyT4oBgHgl3EQf9_ol/content/tmp_files/load_file.txt b/adAyT4oBgHgl3EQf9_ol/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5bc60b1fbe06d90e362a5020bff35a146e6fe054 --- /dev/null +++ b/adAyT4oBgHgl3EQf9_ol/content/tmp_files/load_file.txt @@ -0,0 +1,1667 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf,len=1666 +page_content='THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS CAROLINA ARAUJO, ROYA BEHESHTI, ANA-MARIA CASTRAVET, KELLY JABBUSCH, SVETLANA MAKAROVA, ENRICA MAZZON, AND NIVEDITA VISWANATHAN, WITH AN APPENDIX BY WILL REYNOLDS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Motivated by the problem of classifying toric 2-Fano manifolds, we introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This invariant m(X) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , dim(X)} captures the minimal degree of a dominating family of rational curves on X or, equivalently, the minimal length of a centrally symmet- ric primitive relation for the fan of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We classify smooth projective toric varieties with m(X) ≥ dim(X)−2, and show that projective spaces are the only 2-Fano manifolds among smooth projective toric varieties with m(X) ∈ {1, dim(X) − 2, dim(X) − 1, dim(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Primitive collections 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Notation and background 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The minimal P-dimension 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Some properties of primitive collections 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Primitive collections on toric Fano manifolds 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Toric Fano manifolds with m(X) = 1 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' First case: {x, y, z} is a primitive collection 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Second case: none of {x, y, z} form a primitive collection 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Toric Fano manifolds with m(X) = n − 2 24 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Code for computing primitive collections 26 References 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Introduction Fano varieties are projective varieties with positive first Chern class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Over the complex numbers, this condition is equivalent to the existence of a metric with positive Ricci cur- vature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Basic examples of Fano varieties include projective spaces and Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The positivity condition has further geometric implications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', Fano varieties over the complex numbers are simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This has an analogue on the algebro-geometric side: any Fano variety is covered by rational curves [Mor79], and is in fact rationally con- nected [KMM92;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Cam92], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', there are rational curves connecting any two of its points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In a series of papers, de Jong and Starr introduce and investigate possible candidates for the notion of higher rational connectedness [dHS11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' dS07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' dS06b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' dS06c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Sta06], inspired 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='00883v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='AG] 2 Jan 2023 2 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN by the natural analogue in topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, in [dS06b] they define 2-Fano mani- folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A smooth projective variety X is 2-Fano if it is Fano and its second Chern character ch2(TX) = 1 2c1(TX)2 − c2(TX) is positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', ch2(TX) · S > 0 for every surface S in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In a similar way, one can define k-Fano varieties for any k ≥ 2, and aim at their classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For instance, Pn is n-Fano, and it is conjectured that it is the only n-dimensional n-Fano manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The geometry of higher Fano manifolds has been fairly investigated, and in several special cases they are shown to enjoy the expected nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For instance, 2-Fano manifolds satisfying some mild assumptions are covered by rational surfaces [dS07], and similar results hold for higher Fano manifolds [Suz21], [Nag19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' There is a classification of 2-Fano manifolds of high index [AC13] and, more recently, a classification of homoge- neous 2-Fano manifolds [Ara+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' On the other hand, very few examples of higher Fano manifolds are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Quite strikingly, all known examples of 2-Fano manifolds have Picard rank 1 and relatively large index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It is natural for algebraic geometers to turn to the pool of toric varieties when looking for intuition or examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It is well known that projective spaces are the only projective toric manifolds with Picard rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Thus, a classification of toric 2-Fano manifolds could either provide the first examples of 2-Fano manifolds with higher Picard rank, or it could be an evidence that every 2-Fano manifold has Picard rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Geometric properties of a toric variety can often be checked in the combinatorics of the associated fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This bridge has been exploited in search of new examples of toric 2-Fano manifolds [Nob11], [Nob12], [Sat12], [Sat16], [SS20], [SSS21], [Shr20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Despite the efforts, a complete (computer aided) classification is only known up to dimension 8 [Nob11], [SSS21], and projective spaces remain the only known examples of toric 2-Fano manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The sparsity of higher Fano manifolds leads to the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([SSS21, Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3]) The only toric 2-Fano manifolds are projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In this paper, we propose a new strategy to approach Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We follow the philosophy introduced in [AC12], namely, to investigate 2-Fano manifolds by studying their minimal dominating families of rational curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By [CFH14], minimal dominating families of rational curves on a smooth projective toric variety X correspond to primitive relations of the form (1) x0 + · · · + xm = 0, satisfied by some of the primitive integral generators xi of the corresponding fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' These primitive relations are called centrally symmetric of order m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By [CFH14], a centrally symmetric primitive relation of order m + 1 yields a Pm-bundle structure X◦ → T on a dense open subset X◦ of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If dim(T) ≥ 1, and the complement X \\X◦ has codimension at least 2 in X, then one can construct a complete surface S ⊂ X◦ such that ch2(TX) · S ≤ 0, showing that X is not 2-Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So our basic strategy consists of trying to describe, in a rather explicit way, a suitable birational map ϕ : X ��� Y transforming X into a projective toric variety Y admitting a Pm-bundle structure on a big open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We then hope to be able to compare the second Chern characters ch2(TX) and ch2(TY ) to show that X is not 2-Fano, except if X = Pm and ϕ is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' To follow this strategy, we introduce a new invariant of a smooth projective toric variety X, the minimal projective bundle dimension of X, minimal P-dimension in short, which is THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 3 of independent interest (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='10): m(X) = min � m ∈ Z>0 �� there is a relation as in (1) � ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , dim X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By going through the database of toric Fano manifolds of low dimension and computing their primitive collections, one obtains Table 1, indicating the number of Fano manifolds for each value of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Appendix A contains the code used to compute primitive collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' dim(X) # Fanos #(m=1) #(m=2) #(m=3) #(m=4) #(m=5) #(m=6) 4 124 107 15 1 1 5 866 744 112 8 1 1 6 7622 6333 1174 105 8 1 1 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The minimal P-dimension of toric Fano manifolds of low dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' When m(X) = 1, Casagrande constructs in [Cas03b] a sequence of blowdowns and flips from X to a toric variety admitting a global P1-bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This allows us to make the basic strategy work, yielding the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a smooth toric Fano variety with m(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then X is not 2-Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Table 1 suggests that this result covers “most” toric varieties, and not just the fringe cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Next we turn our attention to toric Fano manifolds X with large values of m(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Projec- tive spaces are the only smooth projective toric varieties admitting a centrally symmetric primitive relation of order dim(X) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In [CFH14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8], Chen, Fu and Hwang classify toric Fano manifolds admitting a centrally symmetric primitive relation of order dim(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' There are three such varieties, and two of them also admit a centrally symmetric primitive relation of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As a consequence, the only n-dimensional toric Fano manifold X with m(X) = n − 1 is the blowup of Pn along a linear Pn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In [BW22], Beheshti and Wormleighton investigate smooth projective toric varieties admitting a centrally symmetric primitive relation of order dim(X)−1, showing that they have Picard rank ρ(X) ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Most of these varieties also admit centrally symmetric primitive relations of order 2 or 3, and we prove the following bound for the remaining ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4 shows that this bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a smooth toric Fano variety with dim(X) = n ≥ 6 and m(X) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If X has a centrally symmetric primitive relation of order n − 1, x0 + x1 + · · · + xn−2 = 0, then ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Moreover, m(X) = n − 2 and the above relation is the only centrally symmetric primitive relation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3 and Batyrev’s description of smooth projective toric varieties with Picard rank 3, we are able to classify n-dimensional smooth toric Fano varieties with m(X) = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' There are eight distinct isomorphism classes when n ≥ 6, which can be explicitly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The following statement summarizes the classification of toric Fano manifolds with m(X) ≥ dim(X) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 4 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We have the following classification of smooth toric Fano varieties with m(X) ≥ dim(X) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (1) The only n-dimensional smooth toric Fano variety X with m(X) = n is Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (2) For n ≥ 3, the only n-dimensional smooth toric Fano variety X with m(X) = n − 1 is the blowup of Pn along a linear Pn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (3) For n ≥ 6, there are eight distinct isomorphism classes of n-dimensional smooth toric Fano varieties X with m(X) = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Namely: (a) X = PS(E) is a Pn−2-bundle over a toric surface S, where (S, E) is one of the following: S = P2 and E = OP2(1) ⊕ O⊕n−2 P2 , S = P2 and E = OP2(1) ⊕ OP2(1) ⊕ O⊕n−3 P2 , S = P2 and E = OP2(2) ⊕ O⊕n−2 P2 , S = P1 × P1 and E = OP1×P1(1, 1) ⊕ O⊕n−2 P1×P1, S = P1 × P1 and E = OP1×P1(1, 0) ⊕ OP1×P1(0, 1) ⊕ O⊕n−3 P1×P1, S = F1 and E = OF1(e + f) ⊕ O⊕n−2 F1 , where e ⊂ F1 is the −1-curve, and f ⊂ F1 is a fiber of F1 → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the first three cases, ρ(X) = 2, while in the latter three cases, ρ(X) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (b) Let Y ≃ PP2� OP2(1) ⊕ O⊕n−2 P2 � be the blowup of Pn along a linear subspace L = Pn−3, and denote by E ⊂ Y the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then X is the blowup of Y along a codimension 2 center Z ⊂ Y , where: Z is the intersection of E with the strict transform of a hyperplane of Pn containing the linear subspace L, or Z is the intersection of the strict transforms of two hyperplanes of Pn, one containing the linear subspace L, and the other one not containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In both cases, ρ(X) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The projective space Pn is the only smooth n-dimensional toric 2-Fano variety with m(X) ∈ {1, n − 2, n − 1, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In Section 2, we review some results from toric geometry and fix notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, we discuss centrally symmetric primitive relations on smooth projective toric varieties, describing explicitly their open subsets admitting a projective space bundle structure (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In Section 3, we study smooth toric Fano varieties with m(X) = 1, and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In Section 4, we investigate smooth projective n-dimensional toric varieties admitting a centrally symmetric primitive relation of order n − 1, and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In Section 5, we use this result, together with Batyrev’s description of smooth projective toric varieties with Picard rank 3, to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This collaboration started as a working group at the ICERM “Women in Algebraic Geometry Collaborative Research Workshop” in July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A great part of this work was developed during the follow-up “Collaborate@ICERM” meeting in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We thank ICERM for the financial support and the great working conditions provided to us during our visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We are very grateful to Cinzia Casagrande and Milena Hering for many enlightening discussions and explanations about toric varieties, and to THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 5 Will Reynolds for running his code and providing us with precious data about primitive relations of toric Fano manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Carolina Araujo was partially supported by CAPES/COFECUB, CNPq and FAPERJ Research Fellowships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Roya Beheshti was supported by NSF grant DMS-2101935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Ana- Maria Castravet was partially supported by the ANR 20-CE40-0023 grant FanoHK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Enrica Mazzon was supported by the collaborative research center SFB 1085 Higher Invariants Interactions between Arithmetic Geometry and Global Analysis funded by the Deutsche Forschungsgemeinschaft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Nivedita Viswanathan was supported by the EPSRC New Hori- zons Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='EP/V048619/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Primitive collections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Notation and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A toric variety is a normal complex variety X that contains a torus T = (C∗)n as a dense open subset, together with an action of T on X that extends the natural action of T on itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' There is a one-to-one correspondence between n-dimensional toric varieties and fans in Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let N be a free abelian group of rank n, and consider the vector space NQ = N ⊗Z Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A fan in NQ is a nonempty finite collection Σ of strongly convex polyhedral cones in NQ such that every face of a cone in Σ is also a cone in Σ, and the intersection of two cones in Σ is a face of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We write δ ≺ τ to express that δ is a face of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' One-dimensional cones in Σ are called rays, and each ray is generated by a primitive vector in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The set of primitive vectors of N generating rays of Σ is denoted by G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We will write a cone τ ∈ Σ in terms of its primitive generators, τ = ⟨v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vl⟩, saying that the vi’s generate τ, and setting G(τ) := {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vl} ⊆ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We denote by XΣ the toric variety corresponding to a fan Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Conversely, given a toric variety X, we denote by ΣX the fan associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' There is a one-to-one inclusion- reversing correspondence between cones in Σ and T-orbit closures in XΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Given a cone τ ∈ Σ, we write V (τ) ⊂ XΣ for the corresponding T-orbit closure, or V (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vl) when G(τ) = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Note that dim(τ) = codimXΣ V (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We refer to [Ful93] and [CLS11] for the background on toric varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In this paper, we are mostly interested in smooth and proper toric varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The smooth- ness conditions translates into the fan Σ being regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', for each cone τ ∈ Σ, the set of generators G(τ) is part of a basis of N ([CLS11, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The properness condition translates into the fan Σ being complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', its support being the whole NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In what follows, smooth and proper toric varieties will be simply called toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We would like to classify toric 2-Fano manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Given a toric manifold X, there is an exact sequence ([CLS11, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1]): 0 → Ω1 X → N ∨ ⊗Z OX → � v∈G(ΣX) OV (v) → 0 , from which one easily computes: c1(X) = � v∈G(ΣX) V (v) and ch2(X) = 1 2 � v∈G(ΣX) V (v)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Bat91, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6]) Let Σ be a regular complete fan in NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A primitive collection P ⊆ G(Σ) is a nonempty set of primitive vectors of N that does not generate a cone of Σ, but such that any proper subset of P does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Equivalently, P = 6 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr} ⊆ G(Σ) is a primitive collection if and only if ⟨v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr⟩ /∈ Σ and ⟨v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ˇvi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr⟩ ∈ Σ for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We denote by PC(Σ) the set of primitive collections of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For a toric manifold X, we will talk about primitive collections of X and write PC(X), meaning PC(ΣX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let Σ be a regular complete fan in NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Given a primitive collection P = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr} ∈ PC(Σ), let σ(P) = ⟨w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ws⟩ be the minimal cone of Σ such that v1 + · · · + vr ∈ σ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then there is a relation r(P): v1 + · · · + vr = µ1w1 + · · · + µsws, where µj ∈ Z≥0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We call r(P) the primitive relation associated to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We define the order of P as ord(P) = |P| = r, while the degree of P as deg(P) = r − �s j=1 µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By [Bat91, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1], for any primitive collection P, we have P ∩ σ(P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr} ∩ {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ws} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let Σ be a regular complete fan in NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A primitive collection P = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xk} of Σ is called centrally symmetric if σ(P) = {0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' r(P): x0 + · · · + xk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let Σ be a regular complete fan in NQ, and let P, Q be two distinct centrally symmetric primitive collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then P ∩ Q = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Write r(P): x0 + · · · + xk = 0 and r(Q): y0 + · · · + yl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that P ∩ Q ̸= ∅, then without loss of generality we may assume that x0 = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' But then subtracting this vector from both relations, we get x1 + · · · + xk = y1 + · · · + yl, which shows that interiors of two distinct cones intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let Σ be a regular complete fan in NQ, and let P, Q be two distinct centrally symmetric primitive collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then Span P ∩Span Q = {0}, in particular |P|+|Q|−2 ≤ dim NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Write r(P): x0 + · · · + xk = 0 and r(Q): y0 + · · · + yl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Take any vector v ∈ Span P ∩ Span Q, so we can write it as v = � aixi = � bjyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By possibly adding r(P) and r(Q) to the sums, we can get that all ai, bj ≥ 0, and up to relabelling the ai, bj, we can assume a0 = b0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' But this shows that v is in the intersection of two cones, ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xk⟩ ∩ ⟨y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , yl⟩, and the sets of generators are disjoint by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4, so ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xk⟩ ∩ ⟨y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , yl⟩ = {0} and v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The last claim follows from considering the dimensions of Span P and Span Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Let A1(XΣ) be the group of algebraic 1-cycles on XΣ modulo numerical equivalence, and set N1(XΣ) = A1(XΣ) ⊗Z Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The Mori cone NE(XΣ) ⊂ N1(XΣ) is the cone generated THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 7 by the classes of effective curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A primitive integral class generating an extremal ray of NE(XΣ) is called an extremal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' There is an exact sequence: 0 > A1(XΣ) > ZG(Σ) > N > 0, [C] > � C · V (v) � v∈G(Σ) � νv � v∈G(Σ) > � v∈G(Σ) νvv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Thus the elements of A1(XΣ) are identified with integral relations between the elements of G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If the class [C] corresponds to the relation � v νvv = 0, then we have −KXΣ · C = � v νv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Cas03a, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4]) Let Σ be a regular complete fan in NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A relation α1x1 + · · · + αlxl − β1y1 − · · · − βmym = 0, with αi, βj ∈ Z>0, defines an effective class in N1(XΣ) provided that ⟨y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ym⟩ is a cone of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We will usually write the above relation as α1x1 + · · · + αlxl = β1y1 + · · · + βmym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that primitive relations correspond to effective curve classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By abuse of notation, we will identify a primitive relation r(P) with the corresponding curve class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Note that deg(P) = −KXΣ ·r(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the projective case we have the following description of NE(XΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Bat91, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='15]) Let Σ be a regular complete fan in NQ, and assume that XΣ is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then the Mori cone is generated by primitive relations: NE(XΣ) = � P∈PC(XΣ) Q≥0 r(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Rei83, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4]) Let Σ be a regular complete fan in NQ, and assume that XΣ is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let γ be an extremal class in NE(XΣ) whose corresponding primitive relation is r(P): v1 + · · · + vr = µ1w1 + · · · + µsws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let τ = ⟨z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , zl⟩ be a cone of Σ such that G(τ) ∩ P = G(τ) ∩ G(σ(P)) = ∅, and such that ⟨σ(P), τ⟩ = ⟨w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ws, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , zl⟩ is a cone of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then, for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , r, ⟨P \\ {vi}, σ(P), τ⟩ = ⟨v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ˇvi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr, w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ws, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , zl⟩ is also a cone of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The minimal P-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric manifold with regular complete fan ΣX in NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In this section, we discuss centrally symmetric primitive collections, introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Bat91, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2]) If X is projective, then ΣX has a centrally symmetric primitive collection of order k + 1 (2) r(P): x0 + · · · + xk = 0 for some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , dim(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 8 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For a projective toric manifold X, we define the minimal P-dimension as m(X) := min � m ∈ Z>0 ��� ΣX has a centrally symmetric primitive collection of order m + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The next remark explains the terminology of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='10 and highlights the signifi- cance of studying centrally symmetric primitive collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In [CFH14], Chen, Fu and Hwang provide a new geometric proof of Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='9 by relating centrally symmetric primitive collections to minimal dominating fam- ilies of rational curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We review some aspects of the theory of rational curves on varieties and refer to [Kol96] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Given a smooth and proper uniruled variety X, there is a scheme RatCurvesn(X) parametrizing rational curves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A dominating family of ra- tional curves on X is an irreducible component of RatCurvesn(X) parametrizing rational curves that sweep out a dense open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A dominating family of rational curves H is said to be minimal if, for a general point x ∈ X, the subvariety of H parametrizing curves through x is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' When X is projective, there always exists a minimal domi- nating family of rational curves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For instance, one can take H to be a dominating family of rational curves on X having minimal degree with respect to some fixed ample line bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' When X = XΣ is a toric variety, there is a one-to-one correspondence between minimal dominating families of rational curves H on X and centrally symmetric primitive collections of Σ ([CFH14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Moreover, if the centrally symmetric primitive collection has order k + 1 as in Equation (2) above, then there is a dense T-invariant open subset U of X and a Pk-bundle π: U → W such that the general curve parametrized by H is a line on a general fiber π ([CFH14, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows from this discussion that the minimal P-dimension m(X) is the smallest integer k such that X admits a generic Pk-bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We have m(X) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , n = dim(X)}, and m(X) = n if and only if X ≃ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By [CFH14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8], there are three toric Fano manifolds admitting a centrally symmetric primitive relation of order n = dim(X), namely: Pn−1 × P1, the blowup of Pn−1 × P1 along a linear Pn−2, and the blowup of Pn along a linear Pn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The first two varieties also admit a generic P1-bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As a consequence, the only n-dimensional toric Fano manifold X with m(X) = n − 1 is the blowup of Pn along a linear Pn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In Section 5, we shall classify n-dimensional toric Fano manifolds X with m(X) = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let P ∈ PC(X) be a centrally symmetric primitive collection of order k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As explained in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='11, P induces a Pk-bundle structure on a dense T-invariant open subset U of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In [CFH14, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6], the T-invariant open subset U was taken as small as possible, namely, U ∼= Pk × (C∗)n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For our purposes, we want to take U as big as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So our next goal is to describe explicitly the biggest T-invariant open subset of X on which P induces a Pk-bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let P = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xk} ∈ PC(X) be a centrally symmetric primitive collec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Denote by EP the set of cones σ = ⟨v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr⟩ ∈ ΣX such that P ∩ G(σ) = ∅, and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xjs} ∈ PC(X) for some s ≥ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', EP := {σ ∈ ΣX | P ∩ G(σ) = ∅ and ∃P ′ ⊊ P such that P ′ ∪ G(σ) ∈ PC(X)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 9 We write V (EP) := � σ∈EP V (σ) ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let P = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xk} ∈ PC(X) be a centrally symmetric primitive collection, and let V (EP) be as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then the open subset U = X \\ V (EP) admits a Pk-bundle structure over a smooth toric variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In order to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13, we first prove two auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let P = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xk} ∈ PC(X) be a centrally symmetric primitive collec- tion, let V (EP) be as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12, and set U = X \\ V (EP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then the fan ΣU of U consists of all cones of ΣX of the form (3) τ ′ = ⟨τ, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xjm⟩, where 0 ≤ m ≤ k, and τ ∈ ΣX is such that ⟨τ, P \\ {xi}⟩ ∈ Σ for every i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (When m = 0, Equation (3) means that τ ′ = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Recall that a cone σ ∈ ΣX corresponds to a T-orbit, which is dense and open in V (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence, a cone σ ∈ ΣX is in ΣU if and only if the corresponding orbit does not intersect V (EP), which is equivalent to saying that V (σ) ̸⊆ V (EP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It is immediate that the cones of the form (3) define a fan Σ′ ⊂ ΣX in NQ, and XΣ′ is a dense open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We now prove that the toric variety XΣ′ coincides with U by showing that a cone σ ∈ ΣX is of the form (3) if and only if V (σ) ̸⊆ V (EP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Consider σ ∈ ΣX \\ Σ′, which means that ⟨G(σ) ∪ P \\ {xi}⟩ /∈ ΣX for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then the set G(σ) ∪ P \\ {xi} contains a primitive collection S, so the cone τ := ⟨S \\ P⟩ is in EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' But notice that τ ≺ σ, so V (σ) ⊆ V (τ) ⊆ V (EP) and hence σ is not in ΣU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Conversely, if σ ∈ ΣX \\ ΣU, then V (σ) ⊆ V (EP), hence there exists τ ∈ EP such that V (σ) ⊆ V (τ) and G(τ) ∪ P ′ ∈ PC(X) for some P ′ ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since G(τ) ⊆ G(σ), we conclude that ⟨G(σ) ∪ P ′⟩ /∈ ΣX, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', σ ̸∈ Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Consider the sequences 0 NP := ker(φ) N N = N/Z⟨x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xk⟩ 0, 0 (NP)Q NQ N Q 0, Σ0 ΣU ΣU, φ φQ where φ is the quotient map, the fan Σ0 of (NP)Q ≃ Qk+1 is the subfan of ΣU of cones of the form (3) with τ = {0} (in particular note that XΣ0 ≃ Pk), and ΣU = {φQ(σ) | σ ∈ ΣU}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let the notation be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ΣU is a toric fan, and the linear map φQ is compatible with the fans ΣU and ΣU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The cones of ΣU are exactly φQ(τ) for τ ∈ ΣU such that G(τ) ∩ P = ∅, so for simplicity we only consider these τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It is immediate that the cones of ΣU are rational polyhedral, and that the faces of φQ(τ) are φQ(δ), for all subcones δ ≺ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 10 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN We need to show that the cone φQ(τ) is strongly convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', if y ∈ φQ(τ) and −y ∈ φQ(τ), then y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This follows automatically from the fact that the images of the generators φQ(G(τ)) = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vr} are linearly independent, which we prove by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If they are linearly dependent, then there exist a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ar ∈ Q, not all 0, such that Σr i=1aivi = 0 in N Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This implies that there exist bj ∈ Q for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , k such that Σr i=1aivi = �k j=1 bjxj, which is a contradiction since ⟨G(τ) ∪ P \\ {x0}⟩ ∈ Σ and hence its primitive generators are linearly independent in NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ΣU and ΣU are compatible with φQ as we have φQ(τ ′) ∈ ΣU for any τ ′ ∈ ΣU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let the notation be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let ˆΣU be the collection of fans of the form (3) above with m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows from the description of the cones of ΣU in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='14 that (1) φQ maps each cone ˆτ ∈ ˆΣU bijectively to a cone τ ∈ ΣU such that φ(ˆτ ∩N) = τ ∩N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Furthermore, the map ˆτ �→ τ defines a bijection ˆΣU → ΣU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (2) given cones ˆτ ∈ ˆΣU and τ0 ∈ Σ0, the sum ˆτ + τ0 lies in ΣU and every cone of ΣU arises in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the notation of [CLS11, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='18], we say that ΣU is split by ΣU and Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We conclude by [CLS11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='19] that U = X \\ V (EP) is a locally trivial fiber bundle over XΣU with fiber XΣ0 ≃ Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows automatically that XΣU is smooth, since it is the base of a locally trivial fibration with a smooth total space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Some properties of primitive collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Before focusing on toric Fano mani- folds, we collect here two useful properties of primitive collections of arbitrary toric man- ifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The first one, by Sato, describes the behaviour of primitive collections under a smooth toric blowdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The second one, by Batyrev, describes primitive collections on toric manifolds of Picard rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Sat00, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='9]) Let X be a toric manifold, and let f : X → Y be the contraction associated to an extremal class in NE(X), corresponding to a primitive relation of the form r(Q): t1 + · · · + ts = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then the fan ΣY is obtained from ΣX by removing the ray generated by z, and X is the blowup of Y along V (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Furthermore, the primitive collections of Y are precisely the following PY ∈ PC(Y ): PY = PX for some PX ∈ PC(X) such that z /∈ PX and PX ̸= Q = {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ts};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' PY = (PX \\ {z}) ∪ {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , tr} for some PX ∈ PC(X) such that z ∈ PX and (PX \\ {z}) ∪ S /∈ PC(X) for any subset S ⊊ {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , tr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Bat91, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6]) Let X be a projective toric manifold with ρ(X) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then the number of primitive collections of ΣX is either l = 3 or l = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Moreover, the set of generators G(ΣX) can be written as a disjoint union of l nonempty subsets G(ΣX) = X0 ⊔ · · · ⊔ Xl−1 that define primitive collections and relations as follows: THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 11 Case l = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Each X0, X1, X2 is a primitive collection, and the corresponding primitive relations are extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Case l = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' There are five primitive collections of the form Xi ⊔ Xi+1, 0 ≤ i ≤ 4, where X5 := X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' To describe the primitive relations of X, we use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We fix a labelling (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , vk) for the elements of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ck) ∈ Zk, then c · Xi stands for c1v1 + · · · + ckvk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Moreover, we set 1 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then there are vectors c and b of nonnegative integers such that at least one entry in c is zero (up to relabelling, we may assume that c1 = 0), and the primitive relations of X are the following: r0 : 1 · X0 + 1 · X1 = c · X2 + (b + 1) · X3 r1 : 1 · X1 + 1 · X2 = 1 · X4, r2 : 1 · X2 + 1 · X3 = 0, r3 : 1 · X3 + 1 · X4 = 1 · X1, r4 : 1 · X4 + 1 · X0 = c · X2 + b · X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The relations r0, r1 and r3 are extremal, while r2 = r1 + r3 and r4 = r0 + r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In [SS20, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2], Sato and Suyama use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17 to show that projective spaces are the only toric 2-Fano manifolds with Picard rank ρ ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Primitive collections on toric Fano manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a projective toric man- ifold with regular complete fan ΣX in NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Bat99, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6]) The toric variety X is Fano if and only if all primitive collections of ΣX have strictly positive degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Cas03a, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4]) Assume that X is Fano, and let P ∈ PC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If deg(P) = 1, then the corresponding curve class is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that X is Fano, and let x ∈ G(ΣX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (1) There is at most one primitive collection of order 2 and degree 2 containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If it exists, then it is of the form x + (−x) = 0, and m(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (2) ([Cas03b, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3]) There are at most two primitive collections of order 2 and degree 1 containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If there are exactly two of them, then they are of the form x + y = (−w) and x + w = (−y), and m(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that X is Fano and m(X) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then any x ∈ G(ΣX) is contained in at most one primitive collection of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If there is such a primitive collection, then it is of the form x + y = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let x ∈ G(ΣX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We say that y ∈ G(ΣX) is an opponent of x if ⟨x, y⟩ /∈ ΣX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that X is Fano and m(X) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='22, each vector x ∈ G(ΣX) has at most one opponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If such an opponent exists, we denote it by x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that X is Fano and m(X) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Consider a pair of opponents x, x′ ∈ G(ΣX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If there exist y, z ∈ G(ΣX) such that x + x′ = y + z, then z = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 12 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If y = x or y = x′, the claim follows automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So we assume otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Note that {x, x′} ∈ PC(X), and y and z do not form a cone, as otherwise x + x′ = y + z would give us a primitive relation of degree 0, which is impossible for a toric Fano manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that X is Fano and m(X) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that there exist x, y, z, u, v in G(ΣX) such that (∗) x + y + z = u + v and such that ⟨u, v⟩ ∈ ΣX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then exactly one of the following must happen: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The vectors x, y, z are pairwise distinct, and {x, y, z} is a primitive collection with primitive relation (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, the corresponding curve class is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Up to relabeling, v = z, y = x′ and x + x′ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume two of {x, y, z} do not form a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For example, assume x, y do not form a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then y = x′, the opponent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let x + x′ = α, for some α ∈ G(ΣX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We have that α+z = u+v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As ⟨u, v⟩ ∈ ΣX, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6 implies that α+z = u+v corresponds to an effective class of degree 0, which therefore implies that {α, z} = {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Up to relabeling, we may assume v = z, and hence, x + x′ = u and we are in the situation b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume now that any two vectors in {x, y, z} form a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then x, y, z are mutually disjoint and ⟨x, y, z⟩ /∈ ΣX, as otherwise by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6 we obtain an effective curve class of degree −1, contradicting the fact that X is Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that {x, y, z} is a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since ⟨u, v⟩ ∈ ΣX, it follows that (∗) is the associated primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='20 now implies that the corresponding curve class is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Toric Fano manifolds with m(X) = 1 In this section, we study toric Fano manifolds with m(X) = 1, and follow the strategy outlined in the introduction to show that they cannot be 2-Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For any x ∈ G(ΣX), the set of primitive collections containing x is denoted by PCx(X) = {P ∈ PC(X) | x ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([Cas03b, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1]) Assume that X is a toric Fano manifold and that P = {x, −x} ∈ PC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (1) Any Q ∈ PCx(X) \\ {P} has degree 1 (hence is extremal by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='20), and r(Q) is of the form r(Q): x + y1 + · · · + yh � �� � ∈ ⟨Q\\{x}⟩ = z1 + · · · + zh � �� � ∈σ(Q) , where we denote ⟨Q \\ {x}⟩ := ⟨y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , yh⟩ and σ(Q) := ⟨z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , zh⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (2) For any R ∈ PCx(X) \\ {P, Q}, we have V (R \\ {x}) ∩ V (Q \\ {x}) = ∅ and V (R \\ {x}) ∩ V (σ(Q)) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (3) For any Q ∈ PCx(X) \\ {P} with r(Q): x + y1 + · · · + yh = z1 + · · · + zh we have Q′ = {−x, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , zh} ∈ PC−x(X), Q′ has degree 1 (hence is extremal) and r(Q′): − x + z1 + · · · + zh � �� � ∈⟨Q′\\{−x}⟩=σ(Q) = y1 + · · · + yh � �� � ∈σ(Q′)=⟨Q\\{x}⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 13 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric manifold, and P = {x, −x} ∈ PC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' With Nota- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12, (4) EP = {⟨Q \\ {v}⟩ | Q ∈ PCv(X) \\ {P}, v = ±x} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If moreover X is Fano, then V (EP) has 0, 2 or 4 components of codimension 1 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric manifold, and P = {x, −x} ∈ PC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The description of V (EP) in Equation (4) follows from Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the Fano case, the number of components of codimension 1 of V (EP) equals the number of primitive collections of order 2 and degree 1 containing x or −x, which is 0, 2 or 4 by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='21 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric Fano manifold, and P = {x, −x} ∈ PC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then there exists a birational morphism f : X → Y such that PY := {x, −x} ∈ PC(Y ), V (EPY ) has codimension ≥ 2 in Y , f is a composition of at most two blow-downs with disjoint centers and smooth target: Exc(f) = � Q∈PCx(X)\\{P} : ord(Q)=2 V (σ(Q)) ⊂ X, f(Exc(f)) = � Q∈PCx(X)\\{P} : ord(Q)=2 V (Q) ⊂ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If V (EP) has codimension ≥ 2 in X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', if P is the unique primitive collection of order 2 containing x, then the statement holds with f = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume now that V (EP) has 2 components of codimension 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' we have primitive relations r(P): x + (−x) = 0, r(Q1): x + y = z, r(Q′ 1): − x + z = y, and, for any other R ∈ PCx(X) ∪ PC−x(X), one has ord(R \\ {±x}) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let f1 : X → Y be the smooth blow-down induced by the extremal ray of NE(X) corresponding to r(Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1, PCx(Y ) ∪ PC−x(Y ) consists of PY = {x, −x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' RY = RX for some RX ∈ PCx(X)∪PC−x(X)\\{Q1} such that z /∈ RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular we have ord(RY \\ {±x}) ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' RY = (RX\\{z})∪{x, y} for some RX ∈ PCz(X) such that (RX\\{z})∪{x} /∈ PC(X) and (RX \\ {z}) ∪ {y} /∈ PC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, we have ⟨RY \\ {x}⟩ = ⟨RX \\ {z}, y⟩, so ord(RY \\ {x}) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that V (EPY ) has codimension ≥ 2 in Y , so the proposition holds with f = f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume now that V (EP) has 4 components of codimension 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', we have r(P): x + (−x) = 0 r(Q1): x + y = −w r(Q′ 1): − x + (−w) = y y + (−y) = 0 r(Q2): x + w = −y r(Q′ 2): − x + (−y) = w w + (−w) = 0 w + y = −x −y + (−w) = x 14 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By [Cas03b, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1487, Case (3)], any other primitive collection R ∈ PC(X) is disjoint from {x, −x, y, −y, w, −w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let f1 : X → X1 be the smooth blow-down induced by the extremal ray of NE(X) corresponding to r(Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16 the primitive collections of X1 containing x or −x are only PX1 = P (Q2)X1 = Q2 (Q′ 2)X1 = Q′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that r(Q2) corresponds to an extremal curve class in NE(X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let f2 : X1 → X2 be the smooth blow-down induced by the extremal ray of NE(X) corresponding to r(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16 PX2 = {x, −x} is the only primitive collection in PCx(X2)∪PC−x(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This implies that V (EPX2) = ∅, so the proposition holds with Y = X2 and f = f2 ◦ f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let Y be a projective toric manifold of dimension ≥ 3, and P = {x, −x} ∈ PC(Y ) a centrally symmetric primitive collection of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let V (EP) ⊂ Y be the closed subset defined in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12, set U := Y \\ V (EP), and let π : U → W be the P1-bundle given by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that V (EP) has codimension ≥ 2 in Y , and let Z ⊂ Y be any given closed subset of codimension ≥ 2 in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Note that a toric manifold is rational, hence rationally connected, so by [Kol96, Proposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7], there is a smooth (very free) rational curve C ⊂ U \\ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Consider the surface S := π−1(π(C)) ⊂ U, and let n: ˜S → S be its normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ˜S is a Hirzebruch surface with P1-bundle structure ˜π : ˜S → P1 induced by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The curve class of the image of a fiber of ˜π on Y corresponds to the centrally symmetric relation x + (−x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By taking C general, we may assume that π(C) and π(Z) meet transversely in at most finitely many general points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence S and Z meet transversely in at most finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let Y be a projective toric manifold of dimension ≥ 3, and PY = {x, −x} ∈ PC(Y ) a centrally symmetric primitive collection of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that V (EP) has codimension ≥ 2 in Y , and let S ⊂ Y be as in Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then S · ch2(Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let S = π−1(π(C)) be the surface from Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4, n: ˜S → S its normaliza- tion, ˜π: ˜S → P1 the P1-bundle structure induced by π, and F a fiber of ˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Our goal is to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' V(Ep) S C (Z) J(C)THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 15 compute S · ch2(Y ) = 1 2 � v∈G(ΣY ) S · V (v)2 = 1 2 � v∈G(ΣY ) � n∗V (v) �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Recall that the curve class of the image of F in Y is associated to the relation x+(−x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By restricting the divisors n∗V (v) to F, we have: � � � � � n∗V (v) · F = 0 if v ̸= x, −x, n∗V (x) · F = 1, n∗V (−x) · F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence there are sections σ, σ′ of ˜π: ˜S → P1, and α, β, γ ∈ Z such that, on ˜S, � � � � � n∗V (v) = αF if v ̸= x, −x, n∗V (x) = σ + βF, n∗V (−x) = σ′ + γF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Therefore � v∈G(ΣY ) � n∗V (v) �2 = � v̸=x,−x \x18\x18\x18\x18\x18\x18 � n∗V (v) �2 + � n∗V (x) �2 + � n∗V (−x) �2 = σ2 + 2β + σ′2 + 2γ = σ2 − 2(σ · σ′) + σ′2 as σ · σ′ + β + γ = n∗V (x) · n∗V (−x) = 0 = (σ − σ′)2 = 0 as σ − σ′ is a multiple of F, and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ([dS06a, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1]) Consider the blowup diagram E ⊂ j> X := BlZ Y Z π:=f|E∨ ⊂ > Y f ∨ where both Y and Z are smooth projective varieties and codimY Z = c ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then we have the following relation between the 2nd Chern characters of X and Y : ch2(X) = f ∗ ch2(Y ) + c + 1 2 E2 − j∗π∗ c1(NZ/Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6, assume that c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let S ⊂ Y be a surface that intersects Z at most transversely at k ≥ 0 points, and let SX ⊂ X be its strict transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ch2(X) · SX = ch2(Y ) · S − 3 2 · k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Write S ∩Z = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , pk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then SX is isomorphic to the blowup of S at p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , pk, SX ∩ E = ∪k i=1ei, where ei ≃ P1 is the exceptional curve over pi, and (e2 i )SX = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By 16 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6, ch2(X) · SX = f ∗ ch2(Y ) · SX + c + 1 2 E2 · SX − j∗π∗ c1(NZ/Y ) · SX = ch2(Y ) · S + 3 2(E|SX)2 − π∗ c1(NZ/Y ) · SX|E = ch2(Y ) · S + 3 2 k � i=1 (ei)2 − π∗ c1(NZ/Y ) · k � i=1 ei = ch2(Y ) · S − 3 2k − k � i=1 ((((((( c1(NZ/Y ) · pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ We are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2: a toric Fano manifold X with m(X) = 1 is not 2-Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric Fano manifold with m(X) = 1, and fix a centrally symmetric primitive relation r(P): x + (−x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let f : X → Y be as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3, and π: U = Y \\ V (EPY ) → W the P1-bundle structure induced by r(PY ): x + (−x) = 0 (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Equation (5), Z := f(Exc(f)) has codimension ≥ 2 in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let S ⊂ Y be the surface given by Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then S and Z meet transversely in at most finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5 that ch2(Y ) · S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let SX be the strict transform of S in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7, ch2(X) · SX ≤ ch2(Y ) · S = 0, and so X is not 2-Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3 In this section, we work in the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3: (I) X is a Fano manifold of dimension n ≥ 6 with fan Σ = ΣX in NQ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (II) m(X) ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (III) r(P): x0 + · · · + xn−2 = 0 is a centrally symmetric primitive relation in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The equality m(X) = n − 2, as well as uniqueness of the centrally symmetric primitive collection, follows immediately from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Now the goal is to prove that ρ(X) ≤ 3, and this will follow from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='11, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='18, we conclude that the projective space Pn is the only smooth n-dimensional toric 2-Fano variety with m(X) ≥ n − 2 (Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let P := {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xn−2} and set Γ = Span P ⊂ NQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Consider the quotient π: NQ ≃ Qn → Qn/Γ ≃ Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since X is complete, the support of Σ is equal to NQ, and hence we can find generators (IV) x, y, z ∈ G(Σ) \\ P, for which (V) 0 ∈ Conv(π(x), π(y), π(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The nonnegative span ⟨x, y, z⟩ is not a cone in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We will argue by contradiction and assume that ⟨x, y, z⟩ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By (V), we can find a nonnegative triple of constants (c1, c2, c3) such that c1π(x) + c2π(y) + c3π(z) = 0, or in other words v := c1x + c2y + c3z = a0x0 + · · · + an−2xn−2 for some constants ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Note that by adding some multiple of r(P) (III) to the right hand side, we can assume the ai’s are nonnegative and such that at least one of them, say aj, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that v lies in two cones of Σ, namely v ∈ ⟨x, y, z⟩ ∩ ⟨x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ˇxj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xn−2⟩, which is impossible since {x, y, z} ∩ S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Since {x, y, z} does not span a cone, we conclude that (1) either it is a primitive collection, (2) or two of these vectors do not form a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The former case is the more technical one, and we start with it in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' After we are done analyzing it, we can assume that none of the triples {x, y, z} as in (IV) and (V) form a primitive collection, and this case will be treated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' First case: {x, y, z} is a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We recall that if X is a projective toric Fano manifold and m(X) > 1, then any x ∈ G(Σ) has at most one opponent by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='22, where the opponent of x is an element x′ ∈ G(Σ) such that {x, x′} ∈ PC(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' When we write {x, x′}, we mean either the set of two elements if x′ exists, or the singleton {x} if x′ does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting (I)—(V), pick a generator u ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then (V) and plane geometry imply that the convex hull of π(u) together with two of the vectors π(x), π(y), π(z) contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting (I)—(V), assume in addition that Q := {x, y, z} is a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then the corresponding primitive relation is either r(Q): x + y + z = xi + xj for possibly equal xi, xj ∈ P, or r(Q): x + y + z = v for some v ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since X is Fano (I), the degree of the primitive relation x + y + z = A is positive, so we can have only three possibilities for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The first one with A = 0 is actually not possible by our assumption (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The second is A = v, and we cannot say much about v at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The last possibility is r(Q): x + y + z = u + v for some, possibly equal, u, v ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='20, this is an extremal primi- tive relation, so by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 applied to r(Q) and τ = {0}, we get that ⟨u, x, y⟩, ⟨u, y, z⟩, ⟨u, x, z⟩ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2, we may assume without loss of generality that 0 ∈ Conv(π(u), π(x), π(y)), hence by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1, we get u ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The same argument applies to conclude v ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Hence we have two cases to consider: when deg(Q) is 1 and when it is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 18 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In this case, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3, we have (VI) a primitive relation r(Q): x + y + z = xi + xj for possibly equal xi, xj ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting (I)—(VI), assume in addition that G(Σ) is contained in P ∪ {x, y, z, x′, y′, z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then x′, y′, z′ do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Consequently, ρ(X) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By contradiction, assume that x′ ∈ G(Σ) exists, so {x, x′} is a primitive collection, and let x + x′ = α be the corresponding primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Clearly α ̸= x, x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The relation r(Q) (VI) gives α + y + z = xi + xj + x′, which shows α ̸∈ P ∪ {y, z} since otherwise the left hand side (LHS) forms a cone and we get an effective class of degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Indeed, if α ∈ {y, z} it is clear that the LHS would form a cone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' if α ∈ P applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to r(Q) and τ = ⟨α⟩ would follow that the LHS is a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So without loss of generality, we can assume x + x′ = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Applying the same argument to y′, we get that y +y′ = x′ or y +y′ = z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The former would imply x+y = 0, which is not possible by (II), so y +y′ = z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Again, applying the same argument to z′, we get z +z′ = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Summing the three primitive relations, we obtain x+y+z = 0, which contradicts (VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ This leaves us with the case when (VII) there exists u ∈ G(Σ) \\ (P ∪ {x, y, z, x′, y′, z′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2, we can assume without loss of generality that (VIII) 0 ∈ Conv(π(x), π(y), π(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume (I)—(VIII), then R := {x, y, u} is a primitive collection with primi- tive relation r(R): x + y + u = v for some v ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1, we have ⟨x, y, u⟩ /∈ Σ, and since u ̸= x′, y′, we conclude that {x, y, u} is a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3, we can have either r(R): x + y + u = xk + xl or r(R): x + y + u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the former case, combining r(R) with r(Q) (VI) provides us with a relation u + xi + xj = z + xk + xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to r(Q) (VI) and τ = ⟨xk, xl⟩, we get ⟨z, xk, xl⟩ ∈ Σ (here we are using the assumption dim(X) = n ≥ 6 (I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6, we get an effective curve class of degree 0, contradicting the Fano assumption (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ We will write down the result of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5 as an additional assumption, remembering that it is implied by the previous assumptions: (IX) We have a primitive relation r(R): x + y + u = v for some v ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume (I)—(IX), then G(Σ) ⊂ P ∪ {x, y, z, u, x′, y′, u′, v′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, z′ ∈ P ∪ {x, y, z, u, x′, y′, u′, v′}, and since v ̸= x, y, u, v′, we have that v ∈ P or v ∈ {z, x′, y′, u′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Take any w ∈ G(Σ) \\ (P ∪ {x, y, z, u, x′, y′, u′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2, the convex hull of π(w) together with two of π(x), π(y), π(u) contains 0, yielding an analog of (VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 19 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1 and from w ̸= x′, y′, u′, it follows that one of {w, x, u}, {w, y, u}, {w, x, y} is a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We will prove that the corresponding primitive relation has the form w + x + u = b or w+y+u = b or w+x+y = b for some b ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume for a contradiction that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3, we have that one of w+x+u, w+y+u, or w+x+y equals xk+xl, for possibly equal xk, xl ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence there exist a, b ∈ G(Σ) (one of which is w ̸= z) such that either x+a+b or y +a+b equals xk +xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume x+a+b = xk +xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Combining this with r(Q) (VI), it follows that y +z +xk +xl = a+b+xi +xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to r(Q) (VI) and τ = ⟨xk, xl⟩, we get ⟨y, z, xk, xl⟩ ∈ Σ (here we are using the assumption dim(X) = n ≥ 6 (I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6, we get an effective curve class of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The class is non-trivial since w ̸= y, z, xk, xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This contradicts the assumption that X is Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The case y + a + b = xk + xl is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So we have a primitive relation of the form w+x+u = b or w+y+u = b or w+x+y = b for some b ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Combining this with r(R) (IX), we get b + y = w + v or b + x = w + v or b + u = w + v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' All possibilities imply that w = v′, as otherwise, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6, we obtain an effective class of degree 0 (non-trivial since w, v ̸= y, u), which contradicts the Fano assumption (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume (I)—(IX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then x, y don’t have opponents, G(Σ) ⊂ P ∪ {x, y, z, u, u′, v′}, and we have a trichotomy: v ∈ P or v = z or v = u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We will only show that x′ does not exist, and the argument for y′ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Suppose to the contrary that x′ ∈ G(Σ), and let x + x′ = α be the corresponding primitive relation, so α ̸= x, x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then we can substitute x = α − x′ into r(Q) (VI) to get α + y + z = x′ + xi + xj, which shows α /∈ P ∪ {y, z}, as otherwise we would get an effective curve class of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Substituting x = α − x′ into r(R) (IX) gives the relation (6) α + y + u = v + x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Note that ⟨v, x′⟩ ∈ Σ by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We claim that Equation (6) is an extremal primitive relation by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume not, then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='26 implies that one of {α, y, u} is in {v, x′} and the remaining two vectors are opponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since u ̸= y′, then either α, y are opponents and u ∈ {v, x′}, or α, u are opponents and y ∈ {v, x′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' From (IX), we have u ̸= v and y ̸= v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (VII) means u ̸= x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' and (VI) implies y ̸= x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So both cases are impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So Equation (6) is an extremal primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, α ̸= y′, u, u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8, v forms a cone with α, hence α ̸= v′, which contradicts Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume (I)—(IX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then we have G(Σ) ⊂ P ∪ {x, y, z, u, v′} and a dichotomy: v ∈ P or v = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If moreover u′ exists, we have u + u′ = z, v = xi and u = x′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 20 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that u′ exists and let u + u′ = α be the corresponding primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then substituting it into r(R) (IX) gives a relation (7) x + y + α = u′ + v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By r(R) (IX), v ̸= u, so ⟨u′, v⟩ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since there are no x′ and y′, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='26 that Equation (7) is an extremal primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that α /∈ {u, u′, x, y, v′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Moreover, α /∈ P, as otherwise applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to r(Q) and τ = ⟨α⟩ would imply ⟨x, y, α⟩ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7, the only possibility is that α = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Therefore u′ + v = xi + xj by r(Q) (VI), so we have, after possibly relabeling, that v = xi, u′ = xj, which by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7 proves the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If instead u′ does not exist, the statement follows directly from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume (I)—(IX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then z doesn’t have an opponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We argue by contradiction and assume that z′ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Clearly z′ ̸= x, y, z by r(Q) (VI) and z′ ̸= u by (VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Furthermore, z′ ̸= u′, otherwise we have z = u by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='22, which contradicts (VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Finally, z′ /∈ P, otherwise z′ = xk implies z = x′ k, and r(Q) (VI) becomes x + y + x′ k = xi + xj, but applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to r(Q) and τ = ⟨xk⟩, we get ⟨xk, x′ k⟩ ∈ Σ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Thus by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8, the only possibility is z′ = v′, so z = v by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8, this implies G(Σ) ⊂ P ∪ {x, y, z, u, z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Consider the primitive relation z + z′ = β ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We will show that β = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Indeed, it is clear that β ̸= z, z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Combining z + z′ = β with r(Q) (VI), we have x + y + β = xi + xj + z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If β ∈ {x, y}, the left hand side is a cone, and we get a non-trivial effective relation of degree 0, which is impossible by (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If β ∈ P, applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to r(Q) and τ = ⟨β⟩ implies that ⟨x, y, β⟩ ∈ Σ, and we obtain a contradiction as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' But now, substituting z +z′ = u into r(R) (IX) yields x+y +z′ = 0, hence contradicting m(X) ≥ 3 (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume (I)—(VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let us summarize the consequences of (I)—(VII): (VIII) 0 ∈ Conv(π(x), π(y), π(u)) after possibly relabeling x, y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (IX) we have a primitive relation r(R): x + y + u = v (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' x, y, z don’t have opponents (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' G(Σ) ⊂ P ∪ {x, y, z, u, v′} (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8), so ρ(X) ≤ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' we have v ∈ P or v = z (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8), so we can consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Case v = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='9, v′ = z′ does not exist, hence it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 that G(Σ) ⊂ P ∪ {x, y, z, u}, so ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Case v ∈ P, say v = xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If v = xl doesn’t have an opponent, then G(Σ) ⊂ P ∪ {x, y, z, u} and ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So the tricky case is when v = xl has an opponent x′ l /∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let v + v′ = β ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By r(R), we have that x + y + u + v′ = v + v′ = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since m(X) ≥ 3, β ̸= x, y, u, v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence β ∈ P ∪ {z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If β ∈ P, let β = xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then v′ = xk − xl and k ̸= l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Using r(P), we obtain that v′ = 2xk + � t̸=k,l xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As the vectors on the right hand side form a cone, we get an effective curve class of degree 2 − n, which is impossible by (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 21 So β = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By r(Q), we obtain that x + y + v′ = xi + xj − xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If l ̸= i, j, then again, by using r(P) (III), we may write xi + xj − xl as xi + xj + � t̸=l xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As the vectors on the right hand side form a cone, we obtain an effective curve class of degree 3 − n, which is impossible by (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So l = i or l = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Up to symmetry, we may assume l = i, so v = xi, xi +x′ i = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By r(Q), we have that x + y + x′ i = xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Combining this with r(R), we obtain that x′ i + xi = u + xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Furthermore, u ̸= v′ = x′ i and u ̸= xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If u, xj form a cone, we obtain an effective non-trivial curve class of degree 0, which is impossible by (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So u = x′ j, v = xi and we have two primitive relations xi + x′ i = z and xj + x′ j = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then either i = j, in which case ρ(X) ≤ 3, or i ̸= j, and we notice that they are both degree 1 and hence extremal, so we can perform the contraction associated to one of them, say we contract the curve class xi + x′ i = z: X Y, V (z) V (xi, x′ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16, r(PY ): x0 + · · · + xn−2 = 0, r(RY ): x + y + x′ j = xi, r(Q′): x + y + x′ i = xj, r(Q′′): xj + x′ j = xi + x′ i are primitive relations in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since ρ(Y ) = 3, we apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We adopt the same notation as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17 and observe that we are in the case l = 5, hence G(Σ) = ⊔4 h=0Xh = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , ˇxj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xn−2} ⊔ {xj} ⊔ {x, y} ⊔ {x′ j} ⊔ {x′ i}, and either X2 ⊔X3 = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xn−2}, or c = b = 0 and X4 ⊔X0 = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xn−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' However, all possibilities for {xj} lead to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Indeed: if X3 = {xj}, then we have r3 : 1 · X3 + 1 · X4 = xj + x′ j = xi + x′ i ̸= 1 · Xh for all h = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' if X2 = {xj}, then we have r1 : 1 · X1 + 1 · X2 = x′ j + xj = xi + x′ i ̸= 1 · Xh for all h = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' if X4 = {xj}, then we have r3 : 1 · X3 + 1 · X4 = x′ j + xj = xi + x′ i ̸= 1 · Xh for all h = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' if X0 = {xj}, then we have r0 : 1·X0+1·X1 = xj+x′ j = xi+x′ i ̸= \x18\x18\x18 \x18 c · X2+(��b+1)X3 = 1 · X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This concludes the last case, and we get that ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a projective toric Fano manifold of dim(X) = n ≥ 5, m(X) ≥ 3, which admits a primitive relation r(P): x0 + x1 + · · · + xn−2 = 0 (I)—(III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let x, y, z ∈ G(Σ) \\ P be such that 0 ∈ Conv(π(x), π(y), π(z)) (IV)—(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume in addition that {x, y, z} is a primitive collection of degree 1 (VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 22 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We recall that assumptions (I)—(III) imply (IV)—(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then either G(Σ) ⊂ P ∪ {x, y, z, x′, y′, z′}, in which case ρ(X) ≤ 2 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4, or there exists a vector u ∈ G(Σ) \\ (P ∪ {x, y, z, x′, y′, z′}) (VII), in which case Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='10 implies ρ(X) ≤ 3, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Still working in the setting (I)—(V), we assume that {x, y, z} is a prim- itive collection whose primitive relation has degree 2, and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='11, we can exclude the case when degree one primitive collections as in (IV)—(VI) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In other words, these are the additional assumptions for this part of the proof: (X) we have a primitive relation r(Q): x + y + z = v for some v ∈ G(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (XI) there is no primitive collection {a, b, c} ⊂ G(Σ) \\ P with 0 ∈ Conv(π(a), π(b), π(c)) whose primitive relation has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting (I)—(V) and (X)—(XI), we have G(Σ) ⊂ P ∪ {x, y, z, x′, y′, z′, v′} and v ∈ P ∪ {x′, y′, z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If there is a generator u ∈ G(Σ) \\ (P ∪ {x, y, z, x′, y′, z′}), then by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2 and (XI) we can assume that we have a primitive relation of the form x + y + u = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Combining it with (X), we get u + v = w + z, so u = v′, otherwise ⟨u, v⟩ ∈ Σ and we get an effective non-trivial curve class of degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In particular, since v ̸= v′, we have v ∈ P ∪ {x′, y′, z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting (I)—(V) and (X)—(XI), assume that v = xm ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then x′ m does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The primitive relation r(Q) (X) becomes x + y + z = xm, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12, we have G(Σ) ⊂ P ∪ {x, y, z, x′, y′, z′, x′ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume to the contrary that we have x′ m ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Clearly, x′ m /∈ P because xm forms a 2-dimensional cone in Σ with any other generator xi ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2, x′ m makes a primitive collection with two of x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Without loss of generality, assume we have a primitive relation of the form x + y + x′ m = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Combining it with r(Q) (X), we get x′ m + xm = z + w, so w = z′ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let α = xm+x′ m = z+z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then α ̸∈ P because x′ m = α−xm is not in Span P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Now substituting z = α − z′ into r(Q) (X) yields (8) x + y + α = xm + z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since xm ̸= z by r(Q) (X), we have ⟨xm, z′⟩ ∈ Σ, so Equation (8) is an extremal primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This implies that α ̸= x, y, z, x′, y′, z′, x′ m, a contradiction with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting (I)—(V) and (X)—(XI), assume that v = xm ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13, we have G(Σ) ⊂ P ∪ {x, y, z, x′, y′, z′}, and, as before, the primitive relation (X) is r(Q): x + y + z = xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 23 We show that at most one of x′, y′, z′ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Suppose to the contrary that for example x′, y′ ∈ G(Σ), and let α = x + x′, β = y + y′ ∈ G(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since α + y + z = xm + x′ and ⟨xm, x′⟩ ∈ Σ, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='26 shows that this is an extremal primitive relation: indeed, either α and y are opponents and z ∈ {xm, x′}, or α and z are opponents and y ∈ {xm, x′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' But y, z /∈ P ∪ {x′}, so this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that α ̸∈ {x, x′, y, y′, z, z′}, so α = xl ∈ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' similarly, β = xk ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' But we have α + β + z = x′ + y′ + xm, which is not possible since we show that the right hand side forms a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Indeed, applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to x + x′ = xl and τ = ⟨xm, xk⟩ (we use here that n ≥ 5), we obtain ⟨xm, xk, x′⟩ ∈ Σ, and applying then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 to y +y′ = xk and τ = ⟨xm, x′⟩ we have that ⟨xm, x′, y′⟩ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So, after possibly relabelling x, y, z, we have G(Σ) ⊂ P ∪{x, y, z, x′}, hence ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting (I)—(V) and (X)—(XI), assume that v = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We have r(Q): x+y+z = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We know G(Σ) ⊂ P∪{x, y, z, x′, y′, z′} by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So it is enough to show y′ and z′ do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Suppose to the contrary that, for example, y′ ∈ G(Σ), and let β = y + y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then β + x + z = x′ + y′ and ⟨x′, y′⟩ ∈ Σ, so again, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='26, this is an extremal primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Indeed, assume the relation is not primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='26, either β and x are opponents and z ∈ {x′, y′}, or β and z are opponents and x ∈ {x′, y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' But z, x /∈ {x′, y′}, since {x, y, z} is a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hence β + x + z = y′ + x′ is a primitive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='8 that ⟨x, x′⟩ ∈ Σ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a projective toric Fano manifold of dim(X) = n ≥ 5, m(X) ≥ 3, which admits a primitive relation r(P): x0 + x1 + · · · + xn−2 = 0 (I)—(III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that any primitive collection {x, y, z} such that x, y, z ∈ G(Σ)\\P and 0 ∈ Conv(π(x), π(y), π(z)) (IV) —(V) has degree 2 (XI), and that there exists such a triple x, y, z (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The statement follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='12, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='14 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Second case: none of {x, y, z} form a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric Fano manifold of dim(X) = n ≥ 5, m(X) ≥ 3, which admits a primitive relation r(P): x0 + x1 + · · · + xn−2 = 0 (I)—(III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume in addition that none of the triples {x, y, z} ⊆ G(Σ)\\P such that 0 ∈ Conv(π(x), π(y), π(z)) (IV)—(V) form a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Before proving Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17, we will formulate the following useful lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17, take any triple {x, y, z} ⊆ G(Σ) \\ P with ⟨x, y⟩ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assume that 0 ∈ Conv(π(x), π(y), π(z)), or equivalently, π(z) ∈ ⟨−π(x), −π(y)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then z = x′ or z = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1, x, y, z do not span a cone, and by assumption, they do not form a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So two of the vectors must not form a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Since we assumed that ⟨x, y⟩ ∈ Σ, we must have z = x′ or z = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ 24 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We select x, y ∈ G(Σ) with ⟨x, y⟩ ∈ Σ and such that the cone generated by π(x) and π(y) is maximal among cones in Q2 ≃ Qn/Γ coming from such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If there is z ∈ G(Σ) such that π(z) is outside the cone ⟨π(x), π(y)⟩, then we show that z = x′ or z = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Indeed, the case π(z) ∈ ⟨−π(x), −π(y)⟩ is covered by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' π(z) ∈ ⟨π(x)⟩ or π(z) ∈ ⟨π(y)⟩ cannot happen by an argument similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' and in the remaining case, π(z) is in ⟨π(x), −π(y)⟩ or ⟨−π(x), π(y)⟩, then we use maximality of the cone generated by π(x) and π(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let v be such that π(v) is in the open half plane determined by Span π(x) not containing π(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Up to relabelling of x, y, we can assume that such a v exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If v = x′, then x + x′ = y′, since π(x + x′) is non-zero and is outside of ⟨π(y), π(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So ⟨π(x), π(y)⟩ ⊂ ⟨−π(y), −π(x′)⟩ ∪ ⟨−π(x′), −π(y′)⟩ ∪ ⟨−π(y′), −π(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='18 that G(Σ) = P ∪ {x, y, x′, y′}, which concludes this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If now v = y′, in case π(v) ∈ ⟨−π(x), −π(y)⟩, we notice that y + y′ = x′ as above, and in case π(v) ∈ ⟨π(x), −π(y)⟩, by completeness, we find w such that π(w) ∈ ⟨−π(x), π(y)⟩ ∪ ⟨−π(x), −π(y)⟩ and notice w = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In either case, we get Q2 = ⟨−π(x), −π(y)⟩ ∪ ⟨−π(y), −π(x′)⟩ ∪ ⟨−π(x′), −π(y′)⟩ ∪ ⟨−π(y′), −π(x)⟩, and now Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='18 gives G(Σ) = P ∪ {x, y, x′, y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Toric Fano manifolds with m(X) = n − 2 In this section, we classify all n-dimensional toric Fano manifolds X with m(X) = n − 2 and n ≥ 6 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='3, we know that ρ(X) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2 classify n-dimensional toric Fano manifolds X with m(X) = n − 2, n ≥ 5, and Picard rank ρ(X) = 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Together, these results yield a classification of toric Fano manifolds with m(X) = n − 2 and n ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric Fano manifold of dimension n ≥ 5, m(X) = n − 2 and ρ(X) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then X ≃ PP2(E), where E is one of the following vector bundles on P2: E = OP2(1) ⊕ O⊕n−2 P2 , E = OP2(1) ⊕ OP2(1) ⊕ O⊕n−3 P2 , E = OP2(2) ⊕ O⊕n−2 P2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be an n-dimensional toric Fano manifold with m(X) = n − 2 and ρ(X) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We recall that toric manifolds with Picard rank 2 are classified by [Kle88]: they are projective space bundles over projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The assumption m(X) = n − 2 implies that X is a Pn−2-bundle over P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We can write X = P(E), where E = OP2(a0) ⊕ OP2(a1) ⊕ · · · ⊕ OP2(an−2) and a0 ≥ a1 ≥ · · ≥ an−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The Fano assumption on X is equivalent to saying that �n−2 i=0 ai ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Thus we have the following cases: (1) E = O⊕n−1 P2 and X ≃ Pn−2 × P2, (2) E = OP2(1) ⊕ O⊕n−2 P2 , (3) E = OP2(1) ⊕ OP2(1) ⊕ O⊕n−3 P2 , (4) E = OP2(2) ⊕ O⊕n−2 P2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In Case (1), we have m(X) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Cases (2)—(4) provide the complete list of toric Fano manifolds of Picard rank 2 and m(X) = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS 25 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Let X be a toric Fano manifold of dimension n ≥ 5, m(X) = n − 2 and ρ(X) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then one of the following holds: (1) X = PS(E) is a Pn−2-bundle over a toric surface S, where (S, E) is one of the following: S = P1 × P1 and E = OP1×P1(1, 1) ⊕ O⊕n−2 P1×P1, S = P1 × P1 and E = OP1×P1(1, 0) ⊕ OP1×P1(0, 1) ⊕ O⊕n−3 P1×P1, S = F1 and E = OF1(e+f)⊕O⊕n−2 F1 , where e ⊂ F1 is the −1-curve, and f ⊂ F1 is a fiber of F1 → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (2) Let Y ≃ PP2� OP2(1)⊕O⊕n−2 P2 � be the blowup of Pn along a linear subspace L = Pn−3, and denote by E ⊂ Y the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then X is the blowup of Y along a codimension 2 center Z ⊂ Y , where: Z is the intersection of E with the strict transform of a hyperplane of Pn containing the linear subspace L, or Z is the intersection of the strict transforms of two hyperplanes of Pn, one containing the linear subspace L, and the other one not containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We apply Batyrev’s classification of toric manifolds with ρ(X) = 3, stated in Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We adopt the same notation as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17, and treat separately the cases when the number of primitive collections is l = 3 and l = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Case l = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As G(Σ) = X0 ⊔ X1 ⊔ X2 and m(X) = n − 2, we have X0 = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xn−2}, X1 = {v1, v2} and X2 = {z1, z2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The corresponding primitive relations are all extremal by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13, X is a Pn−2-bundle over a surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Up to relabelling in X0, X1 and X2, three possible choices for the remaining two primitive relations are (noting that m(X) = n − 2 > 1): � v1 + v2 = x0, z1 + z2 = v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' � v1 + v2 = x0, z1 + z2 = x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' � v1 + v2 = x0, z1 + z2 = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that S is isomorphic to F1 when r(X1) and r(X2) are as in the first column above, or to P1 × P1 in two other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Case l = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We denote by li = |Xi| ≥ 1 the cardinality of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If (c, b) = (0, 0), then both r2 : 1 · X2 + 1 · X3 = 0 and r4 : 1 · X4 + 1 · X0 = 0 are centrally symmetric primitive relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By the assumption m(X) = n − 2, we have 2n − 2 ≤ l2 + l3 + l4 + l0 ≤ |G(Σ)| − 1 = n + 2, which implies n ≤ 4, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' So we have (c, b) ̸= (0, 0), and P := X2 ⊔ X3 is the only centrally symmetric primitive collection, so l2 + l3 = n − 1 and l0 + l1 + l4 = 4, with l0, l1, l4 ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' As deg(r0) > 0, we get the inequality 3 ≥ l0 + l1 > � ci + � bj + l3, which is only satisfied when l3 = 1, l0 + l1 = 3 and exactly one entry in (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , cl2, b1) equals one, while the others are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Up to relabelling, there are two cases: c1 = 1 or b1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' We then have l4 = 4 − (l0 + l1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' From deg(r3) > 0, we get l3 + l4 > l1, 26 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN which means 2 > l1, and this ensures l1 = 1, hence (l0, l1, l4) = (2, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' To sum up, we denote X0 = {v1, v2}, X1 = {y}, X2 = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , xn−2}, X3 = {x0} and X4 = {z}, and get the following two possibilities for X: b1 = 1 and cj = 0 for every j: r0 : v1 + v2 + y = 2x0, r1 : y + x1 + · · · + xn−2 = z, r2 : x0 + · · · + xn−2 = 0, r3 : x0 + z = y, r4 : z + v1 + v2 = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' c1 = 1, b1 = 0 and cj = 0 for every j > 1: r0 : v1 + v2 + y = x1 + x0, r1 : y + x1 + · · · + xn−2 = z, r2 : x0 + · · · + xn−2 = 0, r3 : x0 + z = y, r4 : z + v1 + v2 = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='13, the open U = X \\ � V (y) ∪ V (z) � has a Pn−2-bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The relation r3 corresponds to an extremal curve class by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='20, which induces a smooth blow-down h: X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='16, the primitive relations in Y in cases b1 = 1 and c1 = 1 are, respectively: � x0 + · · · + xn−2 = 0, z + v1 + v2 = x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' � x0 + · · · + xn−2 = 0, z + v1 + v2 = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' It follows that Y ≃ P � OP2(1) ⊕ O⊕n−2 P2 � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', Y is the blowup of Pn along a linear subspace L = V (z, v1, v2) ≃ Pn−3 ⊂ Pn, with exceptional divisor E = V (x0) ⊂ Y in the first case, and E = V (x1) ⊂ Y in the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The center of the blowup h: X → Y is V (x0, z) ⊂ Y , yielding the two varieties described in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Code for computing primitive collections by Will Reynolds Let Σ be the fan of a toric manifold of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For each integer k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , n}, we denote by Σ(k) the subset of Σ consisting of the k-dimensional cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' To determine whether a given subset P ⊆ G(Σ) is a primitive collection, we proceed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' First make sure there does not exist σ ∈ Σ(n) with P ⊆ G(σ), and if there does, stop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' otherwise, for each v ∈ P, make sure there exist σ ∈ Σ(n) with P \\ {v} ⊆ G(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In the special case |P| = 2 it suffices to check that there does not exist σ ∈ Σ(n) with P ⊆ G(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Note that, by definition, if P is a primitive collection and P ⊆ Q, then Q is not a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Therefore, when looking for primitive collections, we go through subsets of G(Σ) in increasing cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Assuming an implementation of the above basic algorithm, a reasonably efficient way to list all of the primitive collections of a fan is to arrive at such a list by eliminating P ⊆ G(Σ) which are not primitive collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The first step is to remove any P with |P| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Then for each P we check whether it is a primitive collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If it is, we keep it and remove all sets containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' If it is not, we remove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' One way of implementing this method of listing primitive collections is implemented in pseudocode in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This algorithm is impractical if G(Σ) is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In practice, on a modern laptop, it works reasonably well up to about |G(Σ)| = 17, partly because of the combination of the following factors: first, eliminating all of the supersets of any P with |P| = 2 cuts down the remaining search space significantly, and second, the relative abundance of primitive collections of size 2, at least among toric Fano varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For example, the 124 toric Fano 4-folds altogether have 785 primitive collections, of which 566 have cardinality 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This last factor makes the computation of the value of m(X) for a given toric Fano variety X easier as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Of the toric Fano varieties of a given dimension n (for n ≤ 6) those X with m(X) = 1 make up an overwhelming majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' This means that computing m(X) is usually extremely fast, even in the most straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' The following table summarizes the data for dim(X) ∈ {4, 5, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' dim(X) # Fanos #(m=1) #(m=2) #(m=3) #(m=4) #(m=5) #(m=6) 4 124 107 15 1 1 5 866 744 112 8 1 1 6 7622 6333 1174 105 8 1 1 27 Algorithm 1: List primitive collections of a given fan Input: fan Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' n ← dim Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' PC ← {P ⊆ G(Σ) : |P| > 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' for P ∈ PC satisfying |P| = 2 do if there exists σ ∈ Σ(n) such that P ⊆ G(σ) then PC ← PC \\ {P};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' else PC ← PC \\ {Q ∈ G(Σ) : P ⊊ Q};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' end end for i ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' , n} do for P ∈ PC satisfying |P| = i do if there exists σ ∈ Σ(n) such that P ⊆ G(σ) then PC ← PC \\ {P};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' else b ← True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' for v ∈ P do if there does not exist σ ∈ Σ(n) with P \\ {v} ⊆ G(σ) then b ← False;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' end end if b then PC ← PC \\ {Q ⊆ G(Σ) : P ⊊ Q};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' else PC ← PC \\ {P};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' end end end end Output: PC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' For convenience, we also provide Macaulay2 code implementing the algorithm for com- puting primitive collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' coneExistenceCheck = (S, fan) -> ( for cone in fan do ( if isSubset(S, cone) then ( return true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 28 properSubsetCheck = (S, fan) -> ( for ray in S do ( if coneExistenceCheck(S-set{ray}, fan) == false then ( return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' return true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' isPrimitiveCollection = (P, Var) -> ( if coneExistenceCheck(P, orbits(Var, 0)) then ( return false) else ( return properSubsetCheck(P, orbits(Var, 0) );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' supsetsOfPrimColl = (E, B) -> ( return set{for P in E-set{B} when isSubset(B, P) list P};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' primitiveCollections = (Var) -> ( n = length rays Var;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' primColls = select(subsets(toList(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='.n-1)), x -> length x > 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' for P in subsets(toList(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='.n-1), 2) do ( if coneExistenceCheck(P, orbits(Var, 0)) == false then ( primColls = primColls - supsetsOfPrimColl(primColls, P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=') else ( primColls = primColls - set{P};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' for i in toList(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='.n) do ( for P in subsets(toList(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='.n-1), i) do ( if member(P, primColls) == false then continue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' if isPrimitiveCollection(P, Var) then ( primColls = primColls - supsetsOfPrimColl(primColls, P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ) else ( primColls = primColls - set{P};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' return sort primColls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 29 30 ARAUJO, BEHESHTI, CASTRAVET, JABBUSCH, MAKAROVA, MAZZON, AND VISWANATHAN References [AC12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Araujo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Castravet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “Polarized minimal families of rational curves and higher Fano manifolds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 87–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' issn: 0002-9327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' [AC13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Araujo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Castravet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “Classification of 2-Fano manifolds with high index”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: A celebration of algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Clay Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=', Providence, RI, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' [Ara+22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Araujo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Beheshti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Castravet, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Jabbusch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Makarova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Mazzon, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Taylor, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Viswanathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “Higher Fano manifolds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Argentina 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 103–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' issn: 0041-6932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' [Bat91] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Batyrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “On the classification of smooth projective toric varieties”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: Tohoku Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “On the classification of toric Fano 4-folds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Algebraic geometry, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 1999, pp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' [Cam92] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Campana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “Connexit´e rationnelle des vari´et´es de Fano”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' ´Ecole Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' (4) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='5 (1992), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 539–545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' issn: 0012-9593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' [Cas03a] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Casagrande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “Contractible classes in toric varieties”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='1 (2003), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Fu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Hwang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' “Minimal rational curves on complete toric manifolds and applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Edinb.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Cox, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Little, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Schenck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Toric varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' Graduate Studies in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' xxiv+841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content=' isbn: 978-0-8218-4819-7.' metadata={'source': 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Sciences, Loughborough Univer- sity, Loughborough, LE11 3TU, UK and School of Mathematical Sciences, University Park Campus, The University of Nottingham, Nottingham, NG7 2RD, UK Email address: N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='Viswanathan@lboro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='uk Will Reynolds, School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, UK Email address: W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAyT4oBgHgl3EQf9_ol/content/2301.00883v1.pdf'} 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+Jerry Yang +Harvard University +Abstract +Few-shot learning allows pre-trained language +models to adapt to downstream tasks while +using a limited number of training examples. +However, practical applications are limited +when all model parameters must be optimized. +In this work we apply a new technique for pa- +rameter efficient few shot learning while adopt- +ing a strict definition of parameter efficiency. +Our training method combines 1) intermedi- +ate training by reformulating natural language +tasks as entailment tasks (Wang et al., 2021a) +and 2) differentiable optimization of template +and label tokens (Zhang et al., 2021). +We +quantify the tradeoff between parameter effi- +ciency and performance in the few shot regime +and propose a simple model agnostic approach +that can be extended to any task By achieving +competitive performance while only optimiz- +ing 3% of a model’s parameters and allowing +for batched inference, we allow for more effi- +cient practical deployment of models. +1 +Introduction +Large pre-trained language models have demon- +strated adaptability to solve natural language pro- +cessing (NLP) tasks. +Typically, such language +models are adapted to a downstream task through +fine-tuning (Howard and Ruder, 2018). Although +fine-tuning improves performance on downstream +tasks, it is costly because it relies on updating ev- +ery parameter of the model (355 million in the case +of roBERTa)and requires storing a separate copy +of the model for every downstream task. These +storage requirements can become prohibitive, thus +necessitating research into more parameter effi- +cient methods . Alternative fine-tuning methods +that update fewer parameters can have other trade- +offs. +For example, adapter tuning fine-tunes a +small number adapter parameters inserted between +the transformer layers (Houlsby et al., 2019)but +requires optimizing external parameters and still +fine-tunes on the entire training dataset. Other +methods have explored fine-tuning in the few shot +learning case, where a limited number of labeled +training samples are used for fine-tuning. These +approaches have the disadvantages of relying on ex- +treme model size (Brown et al., 2020) (Lester et al., +2021), optimizing all model parameters (Wang +et al., 2021a),(Zhang et al., 2021), or using ex- +ternal architectures (Houlsby et al., 2019) (Li and +Liang, 2021) (Gao et al., 2021). In this project, we +present a simple, extensible method that improves +few-shot performance without any extra parame- +ters by combining two approaches: 1) leveraging +trainable prompt pseudotokens rather than updat- +ing all the model parameters (Zhang et al., 2021), +and 2) reformulating natural language processing +tasks as entailment tasks and applying an interme- +diate training step, enabling better generalization +to downstream tasks. (Wang et al., 2021a). Our +major contributions are as follows. +• Our method achieves competitive few shot +performance while optimizing only 3% of a +model’s parameters reducing storage costs by +a factor of 30. +• We introduce a strict definition of parameter +efficiency which extends the practical uses +of few shot learning by allowing batching of +computation across tasks. +2 +Related Work +2.1 +Finetuning +The standard method for fine-tuning Masked Lan- +guage Models (MLMs) like BERT applies a clas- +sification head to the [CLS] token representation. +The language model learns to update the [CLS] rep- +resentation to better solve the downstream task. A +number of reformulations have been proposed seek- +ing to increase performance and improve parameter +efficiency. +arXiv:2301.13345v1 [cs.CL] 31 Jan 2023 + +2.2 +Prompting +Language models learn a general set of abilities +that can be adapted to specific downstream tasks. +One method is to use task-specific natural language +prompts to guide the language model output. GPT- +3, for example, uses prompts and in-context exam- +ples to achieve good few-shot performance on var- +ious tasks (Brown et al., 2020). GPT-3 leverages +extreme scale (175 Billion parameters) to adapt +to natural language prompts without fine-tuning. +Prompting can be particularly useful for few-shot +learning in the low-data regime. For some tasks, a +well designed prompt can be shown to be equiva- +lent to hundreds or thousands of additional labeled +training points (Le Scao and Rush, 2021). AUTO- +PROMPT uses a gradient-based search to optimize +a discrete prompt (Shin et al., 2020). LMBFF uses +an auxiliary language model to generate a set of +candidate prompts and chooses the best candidate +(Gao et al., 2021). +2.3 +Pattern Exploiting Training +One alternative to standard fine-tuning is to model +the output as a cloze completion task where the +output is the model’s representation of a masked +input token (Schick and Schütze, 2021). Intuitively, +this approach works well because it more closely +matches the training process for MLMs. In the +pre-training task for models such as BERT and +roBERTA, the model is asked to predict the identity +of a masked token based on the hidden representa- +tions of neighboring tokens. +Further work has extended this approach to use +natural language prompts to guide the cloze output +(Gao et al., 2021). Additional work has focused on +training the prompt tokens in continuous space by +optimizing a set of prompt pseudotokens. (Li and +Liang, 2021) (Liu et al., 2021) (Lester et al., 2021). +Additionally in the DART method, the tokens used +as classification labels can be optimized (Zhang +et al., 2021). +2.4 +Entailment Reformulation +Work from (Wang et al., 2021a) focuses on improv- +ing language model performance by formulating +NLP tasks as entailment tasks. Fundamentally, en- +tailment seeks to determine whether for a pair of +inputs (S1, S2), the first sentence entails or contra- +dicts the second. Most standard classification tasks +in NLP can be reformulated as entailment tasks. +For example, a sentiment analysis task can be be +framed as an entailment task using the following +template: +[CLS]S1[SEP]S2[EOS], +(1) +With S2 = "It was great" as the entailment prompt. +Instead of using the [CLS] token representation of +S1 to classify the review as positive or negative as +in standard fine-tuning, we instead concatenate the +text with the prompt and use the [CLS] token rep- +resentation of the concatenated sequence to denote +whether the first sentence entails the second. +For multi-class classification problems we con- +struct a different input for every class and take the +label as the class with the highest entailment score. +A key to the success of the entailment approach +from (Wang et al., 2021a) is an intermediate train- +ing step where the pre-trained language model is +fine-tuned on a natural language inference (NLI) +task like MNLI. Intuitively, the model can be +adapted to be good at one entailment task and then +generalize to perform well on other reformulated +entailment tasks. +2.5 +Parameter Efficiency +Related works adopt various, sometimes contra- +dictory, definitions of parameter efficiency when +applied to language model fine-tuning. Broadly, +these definitions can be grouped into several cate- +gories: +1. reducing the number of model parameters nec- +essary to achieve good few shot adaptability +2. optimizing a small subset of the total model +parameters +3. avoiding external parameters or changes to the +model architecture +Some works on few-shot learning explore tech- +niques allowing smaller models to learn robustly +(Wang et al., 2021a). Large models such as GPT3 +with 175 billion parameters can take advantage of +their scale to perform well at few-shot in-context +learning (Brown et al., 2020). A technique can be +parameter efficient if it allows similar results be be +achieved with a smaller language model e.g. a 340 +million parameter roBERTa model rather than the +175 billion parameter GPT-3 or 11 billion parame- +ter T5. +Parameter efficiency can also aim to optimize a +smaller number of task specific parameters while +keeping most of the language model parameters + +Figure 1: Differential Entailment Approach +frozen. Adapter tuning inserts trainable layers be- +tween the frozen layers of a Transformer language +model. (Houlsby et al., 2019). Prompt tuning op- +timizes a small set of trainable input tokens while +keeping the pre-trained Transformer layers frozen +(Lester et al., 2021). Lite Self Training (LiST) +freezes most of the encoder parameters and only +trains a small number of adapter parameters (Wang +et al., 2021b). LoRA, adds low rank trainable ma- +trices between transformer layers (Hu et al., 2021) +while freezing the pretrained model. +Other works define parameter efficiency as the +lack of a need for parameters external to the model +being fine-tuned. Part of the motivation for differ- +ential prompt tuning (Zhang et al., 2021) is that it +directly optimizes trainable pseudotokens without +the need for an external model such as LSTM in +P-tuning (Liu et al., 2021). Such approaches are +advantageous as they require no modifications to +a pre-trained model’s architecture and do not add +additional inference time like adapters. Delta Tun- +ing explores in depth the performance of different +parameter efficient approaches at different model +scales and in combination with one another (Ding +et al., 2022) +We focus on parameter efficiency in the true few +shot learning regime. Therefore, we do not take +advantage of any additional unlabeled training data. +Iterative PET used this approach to pseudolabel +unlabelled samples and provide extra training ex- +amples to a model (Schick and Schütze, 2021). +LiST iteratively trains a student model on data +pseudolabeled by a teacher model (Wang et al., +2021b). However, these semi-supervised learning +approaches require extra unlabeled training data as +well as additional training computation compared +to true few-shot learning. +3 +Approach +Our main approach is shown in Figure 1. We con- +vert all NLP tasks to the entailment format and +train few shot models from an intermediate train- +ing checkpoint. The entailment approach outlined +in (Wang et al., 2021a) performs traditional fine- +tuning and updates all model parameters via gradi- +ent descent. Instead of performing the computation- +ally expensive update step on all model parameters, +our approach fine-tunes only the prompt and label +tokens in continuous space while keeping the main +language model frozen. By using more expressive +pseudotokens as part of our prompt and by training +only the input parameters, we achieve a parameter +efficient few shot learning method with competitive +few-shot performance. +3.1 +Pseudotokens +With discrete tokens, the label template tokens are +either chosen manually or determined through a +search over tokens in a discrete space. In compar- +ison, our label descriptions are optimized in con- +tinuous space via back-propagation and hence can +attain more expressive, fine-grained representations +to prompt a model for a certain task. Formally, we +define a set of pseudotokens T /∈ V outside of the +normal vocabulary. The pseudotoken embedding +h(T ) is a trainable set of parameters that are opti- +mized via backpropagation. For a given input we +might have the following prompt: +S1[SEP] T0T1T2 it was [LABEL] +We differentiably optimize prompting pseudoto- +kens. We also experiment with allowing the label + +Template +[CLS] The drama discloses nothing [SEP] [T1][T2][T2] P(it) P(was) V(positive) +Embeddings +[CLS] e(The) e(drama) e(discloses) e(nothing) e([SEP) h([T1])h([T2])h([T3]) h(P(it)) h(P(was)) h(V(positive)) +Pre-trained Language Model +(roBERTa) +Hidden States +[CLS] The drama discloses nothing [SEP] [T1][T2][T2] P(it) P(was) V(positive) +Label: Y +CLS +Predict +Entail: Positive +Head +Not Entail: Negativeembedding h([LABEL] to be a pseudotoken with a +trainable embedding. For label and prompt tokens +we experiment with both initializing these pseudo- +tokens embeddings randomly and initializing them +with the embeddings of the original tokens. +3.2 +Parameter Efficiency +We adopt the strictest definition of parameter ef- +ficiency that has practical advantages for down- +stream applications. In Differentiable Entailment +we 1) use a smaller language model compared to +GPT-3 or T5, 2) freeze the main encoder parame- +ters, 3) only fine-tune a limited set of pseudotokens +without any external parameters or architectural +modifications and 4) employ strict few-shot learn- +ing without using any additional training data. +Following the method in Prompt Tuning, we +freeze the main model parameters and only fine- +tune the subset of trainable input tokens (Lester +et al., 2021). In contrast to Prompt Tuning we also +fine tune the model classification head since we are +outputting a specific classification label rather than +using a generative model such as T5. +By freezing the model parameters we can effi- +ciently optimize a smaller set of task-specific pa- +rameters, namely the pseudotoken embeddings as +well as the entailment classification head. In con- +trast to approaches outlined above, which rely on +a large-scale model to make up for a reduction in +trainable parameters (Lester et al., 2021), we use a +smaller language model. With roBERTa-large this +leads to a more than 30x reduction in the number of +trainable parameters. Furthermore, instead of stor- +ing a fine-tuned 355 million parameter model for +each task, we only need to store the task-specific +trainable pseudotoken embeddings and classifica- +tion head. Finally, in contrast to methods which +finetune all the model parameters (Zhang et al., +2021) (Wang et al., 2021a) or methods with exter- +nal parameters (Houlsby et al., 2019) our method +allows the hidden state computation for different +tasks to be batched together since only the spe- +cific prompt embeddings for each tasks need to be +changed. As others have noted: such in batch par- +allel computing has extreme practical application +(Ding et al., 2022). LoRA also allows for multitask +batching, however applying additional low rank +matrices to later transformer layers is more com- +plex than simply swapping out a set of task specific +input embeddings (Hu et al., 2021). +Method +Template +Cloze +S_1 [SEP] it was [MASK] +Entailment +S_1 [SEP] it was great +Differential Prompt +S_1 [SEP][Prompt tokens] great +Differential Label and Prompt +S_1 [SEP][Prompt tokens] [Label token] +Table 1: Example Prompting Templates for a Sentiment +Classification task. For our method we optimize either +a set of prompt pseudotokens and/or a label pseudoto- +ken. +3.3 +Templates +We explore several different approaches to combin- +ing label templates with pseudotokens. For various +tasks, we adapt the standard prompt templates used +in previous works (Zhang et al., 2021) (Wang et al., +2021a). For example, sentiment analysis tasks such +as CR can be prompted for both entailment and +cloze completion in a simple way. In 1, we show +label templates for a sentiment analysis tasks. For +such tasks, the prompt standard template is "it was +great". The cloze completion method concatenates +the prompt to the input sentence and masks out +the label "great", whereas our method concatenates +the template without masking the token of inter- +est and predicts entailment. When training label +templates in the continuous space, we initialize +from the embeddings of the label template tokens +in the standard template. For example, given the +following template: +S1[SEP]T0 . . . Tjit was great +We would train the prompt tokens "it", "was", the +label token "great" and j + 1 additional pseudoto- +kens. +For sentence pair tasks such as Quora Question +Pairs (QQP), we adopt a slightly different template +following (Wang et al., 2021a)(Zhang et al., 2021). +The task is to predict entailment based on the sen- +tence pairs and a set of prompt pseudotokens in- +serted between them. For QQP we use the format +S1[SEP]T0 . . . TjS2 +3.4 +Symmetry of Entailment +In (Wang et al., 2021a), a single label description p +is used for each example in a binary classification +task, e.g. a binary sentiment classification task is +formulated as whether input sentence S1 entails +S2 = "This indicates positive sentiment.". To en- +courage more robust tuning of the label description +parameters and classification head, we experiment + +Figure 2: Entailment allows batching of hidden state computations across tasks +Figure 3: Symmetry for simple data augmentation +with using two label descriptions p1 and p−1 for +binary classification tasks, and augment the dataset +as: +Dtrain = {(xi, p1, yi) ∪ (xi, p−1, −yi)}K +i=1 +(2) +For a positive sentiment example, the two cor- +responding +samples +in +the +training +dataset +would +be +(xi, p1, 1) +and +(xi, p−1, −yi) +where p1 += +This indicates positive sentiment +with +label +1 +(does +entail) +and +p−1 += +This indicates negative sentiment +with +label +0 (does not entail). +4 +Experiments +4.1 +Evaluation +We evaluate our method on the tasks from (Wang +et al., 2021a) which are mainly the subset of the +GLUE and SuperGLUE benchmark tasks that are +compatible with the entailment reformulation. In +addition, we follow the best practices for evalua- +tion of few shot NLP fine-tuning methods (Bragg +et al., 2021). For each experiment we sample 5 +non-overlapping training folds and report average +performance after k-shot training over the entire +test set (Gao et al., 2021). Hyperparameters are +tuned for each task and method. +4.2 +Implementation Details +Models are implemented using the pytorch (Paszke +et al., 2019) and transformers (Wolf et al., 2019) li- +braries with code adapted from (Zhang et al., 2021). +Our pre-trained model is roBERTa large (Liu et al., +2019). Checkpoints for roberta-large-base as well +as checkpoint models are downloaded from hug- +gingface. We experiment with different interme- +diate checkpoints, namely roberta-large-mlni and +a checkpoint trained robustly on a wide variety +of NLI tasks (adversarial NLI /ANLI)(Nie et al., +2020). Experiments were run using approximately +100 GPU hours on a single V100. +4.3 +Results +Table 2 contains main results for single sentence +classification tasks. Table 3 shows results for vari- +ous sentence pair tasks. We compare our approach +with other few shot learning techniques and experi- +ment with various modifications to the differential +entailment method. +4.4 +Intermediate Training +We experiment with different intermediate training +steps. Table 5 shows results for fine-tuning various +checkpoints. The MNLI and ANLI checkpoints +drastically outperform the roberta-base checkpoint +because they have been adapted to perform well on +entailment tasks. The ANLI model was trained on +multiple augmented entailment tasks(Wang et al., +2021b) and offers a further boost in performance. +These results show that the entailment reformu- +lation relies heavily fine-tuning a model that has + +Task Specific Tokens +(5k parameters) +Task A Class Head +(10m params) +a1 +A +b1 +c1 +Pre-trained Encoder +Task B Class Head +B +a2 +(355m params) +(10m params) +b2 +C +c2 +Task B Class Head +(10m params) +Mixed-Task +BatchInput +Stunning, a dazzling tour de force +Prompts +This is Positive [EOS] +This is Negative [EOS] +Positive: +Negative: +VEntail +Entail +Not Entail +VNot EntailSST-2 +MR +CR +MPQA +Subj +CoLa +Full Training Dataset +Majority +50.9 +50 +50 +50 +50 +69.1 +Finetuning +95 +90.8 +89.4 +89.4 +97 +86.2 (1.6) +EFL +96.9 (0.2) +92.5 (0.1) +92.5 (0.4) +90.8 (0.4) +97.1 (0.2) +86.4 (0.5) +Few Shot k = 16 +Fine Tuning +81.4 (3.8) +76.9( 5.9) +75.8 (3.2) +59.0 (3.4) +90.8 (1.8) +70.0 (0.9) +DARTS +93.5 (0.5) +88.2 (1.0) +91.8 (0.5) +85.6 (0.3) +90.7 (1.4) +- +LMBFF +92.3 (1.0) +85.5 (2.8) +91.0 (0.9) +85.8 (1.9) +91.2 (1.1) +69.5 (0.5) +EFL +90.8 (1.0) +86.2 (0.8) +92.3 (0.4) +87.0 (0.6) +80.0 (5.4) +69.4 (0.9) +DE +91.9 (0.5) +87.1 (2.1) +91.5 (1.4) +87.0 (0.9) +89.5 (2.4) +70.3 (2.4) +DE PE +91.1 (0.2) +84.5 (0.3) +91.6 (0.2) +85.9 (0.6) +81.5(0.1) +69.7 (0.3) +Table 2: Main Results: all results use roBERTa-large as the base architecture, the standard deviation across 5 +training folds is given. Differentiable Entailment (DE) is our method fine-tuning all model parameters. Differen- +tiable Entailment Parameter Efficient (DE PE) is our parameter efficient method which only finetunes the trainable +pseudotokens and classification head. +MRPC +QQP +Full Training Dataset +(f1) +(f1) +Majority +81.2 +0 +Finetuning +89.9 (1.7) +89.0 (0.1) +EFL +91.0 (0.8) +89.2 (0.1) +Few Shot k = 16 +Fine Tuning +76.6 (2.5) +60.7 (4.3) +DARTS +78.3 (4.5) +67.8 (3.2) +LMBFF +76.2 (2.3) +67.0 (3.0) +EFL +76.2 (1.3) +67.3 (2.6) +DE +83.3 (0.1) +72.9 (0.3) +DE PE +78.0 (1.5) +72.6 (0.7) +Table 3: Results for sentence pair tasks. NLI tasks such +as MNLI, QNLI and SNLI are excluded from the com- +parison because these datasets are already incorporated +as part of the intermediate training step for the ANLI +model +Tokens +SST2 +0 +90.5 (0.4) +2 +91.1 (0.7) +5 +91.1 (0.2) +20 +90.6 (0.5) +Table 4: Performance Scaling with number of trainable +pseudotokens. Using a set of 5 trainable pseudotokens +performed best. +Base +MNLI +ANLI +SST-2 +50.1 (0.1) +89.8 (1.3) +91.1 (0.2) +MR +51.1 (0.2) +83.6 (0.4) +84.5 (0.3) +Table 5: Importance of Intermediate Training Steps: +Accuracy is shown for finetuning from the roberta- +large checkpoint, a checkpoint trained on MNLI, and +a checkpoint trained on ANLI, a large number of cu- +rated and synthetic NLI examples. The more robustly +trained NLI checkpoint consistently performs better on +downstream tasks. +already been adapted for entailment. +4.5 +Prompting Schemes +We further experiment with different prompting +schemes. We find best performance when we train +the prompt tokens, the label token and an addi- +tional set of task specific pseudotokens. Table 4 +shows scaling with various numbers of prompting +pseudotokens. Using 5 additional pseudotokens +in addition to trainable prompt and label tokens +worked best. +4.6 +Symmetry +By adding an symmetric entailment example for +binary classification tasks during training we can +effectively provide double the training signal (Fig- +ure 3). However, it appears that it is difficult for the +model to learn from the two complementary train- +ing signals in a few shot scenario. Simply adding +the symmetric examples at training time leads to +a drop in performance (Table 6). These results + +SST-2 +MR +CR +DE PE +91.1 (0.2) +84.5 (0.3) +91.6 (0.2) +DE PE Sym +51.1 (3.1) +48.3 (3.3) +52.2(2.4) +Table 6: Few shot learning results on binary classifi- +cation using symmetric entailment scheme. DE PE is +the regular parameter efficient differential entailment +method. Training with both symmetric signals does not +lead to a robust model. +reveal limitations in the model’s actual understand- +ing of the entailment task. When given only the +template with the positive label the model learns +to associate entailment with the positive class and +not entailment with the negative class. When using +additional symmetric examples, this correlation is +reversed and may be too difficult for a model of +this size and ability to parse. Further work could +explore improving this method or ensembling the +outputs of models trained on symmetric examples. +5 +Analysis and Discussion +Our method achieves competitive performance +with other few shot learning techniques while opti- +mizing 30 times fewer parameters. On most single +sentence tasks performance is within a few points +of methods that train all model parameters. When +we relax the constraints on parameter efficiency +performance is directly competitive with other few +shot learning methods. In some cases we exceed +the performance of methods that rely on optimizing +all model parameters or even additional external +architectures. Notable we achieve much stronger +performance on sentence pair tasks such as MRPC +and QQP. We theorize that this may be because +these sentence pair tasks are most similar to the +entailment tasks seen during intermediate training. +Fundamentally, intermediate training is crucial +for parameter efficient performance because it +gives the model a head start in adapting to the re- +formulated task. We see that using a strong NLI +trained intermediate model improves results (Ta- +ble 5). To adapt to a specific entailment task then +requires only a small number of parameter updates. +6 +Conclusion +In this paper we achieve parameter efficient few- +shot learning by combining 1) entailment refor- +mulation of NLP tasks and 2) trainable prompt +pseudotokens in the continuous space. Our Differ- +entiable Entailment approach achieves competitive +results while only training 3% of the parameters +compared to match. We quantify the impact of in- +termediate training steps and different prompting +schemes. By adopting a strict definition of a param- +eter efficiency we achieve few-shot performance +with fewer trainable parameters, no external param- +eters and without scaling up model size or using +unlabeled training data. One major limitation is +that we have to train a separate classification head +for each downstream task, limiting potential gains +in parameter efficiency. Further work could explore +different intermediate training tasks, ensembling +sets of prompts tokens and combining cloze com- +pletion for classification with the entailment refor- +mulation. Given that our method is model agnostic +and efficient it is likely to be broadly applicable to +additional tasks. +7 +Broader Impact +Parameter efficient models, especially with the +method described in this paper have the poten- +tial to allow use of machine learning models on +a more widespread basis. In our approach, batch- +ing computations for different tasks and using a +single forward pass through a model could allow +many models to be run on a single device at a single +team. Such a scheme has advantages in terms of +providing more accessibility to machine learning +models and reduced energy consumption. How- +ever, parameter efficiency also opens that door to +running personalized models that may be injurious +to individual security or privacy. For example, user +specific embeddings could easily be trained to pre- +dict a user’s behavior with a specialized model. We +anticipate that such potential use cases of param- +eter efficient few shot learning should be treated +carefully. +References +Jonathan Bragg, Arman Cohan, Kyle Lo, and Iz Belt- +agy. 2021. +FLEX: Unifying Evaluation for Few- +Shot NLP. +arXiv:2107.07170 [cs]. +ArXiv: +2107.07170. +Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie +Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind +Neelakantan, Pranav Shyam, Girish Sastry, Amanda +Askell, +Sandhini Agarwal, +Ariel Herbert-Voss, +Gretchen Krueger, Tom Henighan, Rewon Child, +Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, +Clemens Winter, Christopher Hesse, Mark Chen, +Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin + +Chess, Jack Clark, Christopher Berner, Sam Mc- +Candlish, Alec Radford, Ilya Sutskever, and Dario +Amodei. 2020. Language models are few-shot learn- +ers. CoRR, abs/2005.14165. +Ning Ding, Yujia Qin, Guang Yang, Fuchao Wei, Zong- +han Yang, Yusheng Su, Shengding Hu, Yulin Chen, +Chi-Min Chan, Weize Chen, et al. 2022. Delta tun- +ing: A comprehensive study of parameter efficient +methods for pre-trained language models. +arXiv +preprint arXiv:2203.06904. +Tianyu Gao, Adam Fisch, and Danqi Chen. 2021. +Making Pre-trained Language Models Better Few- +shot Learners. +arXiv:2012.15723 [cs]. +ArXiv: +2012.15723. +Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, +Bruna Morrone, Quentin de Laroussilhe, Andrea +Gesmundo, Mona Attariyan, and Sylvain Gelly. +2019. Parameter-efficient transfer learning for nlp. +Jeremy Howard and Sebastian Ruder. 2018. Universal +Language Model Fine-tuning for Text Classification. +In Proceedings of the 56th Annual Meeting of the +Association for Computational Linguistics (Volume +1: Long Papers), pages 328–339, Melbourne, Aus- +tralia. Association for Computational Linguistics. +Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan +Allen-Zhu, Yuanzhi Li, Shean Wang, and Weizhu +Chen. 2021. Lora: Low-rank adaptation of large lan- +guage models. CoRR, abs/2106.09685. +Teven Le Scao and Alexander Rush. 2021. How many +data points is a prompt worth? In Proceedings of the +2021 Conference of the North American Chapter of +the Association for Computational Linguistics: Hu- +man Language Technologies, pages 2627–2636, On- +line. Association for Computational Linguistics. +Brian Lester, Rami Al-Rfou, and Noah Constant. +2021. The Power of Scale for Parameter-Efficient +Prompt Tuning. +arXiv:2104.08691 [cs]. +ArXiv: +2104.08691. +Xiang Lisa Li and Percy Liang. 2021. Prefix-Tuning: +Optimizing Continuous Prompts for Generation. +arXiv:2101.00190 [cs]. ArXiv: 2101.00190. +Xiao Liu, Yanan Zheng, Zhengxiao Du, Ming Ding, +Yujie Qian, Zhilin Yang, and Jie Tang. 2021. GPT +Understands, Too. arXiv:2103.10385 [cs]. ArXiv: +2103.10385. +Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- +dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, +Luke Zettlemoyer, and Veselin Stoyanov. 2019. +Roberta: A robustly optimized BERT pretraining ap- +proach. CoRR, abs/1907.11692. +Yixin Nie, Adina Williams, Emily Dinan, Mohit +Bansal, Jason Weston, and Douwe Kiela. 2020. Ad- +versarial NLI: A new benchmark for natural lan- +guage understanding. In Proceedings of the 58th An- +nual Meeting of the Association for Computational +Linguistics. Association for Computational Linguis- +tics. +Adam Paszke, Sam Gross, Francisco Massa, Adam +Lerer, James Bradbury, Gregory Chanan, Trevor +Killeen, Zeming Lin, Natalia Gimelshein, Luca +Antiga, Alban Desmaison, Andreas Köpf, Edward Z. +Yang, Zach DeVito, Martin Raison, Alykhan Tejani, +Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Jun- +jie Bai, and Soumith Chintala. 2019. Pytorch: An +imperative style, high-performance deep learning li- +brary. CoRR, abs/1912.01703. +Timo Schick and Hinrich Schütze. 2021. Exploiting +cloze questions for few shot text classification and +natural language inference. +Taylor Shin, Yasaman Razeghi, Robert L. Logan IV, +Eric Wallace, and Sameer Singh. 2020. +Auto- +prompt: Eliciting knowledge from language mod- +els with automatically generated prompts. +CoRR, +abs/2010.15980. +Sinong Wang, Han Fang, Madian Khabsa, Hanzi +Mao, and Hao Ma. 2021a. +Entailment as Few- +Shot Learner. +arXiv:2104.14690 [cs]. +ArXiv: +2104.14690. +Yaqing Wang, Subhabrata Mukherjee, Xiaodong Liu, +Jing Gao, Ahmed Hassan Awadallah, and Jianfeng +Gao. 2021b. List: Lite self-training makes efficient +few-shot learners. +Thomas Wolf, Lysandre Debut, Victor Sanh, Julien +Chaumond, Clement Delangue, Anthony Moi, Pier- +ric Cistac, Tim Rault, Rémi Louf, Morgan Funtow- +icz, and Jamie Brew. 2019. +Huggingface’s trans- +formers: State-of-the-art natural language process- +ing. CoRR, abs/1910.03771. +Ningyu Zhang, Luoqiu Li, Xiang Chen, Shumin Deng, +Zhen Bi, Chuanqi Tan, Fei Huang, and Huajun +Chen. 2021. +Differentiable Prompt Makes Pre- +trained Language Models Better Few-shot Learners. +arXiv:2108.13161 [cs]. ArXiv: 2108.13161. +A +Hyperparameters +The hyperparameter search space used for all ex- +periments is as follows: +• learning rate [1e-5, 3e-5, 1e-4] +• weight decay [0.0, 0.05, 0.1] +• batch size [8, 16] +• gradient accumulation steps [1, 2] +B +Prompting Templates +The standard prompting templates from (Wang +et al., 2021a) are used for each task. +• SST-2: sentence1[SEP]It was great + +• MR: sentence1[SEP]It was great +• CR: sentence1[SEP]It was great +• MPQA: sentence1[SEP]It was positive +• Subj: sentence1[SEP]It was objective +• CoLA: sentence1[SEP]It was correct +• MRPC: sentence1[SEP]sentence2 +• QQP: sentence1[SEP]sentence2 + diff --git a/bdFQT4oBgHgl3EQfhTaZ/content/tmp_files/load_file.txt b/bdFQT4oBgHgl3EQfhTaZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea0fcb131bb12c8fc30fd056626d3a5b68fa2039 --- /dev/null +++ b/bdFQT4oBgHgl3EQfhTaZ/content/tmp_files/load_file.txt @@ -0,0 +1,544 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf,len=543 +page_content='Differentiable Entailment for Parameter Efficient Few Shot Learning Ethan Kim Harvard University Jerry Yang Harvard University Abstract Few-shot learning allows pre-trained language models to adapt to downstream tasks while using a limited number of training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' However, practical applications are limited when all model parameters must be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In this work we apply a new technique for pa- rameter efficient few shot learning while adopt- ing a strict definition of parameter efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Our training method combines 1) intermedi- ate training by reformulating natural language tasks as entailment tasks (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a) and 2) differentiable optimization of template and label tokens (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We quantify the tradeoff between parameter effi- ciency and performance in the few shot regime and propose a simple model agnostic approach that can be extended to any task By achieving competitive performance while only optimiz- ing 3% of a model’s parameters and allowing for batched inference, we allow for more effi- cient practical deployment of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 1 Introduction Large pre-trained language models have demon- strated adaptability to solve natural language pro- cessing (NLP) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Typically, such language models are adapted to a downstream task through fine-tuning (Howard and Ruder, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Although fine-tuning improves performance on downstream tasks, it is costly because it relies on updating ev- ery parameter of the model (355 million in the case of roBERTa)and requires storing a separate copy of the model for every downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' These storage requirements can become prohibitive, thus necessitating research into more parameter effi- cient methods .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Alternative fine-tuning methods that update fewer parameters can have other trade- offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For example, adapter tuning fine-tunes a small number adapter parameters inserted between the transformer layers (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2019)but requires optimizing external parameters and still fine-tunes on the entire training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Other methods have explored fine-tuning in the few shot learning case, where a limited number of labeled training samples are used for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' These approaches have the disadvantages of relying on ex- treme model size (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2020) (Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021), optimizing all model parameters (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a),(Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021), or using ex- ternal architectures (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2019) (Li and Liang, 2021) (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In this project, we present a simple, extensible method that improves few-shot performance without any extra parame- ters by combining two approaches: 1) leveraging trainable prompt pseudotokens rather than updat- ing all the model parameters (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021), and 2) reformulating natural language processing tasks as entailment tasks and applying an interme- diate training step, enabling better generalization to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Our major contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Our method achieves competitive few shot performance while optimizing only 3% of a model’s parameters reducing storage costs by a factor of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We introduce a strict definition of parameter efficiency which extends the practical uses of few shot learning by allowing batching of computation across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 Finetuning The standard method for fine-tuning Masked Lan- guage Models (MLMs) like BERT applies a clas- sification head to the [CLS] token representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The language model learns to update the [CLS] rep- resentation to better solve the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' A number of reformulations have been proposed seek- ing to increase performance and improve parameter efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='13345v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='CL] 31 Jan 2023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 Prompting Language models learn a general set of abilities that can be adapted to specific downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' One method is to use task-specific natural language prompts to guide the language model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' GPT- 3, for example, uses prompts and in-context exam- ples to achieve good few-shot performance on var- ious tasks (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' GPT-3 leverages extreme scale (175 Billion parameters) to adapt to natural language prompts without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Prompting can be particularly useful for few-shot learning in the low-data regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For some tasks, a well designed prompt can be shown to be equiva- lent to hundreds or thousands of additional labeled training points (Le Scao and Rush, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' AUTO- PROMPT uses a gradient-based search to optimize a discrete prompt (Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' LMBFF uses an auxiliary language model to generate a set of candidate prompts and chooses the best candidate (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 Pattern Exploiting Training One alternative to standard fine-tuning is to model the output as a cloze completion task where the output is the model’s representation of a masked input token (Schick and Schütze, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Intuitively, this approach works well because it more closely matches the training process for MLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In the pre-training task for models such as BERT and roBERTA, the model is asked to predict the identity of a masked token based on the hidden representa- tions of neighboring tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Further work has extended this approach to use natural language prompts to guide the cloze output (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Additional work has focused on training the prompt tokens in continuous space by optimizing a set of prompt pseudotokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' (Li and Liang, 2021) (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021) (Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Additionally in the DART method, the tokens used as classification labels can be optimized (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 Entailment Reformulation Work from (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a) focuses on improv- ing language model performance by formulating NLP tasks as entailment tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Fundamentally, en- tailment seeks to determine whether for a pair of inputs (S1, S2), the first sentence entails or contra- dicts the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Most standard classification tasks in NLP can be reformulated as entailment tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For example, a sentiment analysis task can be be framed as an entailment task using the following template: [CLS]S1[SEP]S2[EOS], (1) With S2 = "It was great" as the entailment prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Instead of using the [CLS] token representation of S1 to classify the review as positive or negative as in standard fine-tuning, we instead concatenate the text with the prompt and use the [CLS] token rep- resentation of the concatenated sequence to denote whether the first sentence entails the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For multi-class classification problems we con- struct a different input for every class and take the label as the class with the highest entailment score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' A key to the success of the entailment approach from (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a) is an intermediate train- ing step where the pre-trained language model is fine-tuned on a natural language inference (NLI) task like MNLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Intuitively, the model can be adapted to be good at one entailment task and then generalize to perform well on other reformulated entailment tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 Parameter Efficiency Related works adopt various, sometimes contra- dictory, definitions of parameter efficiency when applied to language model fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Broadly, these definitions can be grouped into several cate- gories: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' reducing the number of model parameters nec- essary to achieve good few shot adaptability 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' optimizing a small subset of the total model parameters 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' avoiding external parameters or changes to the model architecture Some works on few-shot learning explore tech- niques allowing smaller models to learn robustly (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Large models such as GPT3 with 175 billion parameters can take advantage of their scale to perform well at few-shot in-context learning (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' A technique can be parameter efficient if it allows similar results be be achieved with a smaller language model e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' a 340 million parameter roBERTa model rather than the 175 billion parameter GPT-3 or 11 billion parame- ter T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Parameter efficiency can also aim to optimize a smaller number of task specific parameters while keeping most of the language model parameters Figure 1: Differential Entailment Approach frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Adapter tuning inserts trainable layers be- tween the frozen layers of a Transformer language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Prompt tuning op- timizes a small set of trainable input tokens while keeping the pre-trained Transformer layers frozen (Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Lite Self Training (LiST) freezes most of the encoder parameters and only trains a small number of adapter parameters (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' LoRA, adds low rank trainable ma- trices between transformer layers (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021) while freezing the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Other works define parameter efficiency as the lack of a need for parameters external to the model being fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Part of the motivation for differ- ential prompt tuning (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021) is that it directly optimizes trainable pseudotokens without the need for an external model such as LSTM in P-tuning (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Such approaches are advantageous as they require no modifications to a pre-trained model’s architecture and do not add additional inference time like adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Delta Tun- ing explores in depth the performance of different parameter efficient approaches at different model scales and in combination with one another (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2022) We focus on parameter efficiency in the true few shot learning regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Therefore, we do not take advantage of any additional unlabeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Iterative PET used this approach to pseudolabel unlabelled samples and provide extra training ex- amples to a model (Schick and Schütze, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' LiST iteratively trains a student model on data pseudolabeled by a teacher model (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' However, these semi-supervised learning approaches require extra unlabeled training data as well as additional training computation compared to true few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 3 Approach Our main approach is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We con- vert all NLP tasks to the entailment format and train few shot models from an intermediate train- ing checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The entailment approach outlined in (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a) performs traditional fine- tuning and updates all model parameters via gradi- ent descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Instead of performing the computation- ally expensive update step on all model parameters, our approach fine-tunes only the prompt and label tokens in continuous space while keeping the main language model frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' By using more expressive pseudotokens as part of our prompt and by training only the input parameters, we achieve a parameter efficient few shot learning method with competitive few-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 Pseudotokens With discrete tokens, the label template tokens are either chosen manually or determined through a search over tokens in a discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In compar- ison, our label descriptions are optimized in con- tinuous space via back-propagation and hence can attain more expressive, fine-grained representations to prompt a model for a certain task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Formally, we define a set of pseudotokens T /∈ V outside of the normal vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The pseudotoken embedding h(T ) is a trainable set of parameters that are opti- mized via backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For a given input we might have the following prompt: S1[SEP] T0T1T2 it was [LABEL] We differentiably optimize prompting pseudoto- kens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We also experiment with allowing the label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='[CLS] The drama discloses nothing [SEP] [T1][T2][T2] P(it) P(was) V(positive) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='[CLS] e(The) e(drama) e(discloses) e(nothing) e([SEP) h([T1])h([T2])h([T3]) h(P(it)) h(P(was)) h(V(positive)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Pre-trained Language Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='(roBERTa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Hidden States ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='[CLS] The drama discloses nothing [SEP] [T1][T2][T2] P(it) P(was) V(positive) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Label: Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='CLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Predict ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Entail: Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='Not Entail: Negativeembedding h([LABEL] to be a pseudotoken with a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='trainable embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For label and prompt tokens we experiment with both initializing these pseudo- tokens embeddings randomly and initializing them with the embeddings of the original tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 Parameter Efficiency We adopt the strictest definition of parameter ef- ficiency that has practical advantages for down- stream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In Differentiable Entailment we 1) use a smaller language model compared to GPT-3 or T5, 2) freeze the main encoder parame- ters, 3) only fine-tune a limited set of pseudotokens without any external parameters or architectural modifications and 4) employ strict few-shot learn- ing without using any additional training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Following the method in Prompt Tuning, we freeze the main model parameters and only fine- tune the subset of trainable input tokens (Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In contrast to Prompt Tuning we also fine tune the model classification head since we are outputting a specific classification label rather than using a generative model such as T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' By freezing the model parameters we can effi- ciently optimize a smaller set of task-specific pa- rameters, namely the pseudotoken embeddings as well as the entailment classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In con- trast to approaches outlined above, which rely on a large-scale model to make up for a reduction in trainable parameters (Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021), we use a smaller language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' With roBERTa-large this leads to a more than 30x reduction in the number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Furthermore, instead of stor- ing a fine-tuned 355 million parameter model for each task, we only need to store the task-specific trainable pseudotoken embeddings and classifica- tion head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Finally, in contrast to methods which finetune all the model parameters (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a) or methods with exter- nal parameters (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2019) our method allows the hidden state computation for different tasks to be batched together since only the spe- cific prompt embeddings for each tasks need to be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' As others have noted: such in batch par- allel computing has extreme practical application (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' LoRA also allows for multitask batching, however applying additional low rank matrices to later transformer layers is more com- plex than simply swapping out a set of task specific input embeddings (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Method Template Cloze S_1 [SEP] it was [MASK] Entailment S_1 [SEP] it was great Differential Prompt S_1 [SEP][Prompt tokens] great Differential Label and Prompt S_1 [SEP][Prompt tokens] [Label token] Table 1: Example Prompting Templates for a Sentiment Classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For our method we optimize either a set of prompt pseudotokens and/or a label pseudoto- ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 Templates We explore several different approaches to combin- ing label templates with pseudotokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For various tasks, we adapt the standard prompt templates used in previous works (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For example, sentiment analysis tasks such as CR can be prompted for both entailment and cloze completion in a simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In 1, we show label templates for a sentiment analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For such tasks, the prompt standard template is "it was great".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The cloze completion method concatenates the prompt to the input sentence and masks out the label "great", whereas our method concatenates the template without masking the token of inter- est and predicts entailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' When training label templates in the continuous space, we initialize from the embeddings of the label template tokens in the standard template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For example, given the following template: S1[SEP]T0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Tjit was great We would train the prompt tokens "it", "was", the label token "great" and j + 1 additional pseudoto- kens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For sentence pair tasks such as Quora Question Pairs (QQP), we adopt a slightly different template following (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a)(Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The task is to predict entailment based on the sen- tence pairs and a set of prompt pseudotokens in- serted between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For QQP we use the format S1[SEP]T0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' TjS2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 Symmetry of Entailment In (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a), a single label description p is used for each example in a binary classification task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' a binary sentiment classification task is formulated as whether input sentence S1 entails S2 = "This indicates positive sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' To en- courage more robust tuning of the label description parameters and classification head,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' we experiment Figure 2: Entailment allows batching of hidden state computations across tasks Figure 3: Symmetry for simple data augmentation with using two label descriptions p1 and p−1 for binary classification tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' and augment the dataset as: Dtrain = {(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' yi) ∪ (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' p−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' −yi)}K i=1 (2) For a positive sentiment example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' the two cor- responding samples in the training dataset would be (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 1) and (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' p−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' −yi) where p1 = This indicates positive sentiment with label 1 (does entail) and p−1 = This indicates negative sentiment with label 0 (does not entail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 Evaluation We evaluate our method on the tasks from (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a) which are mainly the subset of the GLUE and SuperGLUE benchmark tasks that are compatible with the entailment reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In addition, we follow the best practices for evalua- tion of few shot NLP fine-tuning methods (Bragg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For each experiment we sample 5 non-overlapping training folds and report average performance after k-shot training over the entire test set (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Hyperparameters are tuned for each task and method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 Implementation Details Models are implemented using the pytorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2019) and transformers (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2019) li- braries with code adapted from (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Our pre-trained model is roBERTa large (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Checkpoints for roberta-large-base as well as checkpoint models are downloaded from hug- gingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We experiment with different interme- diate checkpoints, namely roberta-large-mlni and a checkpoint trained robustly on a wide variety of NLI tasks (adversarial NLI /ANLI)(Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Experiments were run using approximately 100 GPU hours on a single V100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 Results Table 2 contains main results for single sentence classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Table 3 shows results for vari- ous sentence pair tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We compare our approach with other few shot learning techniques and experi- ment with various modifications to the differential entailment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 Intermediate Training We experiment with different intermediate training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Table 5 shows results for fine-tuning various checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The MNLI and ANLI checkpoints drastically outperform the roberta-base checkpoint because they have been adapted to perform well on entailment tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The ANLI model was trained on multiple augmented entailment tasks(Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021b) and offers a further boost in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' These results show that the entailment reformu- lation relies heavily fine-tuning a model that has Task Specific Tokens (5k parameters) Task A Class Head (10m params) a1 A b1 c1 Pre-trained Encoder Task B Class Head B a2 (355m params) (10m params) b2 C c2 Task B Class Head (10m params) Mixed-Task BatchInput Stunning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' a dazzling tour de force Prompts This is Positive [EOS] This is Negative [EOS] Positive: Negative: VEntail Entail Not Entail VNot EntailSST-2 MR CR MPQA Subj CoLa Full Training Dataset Majority 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9 50 50 50 50 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 Finetuning 95 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 97 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6) EFL 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) Few Shot k = 16 Fine Tuning 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9( 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9) DARTS 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='7 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) LMBFF 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) EFL 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9) DE 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) DE PE 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) Table 2: Main Results: all results use roBERTa-large as the base architecture, the standard deviation across 5 training folds is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Differentiable Entailment (DE) is our method fine-tuning all model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Differen- tiable Entailment Parameter Efficient (DE PE) is our parameter efficient method which only finetunes the trainable pseudotokens and classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' MRPC QQP Full Training Dataset (f1) (f1) Majority 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 0 Finetuning 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='7) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) EFL 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) Few Shot k = 16 Fine Tuning 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='7 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) DARTS 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) LMBFF 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0) EFL 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6) DE 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) DE PE 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='7) Table 3: Results for sentence pair tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' NLI tasks such as MNLI, QNLI and SNLI are excluded from the com- parison because these datasets are already incorporated as part of the intermediate training step for the ANLI model Tokens SST2 0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='7) 5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 20 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5) Table 4: Performance Scaling with number of trainable pseudotokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Using a set of 5 trainable pseudotokens performed best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Base MNLI ANLI SST-2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) MR 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) Table 5: Importance of Intermediate Training Steps: Accuracy is shown for finetuning from the roberta- large checkpoint, a checkpoint trained on MNLI, and a checkpoint trained on ANLI, a large number of cu- rated and synthetic NLI examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' The more robustly trained NLI checkpoint consistently performs better on downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' already been adapted for entailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 Prompting Schemes We further experiment with different prompting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We find best performance when we train the prompt tokens, the label token and an addi- tional set of task specific pseudotokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Table 4 shows scaling with various numbers of prompting pseudotokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Using 5 additional pseudotokens in addition to trainable prompt and label tokens worked best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 Symmetry By adding an symmetric entailment example for binary classification tasks during training we can effectively provide double the training signal (Fig- ure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' However, it appears that it is difficult for the model to learn from the two complementary train- ing signals in a few shot scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Simply adding the symmetric examples at training time leads to a drop in performance (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' These results SST-2 MR CR DE PE 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2) DE PE Sym 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='3) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='2(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='4) Table 6: Few shot learning results on binary classifi- cation using symmetric entailment scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' DE PE is the regular parameter efficient differential entailment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Training with both symmetric signals does not lead to a robust model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' reveal limitations in the model’s actual understand- ing of the entailment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' When given only the template with the positive label the model learns to associate entailment with the positive class and not entailment with the negative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' When using additional symmetric examples, this correlation is reversed and may be too difficult for a model of this size and ability to parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Further work could explore improving this method or ensembling the outputs of models trained on symmetric examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 5 Analysis and Discussion Our method achieves competitive performance with other few shot learning techniques while opti- mizing 30 times fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' On most single sentence tasks performance is within a few points of methods that train all model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' When we relax the constraints on parameter efficiency performance is directly competitive with other few shot learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In some cases we exceed the performance of methods that rely on optimizing all model parameters or even additional external architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Notable we achieve much stronger performance on sentence pair tasks such as MRPC and QQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We theorize that this may be because these sentence pair tasks are most similar to the entailment tasks seen during intermediate training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Fundamentally, intermediate training is crucial for parameter efficient performance because it gives the model a head start in adapting to the re- formulated task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We see that using a strong NLI trained intermediate model improves results (Ta- ble 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' To adapt to a specific entailment task then requires only a small number of parameter updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 6 Conclusion In this paper we achieve parameter efficient few- shot learning by combining 1) entailment refor- mulation of NLP tasks and 2) trainable prompt pseudotokens in the continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Our Differ- entiable Entailment approach achieves competitive results while only training 3% of the parameters compared to match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' We quantify the impact of in- termediate training steps and different prompting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' By adopting a strict definition of a param- eter efficiency we achieve few-shot performance with fewer trainable parameters, no external param- eters and without scaling up model size or using unlabeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' One major limitation is that we have to train a separate classification head for each downstream task, limiting potential gains in parameter efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Further work could explore different intermediate training tasks, ensembling sets of prompts tokens and combining cloze com- pletion for classification with the entailment refor- mulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Given that our method is model agnostic and efficient it is likely to be broadly applicable to additional tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 7 Broader Impact Parameter efficient models, especially with the method described in this paper have the poten- tial to allow use of machine learning models on a more widespread basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' In our approach, batch- ing computations for different tasks and using a single forward pass through a model could allow many models to be run on a single device at a single team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Such a scheme has advantages in terms of providing more accessibility to machine learning models and reduced energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' How- ever, parameter efficiency also opens that door to running personalized models that may be injurious to individual security or privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' For example, 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inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Taylor Shin, Yasaman Razeghi, Robert L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Logan IV, Eric Wallace, and Sameer Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Auto- prompt: Eliciting knowledge from language mod- els with automatically generated prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' CoRR, abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Huggingface’s trans- formers: State-of-the-art natural language process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' CoRR, abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='03771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Ningyu Zhang, Luoqiu Li, Xiang Chen, Shumin Deng, Zhen Bi, Chuanqi Tan, Fei Huang, and Huajun Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' Differentiable Prompt Makes Pre- trained Language Models Better Few-shot Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='13161 [cs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' ArXiv: 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='13161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' A Hyperparameters The hyperparameter search space used for all ex- periments is as follows: learning rate [1e-5, 3e-5, 1e-4] weight decay [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content='1] batch size [8, 16] gradient accumulation steps [1, 2] B Prompting Templates The standard prompting templates from (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=', 2021a) are used for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} +page_content=' SST-2: sentence1[SEP]It was great MR: sentence1[SEP]It was great CR: sentence1[SEP]It was great MPQA: sentence1[SEP]It was positive Subj: sentence1[SEP]It was objective CoLA: sentence1[SEP]It was correct MRPC: sentence1[SEP]sentence2 QQP: sentence1[SEP]sentence2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFQT4oBgHgl3EQfhTaZ/content/2301.13345v1.pdf'} diff --git a/cNAzT4oBgHgl3EQfnf3Z/vector_store/index.pkl b/cNAzT4oBgHgl3EQfnf3Z/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b7134299552a24519c6e8f8011743320f82343ef --- /dev/null +++ b/cNAzT4oBgHgl3EQfnf3Z/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8871e57eac995e2330d40c9e76bffcdb442a0ec428df4b47f9b888d4ec96ae54 +size 427690 diff --git a/d9FJT4oBgHgl3EQfSixW/content/tmp_files/2301.11500v1.pdf.txt b/d9FJT4oBgHgl3EQfSixW/content/tmp_files/2301.11500v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e9ca3d92e42647cd0565388ee43cb688541a66d --- /dev/null +++ b/d9FJT4oBgHgl3EQfSixW/content/tmp_files/2301.11500v1.pdf.txt @@ -0,0 +1,6115 @@ +Understanding Incremental Learning of Gradient Descent: +A Fine-grained Analysis of Matrix Sensing +Jikai Jin * +Zhiyuan Li† Kaifeng Lyu‡ Simon S. Du§ Jason D. Lee¶ +January 30, 2023 +Abstract +It is believed that Gradient Descent (GD) induces an implicit bias towards good gener- +alization in training machine learning models. This paper provides a fine-grained analysis +of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank +ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small +initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and +follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns so- +lutions with increasing ranks until it recovers the ground truth matrix. Compared to existing +works which only analyze the first learning phase for rank-1 solutions, our result provides +characterizations for the whole learning process. Moreover, besides the over-parameterized +regime that many prior works focused on, our analysis of the incremental learning procedure +also applies to the under-parameterized regime. Finally, we conduct numerical experiments +to confirm our theoretical findings. +1 +Introduction +Understanding the optimization and generalization properties of optimization algorithms is one +of the central topics in deep learning theory (Zhang et al., 2017; Sun, 2019). It has long been a +mystery why simple algorithms such as Gradient Descent (GD) or Stochastic Gradient Descent +(SGD) can find global minima even for highly non-convex functions (Du et al., 2019), and why +the global minima being found can generalize well (Hardt et al., 2016). +One influential line of works provides theoretical analysis of the implicit bias of GD/SGD. +These results typically exhibit theoretical settings where the low-loss solutions found by GD/SGD +attain certain optimality conditions of a particular generalization metric, e.g., the parameter norm +(or the classifier margin) (Soudry et al., 2018; Gunasekar et al., 2018; Nacson et al., 2019; Lyu +& Li, 2020; Ji & Telgarsky, 2020), the sharpness of local loss landscape (Blanc et al., 2020; +Damian et al., 2021; Li et al., 2022a; Lyu et al., 2022). +*Peking University. Email: jkjin@pku.edu.cn +†Stanford University. Email: zhiyuanli@stanford.edu +‡Princeton University. Email: klyu@cs.princeton.edu +§University of Washington. Email: ssdu@cs.washington.edu +¶Princeton University. Email: jasonlee@princeton.edu +1 +arXiv:2301.11500v1 [cs.LG] 27 Jan 2023 + +Among these works, a line of works seek to characterize the implicit bias even when the train- +ing is away from convergence. Kalimeris et al. (2019) empirically observed that SGD learns +model from simple ones, such as linear classifiers, to more complex ones. As a result, SGD +always tries to fit the training data with minimal model complexity. This behavior, usually re- +ferred to as the simplicity bias or the incremental learning behavior of GD/SGD, can be a hidden +mechanism of deep learning that prevents highly over-parameterized models from overfitting. In +theory, Hu et al. (2020); Lyu et al. (2021); Frei et al. (2021) established that GD on two-layer +nets learns linear classifiers first. +The goal of this paper is to demonstrate this simplicity bias/incremental learning in the matrix +sensing problem, a non-convex optimization problem that arises in a wide range of real-world +applications, e.g., image reconstruction (Zhao et al., 2010; Peng et al., 2014), object detection +(Shen & Wu, 2012; Zou et al., 2013) and array processing systems (Kalogerias & Petropulu, +2013). Moreover, this problem can serve as a standard test-bed of the implicit bias of GD/SGD +in deep learning theory, since it retains many of the key phenomena in deep learning while being +simpler to analyze. +Formally, the matrix sensing problem asks for recovering a ground-truth matrix Z∗ ∈ Rd×d +given m observations y1, . . . , ym. Each observation yi here is resulted from a linear measurement +yi = ⟨Ai, Z∗⟩, where {Ai}1≤i≤m is a collection of symmetric measurement matrices. In this +paper, we focus on the case where Z∗ is symmetric, positive semi-definite (PSD) and low-rank, +i.e., Z∗ ⪰ 0 and rank(Z∗) = r∗ ≪ d. +An intriguing approach to solve this problem is to use the Burer-Monteiro type decomposition +Z∗ = UU ⊤ with U ∈ Rd׈r, and minimize the squared loss with GD: +min +U∈Rd׈r +f(U) := +1 +4m +m +� +i=1 +� +yi − +� +Ai, UU ⊤��2 +. +(1) +In the ideal case, the number of columns of U, denoted as ˆr above, should be set to r∗. However, +r∗ may not be known in advance. This leads to two training regimes that are more likely to +happen: the under-parameterized regime where ˆr ≤ r∗, and the over-parameterized regime +where ˆr > r∗. +The over-parameterized regime may lead to overfitting at first glance, but surprisingly, with +small initialization, GD induces a good implicit bias towards solutions with the exact or ap- +proximate recovery of the ground truth. It was first conjectured in Gunasekar et al. (2017) that +GD with small initialization finds the matrix with the minimum nuclear norm. Gunasekar et al. +(2017) also proved this conjecture for a special case where all measurements are commutable. +However, a series of works point out that this nuclear norm minimization view cannot capture +the incremental learning behavior of GD, which, in the context of matrix sensing, refers to the +phenomenon that GD tends to learn solutions with rank gradually increasing with training steps. +Arora et al. (2019) exhibited this phenomenon when there is only one observation (m = 1). +Gissin et al. (2019); Jiang et al. (2022) studied the full-observation case, where every entry of +the ground truth is measured independently f(U) = +1 +4d2 ∥Z∗ − UU ⊤∥2 +F, and GD is shown to +sequentially recover singular components of the ground truth from the largest singular value to +the smallest one. Li et al. (2020) provided theoretical evidence that the incremental learning +2 + +behavior generally occurs for matrix sensing. They specifically provided a counterexample for +Gunasekar et al. (2017)’s conjecture, where GD converges to a rank-1 solution with a very large +nuclear norm. Razin & Cohen (2020) also pointed out a case where GD drives the norm to +infinity while keeping the rank to be approximately 1. +Despite these progresses, theoretical understanding of the simplicity bias of GD remains lim- +ited. In fact, a vast majority of existing analyses can only show that GD is initially biased towards +a rank-1 solution and cannot be generalized to higher ranks, unless additional assumptions on +GD dynamics are made (Li et al., 2020, Appendix H), (Belabbas, 2020; Jacot et al., 2021; Razin +et al., 2021, 2022). Recently Li et al. (2022b) shows that the implicit bias of Gunasekar et al. +(2017) essentially relies on rewriting gradient flow in the space of U as continuous mirror de- +scent in the space of UU ⊤, which only works a special type of reparametrized model, named +“commuting parametrization”. However, Li et al. (2022b) also shows that matrix sensing with +general (non-commutable) measurements does not fall into this type. +1.1 +Our Contributions +In this paper, we take a step towards understanding the generalization of GD with small initial- +ization by firmly demonstrating the simplicity bias/incremental learning behavior in the matrix +sensing setting, assuming the Restricted Isometry Property (RIP). Our main result is informally +stated below. See Theorem 4.1 for the formal version. +Definition 1.1 (Best Rank-s Solution). We define the best rank-s solution as the unique global +minimizer Z∗ +s of the following constrained optimization problem: +min +Z∈Rd×d +1 +4m +m +� +i=1 +(yi − ⟨Ai, Z⟩)2 +s.t. +Z ⪰ 0, +rank(Z) ≤ s. +(2) +Theorem 1.1 (Informal version of Theorem 4.1). Consider the matrix sensing problem (1) with +rank-r∗ ground-truth matrix Z∗ and measurements {Ai}m +i=1. Assume that the measurements +satisfy the RIP condition (Definition 3.2). With small learning rate µ > 0 and small initial- +ization Uα,0 = αU ∈ Rd׈r, the trajectory of Uα,tU ⊤ +α,t during GD training enters an o(1)- +neighborhood of each of the best rank-s solutions in the order of s = 1, 2, . . . , ˆr ∧ r∗ when +α → 0. Moreover, when ˆr ≤ r∗, we have limt→∞ Uα,tU ⊤ +α,t = Z∗ +ˆr . +It is shown in Li et al. (2018); St¨oger & Soltanolkotabi (2021) that GD exactly recovers the +ground truth under the RIP condition, but our theorem goes beyond them in a number of ways. +First, in the over-parameterized regime (i.e., ˆr > r∗), it implies that GD performs incremental +learning: learning solutions with increasing ranks until it finds the ground truth. Second, this +result also shows that in the under-parameterized regime (i.e., ˆr ≤ r∗), GD exhibits the same +implicit bias, but finally it converges to the best low-rank solution of the matrix sensing loss. By +contrast, to the best of our knowledge, only the over-parameterized setting is analyzed in existing +literature. +Theorem 1.1 can also be considered as a generalization of previous results in Gissin et al. +(2019); Jiang et al. (2022) which show that Uα,tU ⊤ +α,t passes by the best low-rank solutions one +3 + +by one in the full observation case of matrix sensing f(U) = +1 +4d2 ∥Z∗ − UU ⊤∥2 +F. However, +our setting has two major challenges which significantly complicate our analysis. First, since +our setting only gives partial measurements, the decomposition of signal and error terms in +Gissin et al. (2019); Jiang et al. (2022) cannot be applied. Instead, we adopt a different approach +which is motivated by St¨oger & Soltanolkotabi (2021). Second, it is well-known that the optimal +rank-s solution of matrix factorization is Xs (defined in Section 3), but little is known for Z∗ +s. +In Section 4.1 we analyze the landscape of (2), establishing the uniqueness of Z∗ +s and local +landscape properties under the RIP condition. We find that when Uα,tU ⊤ +α,t ≈ Z∗ +s, GD follows +an approximate low-rank trajectory, so that it behaves similarly to GD in the under-parameterized +regime. Using our landscape results, we can finally prove Theorem 1.1. +Organization. We review additional related works in Section 2. In Section 3, we provide an +overview of necessary background and notations. We then present our main results in Section 4 +with proof sketch where we also state some key lemmas that are used in the proof, including +Lemma 4.1 and some landscape results. In Section 5 we present a trajectory analysis of GD and +prove Lemma 4.1. Experimental results are presented in Section 6 which verify our theoretical +findings. Finally, in Section 7, we summarize our main contributions and discuss some promising +future directions. Complete proofs of all results are given in the Appendix. +2 +Related work +Low-rank matrix recovery. The goal of low-rank matrix recovery is to recover an unknown +low-rank matrix from a number of (possibly noisy) measurements. Examples include matrix +sensing (Recht et al., 2010), matrix completion (Cand`es & Recht, 2009; Candes & Plan, 2010) +and robust PCA (Xu et al., 2010; Cand`es et al., 2011). Fornasier et al. (2011); Ngo & Saad +(2012); Wei et al. (2016); Tong et al. (2021) study efficient optimization algorithms with conver- +gence guarantees. Interested readers can refer to Davenport & Romberg (2016) for an overview +of this topic. +Simplicity bias/incremental learning of GD. Besides the works mentioned in the introduction, +there are many other works studying the simplicity bias/incremental learning of GD on tensor +factorization (Razin et al., 2021, 2022), deep linear networks (Gidel et al., 2019), two-layer nets +with orthogonal inputs (Boursier et al., 2022). +Landscape analysis of non-convex low-rank problems. The strict saddle property (Ge et al., +2016, 2015; Lee et al., 2016) was established for non-convex low-rank problems in a unified +framework by Ge et al. (2017). Tu et al. (2016) proved a local PL property for matrix sens- +ing with exact parameterization (i.e. the rank of parameterization and ground-truth matrix are +the same). The optimization geometry of general objective function with Burer-Monteiro type +factorization is studied in Zhu et al. (2018); Li et al. (2019); Zhu et al. (2021). We provide a +comprehensive analysis in this regime for matrix factorization as well as matrix sensing that +improves over their results. +4 + +3 +Preliminaries +In this section, we first list the notations used in this paper, and then provide details of our +theoretical setup and necessary preliminary results. +3.1 +Notations +We write min{a, b} as a ∧ b for short. For any matrix A, we use ∥A∥F to denote the Frobenius +norm of A, use ∥A∥ to denote the spectral norm ∥A∥2, and use σmin(A) to denote the smallest +singular value of A. We use the following notation for Singular Value Decomposition (SVD): +Definition 3.1 (Singular Value Decomposition). For any matrix A ∈ Rd1×d2 of rank r, we +use A = VAΣAW ⊤ +A to denote a Singular Value Decomposition (SVD) of A, where VA ∈ +Rd1×r, WA ∈ Rd2×r satisfy V ⊤ +A VA = I, W ⊤ +AWA = I, and ΣA ∈ Rr×r is diagonal. +For the matrix sensing problem (1), we write the ground-truth matrix as Z∗ = XX⊤ for +some X = [v1, v2, · · · , vr∗] ∈ Rd×r∗ with orthogonal columns from an orthogonal basis {vi : +i ∈ [d]} of Rd. We denote the singular values of X as σ1, σ2, . . . , σr∗, then the singular values +of Z∗ are σ2 +1, σ2 +2, . . . , σ2 +r∗. We set σr∗+1 := 0 for convenience. For simplicity, we only consider +the case where Z∗ has distinct singular values, i.e., σ2 +1 > σ2 +2 > · · · > σ2 +r∗ > 0. We use +κ := +σ2 +1 +min1≤s≤r∗{σ2s−σ2 +s+1} to quantify the degeneracy of the singular values of Z∗. We also use +the notation Xs = [v1, v2, · · · , vs] for the matrix consisting of the first s columns of X and +X⊥ +s = [vs+1, · · · , vd]. Following Definition 3.1, we let VX⊥ +s = +� +vs+1 +∥vs+1∥, · · · , +vd +∥vd∥ +� +. Note that +the best rank-s solution Z∗ +s (Definition 1.1) does not equal XsX⊤ +s in general. +We write the results of the measurements {Ai}m +i=1 as a linear mapping A : Rd×d �→ Rm, +where [A(Z)]i = +1 +√m⟨Ai, Z⟩ for all 1 ≤ i ≤ m. We use A∗ : Rm → Rd×d, A∗(w) = +1 +√m +�m +i=1 wiAi to denote the adjoint operator of A. Our loss function (1) can then be written as +f(U) = 1 +4 +��A +� +Z∗ − UU ⊤���2 +2. The gradient is given by ∇f(U) = A∗ � +y − A(UU ⊤) +� +U = +A∗A +� +XX⊤ − UU ⊤� +U. +In this paper, we consider GD with learning rate µ > 0 starting from U0. The update rule is +Ut+1 = Ut − µ∇f (Ut) = (I + µMt)Ut, +(3) +where Mt := A∗A +� +XXT − UtU T +t +� +. We specifically focus on GD with small initialization: +letting U0 = α ¯U for some matrix ¯U ∈ Rd׈r with ∥ ¯U∥ = 1, we are interested in the trajectory +of GD when α → 0. Sometimes we write Ut as Uα,t to highlight the dependence of the trajectory +on α. +3.2 +Assumptions +For our theoretical analysis of the matrix sensing problem, we make the following standard +assumption in the matrix sensing literature: +5 + +Definition 3.2 (Restricted Isometry Property). We say that a measurement operator A satisfies +the (δ, r)-RIP condition if (1−δ)∥Z∥2 +F ≤ ∥A(Z)∥2 +2 ≤ (1+δ)∥Z∥2 +F for all matrices Z ∈ Rd×d +with rank(Z) ≤ r. +Assumption 3.1. The measurement operator A satisfies the (2r∗ + 1, δ)-RIP property, where +r∗ = rank(Z∗) and δ ≤ 10−12κ−4.5r−1 +∗ . +The RIP condition is the key to ensure the ground truth to be recoverable with partial observa- +tions. An important consequence of RIP is that it guarantees A∗A(Z) = 1 +m +�m +i=1 ⟨Ai, Z⟩ Ai ≈ +Z when Z is low-rank. This is made rigorous in the following proposition. +Proposition 3.1. (St¨oger & Soltanolkotabi, 2021, Lemma 7.3) Suppose that A satisfies (r, δ)- +RIP with r ≥ 2, then for all symmetric Z, we have ∥(A∗A − I)Z∥2 ≤ δ∥Z∥∗, where ∥ · ∥∗ is +the nuclear norm. Moreover, if rank(Z) ≤ r − 1, then ∥(A∗A − I)Z∥2 ≤ √rδ∥Z∥. +We need the following regularity condition on initialization. +Assumption 3.2. For all 1 ≤ s ≤ ˆr ∧ r∗, σmin +� +V ⊤ +Xs ¯U +� +≥ ρ for some constant ρ > 0, where +VXs is defined as Definition 3.1. +The following proposition implies that Assumption 3.2 is satisfied with high probability with +a Gaussian initialization. +Proposition 3.2. Suppose that all entries of ¯U ∈ Rd׈r are independently drawn from N +� +0, 1 +ˆr +� +and ρ = ϵ +√ +ˆr−√ˆr∧r∗−1 +√ +ˆr +≥ +ϵ +2r∗ , then σmin +� +V ⊤ +Xs ¯U +� +≥ ρ holds for all 1 ≤ s ≤ ˆr ∧ r∗ with +probability at least 1 − ˆr +� +Cϵ + e−cˆr� +, where c, C > 0 are universal constants. +Lastly, we make the following assumption on the step size. +Assumption 3.3. The step size µ ≤ 10−4δ∥X∥−2. +3.3 +Procrustes Distance +Our analysis uses the notion of Procrustes distance defined as in Goodall (1991); Tu et al. (2016). +Definition 3.3 (Procrustes Distance). The Procrustes distance between two matrices U1, U2 ∈ +Rd×s (d, s > 0) is defined as the optimal value of the classic orthogonal Procrustes problem: +dist(U1, U2) = +min +R∈Rs×s:R⊤R=I ∥U1 − U2R∥F . +(4) +We note that the Procrustes distance is well-defined because the set of s × s orthogonal +matrices is compact and thus the continuous function ∥U1 − U2R∥F in R can attain its mini- +mum. The Procrustes distance is a pseudometric, i.e., it is symmetric and satisfies the triangle +inequality. +The following lemma is borrowed from Tu et al. (2016), which connects the Procrustes dis- +tance between U1 and U2 with the distance between U1U ⊤ +1 and U2U ⊤ +2 . +Lemma 3.1 (Tu et al. 2016, Lemma 5.4). For any two matrices U1, U2 ∈ Rd×r, we have +��U1U ⊤ +1 − U2U ⊤ +2 +�� +F ≥ (2 +√ +2 − 2)1/2 · σr(U1) · dist(U1, U2). +6 + +4 +Main results +In this section, we present our main theorems and their proof sketches, following the theoretical +setup in Section 3. Full proofs can be found in Appendices C to F. +Theorem 4.1. Under Assumptions 3.1 to 3.3, consider GD (3) with initialization Uα,0 = α ¯U +for solving the matrix sensing problem (1). There exist universal constants c, M, constant C = +C(X, ¯U) and a sequence of time points T 1 +α < T 2 +α < · · · < T ˆr∧r∗ +α +such that for all 1 ≤ s ≤ ˆr∧r∗, +the following holds when α is sufficiently small: +���Uα,T sαU ⊤ +α,T sα − Z∗ +s +��� +F ≤ Cα +1 +Mκ2 , +(5) +where we recall that Z∗ +s is the best rank-s solution defined in Definition 1.1. Moreover, GD +follows an incremental learning procedure: we have limα→0 max1≤t≤T sα σs+1(Uα,t) = 0 for all +1 ≤ s ≤ ˆr ∧ r∗, where σi(A) denotes the i-th largest singular value of a matrix A. +It is guaranteed that Z∗ +s is unique for all 1 ≤ s ≤ ˆr ∧ r∗ under our assumptions (see +Lemma 4.2). In short, Theorem 4.1 states that GD with small initialization discovers the best +rank-s solution (s = 1, 2, · · · , ˆr ∧ r∗) sequentially. In particular, when s = r∗, the best rank-s +solution is exactly the ground truth XX⊤. Hence with over-parameterization (ˆr ≥ r∗), GD can +discover the ground truth. +At a high level, our result characterizes the complete learning dynamics of GD and reveals +an incremental learning mechanism, i.e., GD starts from learning simple solutions and then +gradually increases the complexity of search space until it finds the ground truth. +In the under-parameterized setting, we can further establish the following convergence result: +Theorem 4.2 (Convergence in the under-parameterized regime). Suppose that ˆr ≤ r∗, then there +exists a constant ¯α > 0 such that when α < ¯α, we have limt→+∞ Uα,tU ⊤ +α,t = Z∗ +ˆr . +4.1 +Key lemmas +In this section, we present some key lemmas for proving our main results. First, we can show +that with small initialization, GD can get into a small neighborhood of Z∗ +s. +Lemma 4.1. Under Assumptions 3.1 and 3.2, there exists ˆT s +α > 0 for all α > 0 and 1 ≤ s ≤ ˆr ∧ +r∗ such that limα→0 max1≤t≤ ˆT sα σs+1(Uα,t) = 0. Furthermore, it holds that +���U ˆT sαU ⊤ +ˆT sα − Z∗ +s +��� +F = +O +� +κ3r∗δ∥X∥2� +. +The full proof can be found in Appendix C. Motivated by St¨oger & Soltanolkotabi (2021), +we consider the following decomposition of Ut: +Ut = UtWtW ⊤ +t + UtWt,⊥W ⊤ +t,⊥, +(6) +where Wt := WV ⊤ +XsUt ∈ Rˆr×s is the matrix consisting of the right singular vectors of V ⊤ +XsUt +(Definition 3.1) and Wt,⊥ ∈ Rˆr×(ˆr−s) is any orthogonal complement of Wt, i.e., WtW ⊤ +t ++ +7 + +Wt,⊥W ⊤ +t,⊥ = I. The dependence of Wt, Wt,⊥ on s is omitted but will be clear from the +context. +We will refer to the term UtWtW ⊤ +t +as the parallel component and UtWt,⊥W ⊤ +t,⊥ as the +orthogonal component. The idea is to show that the parallel component grows quickly until +it gets close to the best rank-s solution at some time ˆT s +α (namely UtWtW ⊤ +t U ⊤ +t +≈ Z∗ +s when +t = ˆT s +α). Meanwhile, the orthogonal term grows exponentially slower and stays o(1) before ˆT s +α. +See Section 5 for a detailed proof sketch. +Lemma 4.1 shows that UtU ⊤ +t would enter a neighborhood of Z∗ +s with constant radius. How- +ever, there is still a gap between Lemma 4.1 and Theorem 4.1, since the latter states that UtU ⊤ +t +would actually get o(1)-close to Z∗ +s. +To proceed, we define the under-parameterized matrix sensing loss fs for every 1 ≤ s ≤ r∗: +fs(U) = 1 +4 +���A(Z∗ − UU ⊤) +��� +2 +2 , +U ∈ Rd×s. +(7) +While the function we are minimizing is f (defined in (1)) rather than fs, Lemma 4.1 suggests +that for t ≤ ˆT s +α, Ut is always approximately rank-s, so that we use a low-rank approximation for +U ˆT sα and associate the dynamics locally with the GD dynamics of fs. We will elaborate on how +this is done in Section 4.2. +When dist(U1, U2) = 0, it can be easily shown that fs(U1) = fs(U2) since fs is invariant +to orthogonal transformations. Moreover, we note that the global minimizer of fs is unique up +to orthogonal transformations. +Lemma 4.2. Under Assumption 3.1, if U ∗ +s ∈ Rd×s is a global minimizer of fs, then the set of +global minimizers arg min fs is equal to +� +U ∗ +s R : R ∈ Rs×s, R⊤R = I +� +. +Around the global minimizers, we show that fs satisfies the Restricted Secant Inequality +(RSI) which is useful for optimization analysis. +Definition 4.1. For any U ∈ Rd×s, we use Πs(U) to denote the set of closest global minimizers +of fs to U, namely Πs(U) = arg min{∥U − U ∗ +s ∥F : U ∗ +s ∈ arg min fs}. +Lemma 4.3 (Restricted Secant Inequality). Under Assumption 3.1, if a matrix U ∈ Rd×s satis- +fies ∥U − U ∗ +s ∥F ≤ 10−2κ−1∥X∥ for some U ∗ +s ∈ Πs(U), then we have +⟨∇fs(U), U − U ∗ +s ⟩ ≥ 0.1κ−1∥X∥2∥U − U ∗ +s ∥2 +F . +(8) +Remark 4.1. In general, a function g : Rn �→ R satisfies the RSI condition if for some +µ > 0, ⟨∇g(x), x − π(x)⟩ ≥ µ∥x − π(x)∥2 holds for all x, where π(x) is a projection of +x onto arg min g. This condition can be used to prove linear convergence of GD (Zhang & Yin, +2013), but it is weaker than strong convexity and stronger than Polyak-Łojasiewicz(PL) condi- +tion (Karimi et al., 2016). +We end this subsection with the following lemma which says that all global minimizers of +the fs must be close to Xs under the procrustes distance, which is used in the proof sketch of +Theorem 4.1 in Section 4. +8 + +Lemma 4.4. Under Assumption 3.1, we have dist(U ∗ +s , Xs) ≤ 40δκ∥X∥F for any global mini- +mizer U ∗ +s of fs. Moreover, +��Z∗ +s − XsX⊤ +s +�� +F ≤ 160δκ√r∗∥X∥2. +Corollary 4.1. Under Assumption 3.1, we have σmin(U ∗ +s ) ≥ 1 +2σmin(Xs) = 1 +2σs ≥ 1 +2κ− 1 +2 ∥X∥. +The full proofs for Lemmas 4.3 and 4.4 and Corollary 4.1 can be found in Appendix E. +4.2 +Proof outline +Based on the key lemmas, here we provide the outlines of the proofs for our main theorems and +defer the details to Appendix F. We first prove Theorem 4.2 which can be directly derived by +combining the lemmas in Section 4.1. +Proof :[Proof Sketch of Theorem 4.2] For any global minimizer U ∗ +ˆr of (1), we have +dist(U ∗ +ˆr , Uα, ˆT ˆrα) +≤ (2 +√ +2 − 2)−1/2σ−1 +min (U ∗ +ˆr ) +���Uα, ˆT ˆrαU ⊤ +α, ˆT ˆrα − Z∗ +ˆr +��� +F +≤ O(κ +1 +2 ∥X∥−1) · O(κ3r∗δ∥X∥2) += O(κ3.5r∗δ∥X∥), +where the first inequality is due to Lemma 3.1 and the second one is due to Corollary 4.1 and +Lemma 4.1. +Assumption 3.1 and Lemma 4.3 then imply that Uα, ˆT ˆrα lies in the small neighborhood of +the set of global minimizers of f = fˆr, in which the RSI holds. Following a standard non- +convex optimization analysis (Karimi et al., 2016), we can show that GD converges linearly to +arg min fˆr (in the Procrustes distance), which yields the conclusion. +□ +Now we turn to prove Theorem 4.1. While f is not necessarily local RSI, we use a low-rank +approximation for Ut and associate the dynamics in this neighborhood with the GD dynamics +of fs. +Proof :[Proof sketch of Theorem 4.1] Recall that by Lemma 4.1, Uα, ˆT sα is approximately rank-s. +So there must exist a matrix ¯Uα,0 ∈ Rd׈r with rank( ¯Uα,0) ≤ s such that +¯Uα,0 ¯U ⊤ +α,0 − Uα,T sαU ⊤ +α,T sα = o(1) +as α → 0. +(9) +Indeed, we can let ¯Uα,t be the parallel component of Uα,T sα because the orthogonal component +stays o(1) (see the discussions following (6) and Corollary 5.2 for details). +Let +� ¯Uα,t +� +t≥0 be the trajectory of GD with step size µ, initialized at ¯Uα,0. Since the gradient +of the objective function f is locally Lipschitz, the solution obtained by the two GD trajectories +{ ¯Uα,t}t≥0 and {Uα, ˆT sα+t}t≥0 will remain o(1)-close for at least constantly many steps. Indeed +we can show that they will keep o(1) close for some ¯tα = ω(1) steps, i.e., for all t ∈ [0, ¯tα], +¯Uα,t ¯U ⊤ +α,t − Uα, ˆT sα+tU ⊤ +α, ˆT sα+t = o(1). +(10) +9 + +From (3) it is evident that GD initialized at ¯Uα,0 actually lives in the space of matrices with rank +≤ s. Indeed we can identify its dynamics with another GD on fs (defined in (7)). Concretely, let +ˆUα,0 ∈ Rd×s be a matrix so that ˆUα,0 ˆU ⊤ +α,0 = ¯Uα,0 ¯U ⊤ +α,0, and let { ˆUα,t}t≥0 be the trajectory of +GD that optimizes fs with step size µ starting from ˆUα,0. Then we have ˆUα,t ˆU ⊤ +α,t = ¯Uα,t ¯U ⊤ +α,t +for all t ≥ 0. +We can now apply our landscape results for fs to analyze the GD trajectory { ˆUα,t}t≥0. By (9) +and Lemma 4.1, we have +��� ˆUα,0 ˆU ⊤ +α,0 − Z∗ +s +��� +F = O(κ3r∗δ∥X∥2), so using a similar argument +as in the proof sketch of Theorem 4.2, Corollary 4.1 and Lemma 3.1 imply that the initialization +ˆUα,0 is within an O +� +κ3.5r∗δ∥X∥2� +neighborhood of the set of global minimizers of fs(U). +From Assumption 3.1 and Lemma 4.3 we know that that fs(U) satisfies a local RSI condition in +this neighborhood, so following standard non-convex optimization analysis (Karimi et al., 2016), +we can show that { ˆUα,t}t≥0 converges linearly to its set of global minimizers in the Procrustes +distance. We need to choose a time t such that (10) remains true while this linear convergence +process takes place for sufficiently many steps. This is possible since ¯tα = ω(1); indeed we can +show that there always exists some t = ts +α ≤ ¯tα such that both +��� ˆUα,t ˆU ⊤ +α,t − Uα, ˆT sα+tU ⊤ +α, ˆT sα+t +��� +F +and +��� ˆUα,t − U ∗ +s +��� +F are bounded by O(α +1 +Mκ2 ). Hence +��Uα,tU ⊤ +α,t − Z∗ +s +�� +F = O(α +1 +Mκ2 ) when +t = T s +α := ˆT s +α + ts +α. +For 1 ≤ s < ˆr ∧ r∗ and t ≤ ts +α, since (10) holds and rank( ˆUα,t) ≤ s, we have +max +1≤t≤T sα +σs+1 (Uα,t) → 0 +as α → 0. Finally, by Lemma 4.4 and Assumption 3.1 we have +��Z∗ +s+1 − Xs+1X⊤ +s+1 +�� = +O(δκ√r∗) = O(κ−1∥X∥2), so σs+1(Z∗ +s+1) ≳ σ2 +s+1. Therefore, Uα,tU ⊤ +α,t cannot be close to +Z∗ +s+1 when t ≤ T s +α, so we must have T s+1 +α +> T s +α. This completes the proof of Theorem 4.1. □ +5 +Proof sketch of Lemma 4.1 +In this section, we outline the proof sketch of Lemma 4.1. We divide the GD dynamics intro +three phases and characterize the dynamics separately. Proof details for these three phases can +be found in Appendices C.1, C.2 and C.4. +5.1 +The spectral phase +Starting from a small initialization, GD initially behaves similarly to power iteration since +Ut+1 = (I + µMt)Ut ≈ (I + µM)Ut, where M := A∗A(XX⊤) is a symmetric matrix. Let +M = �d +k=1 ˆσ2 +kˆvkˆv⊤ +k be the eigendecomposition of M, where ˆσ1 ≥ ˆσ2 ≥ · · · ≥ ˆσd ≥ 0. Using +our assumption on δ (Assumption 3.1), we can show that |σi − ˆσi| , 1 ≤ i ≤ s are sufficiently +10 + +small so that ˆσi’s are positive and well-separated. Then we have +UT ≈ (I + µM)T U0 = +d +� +i=1 +(1 + µˆσ2 +i )T ˆviˆv⊤ +i U0 +≈ +s +� +i=1 +(1 + µˆσ2 +i )T ˆviˆv⊤ +i U0, +(11) +where the last step holds because (1 + µˆσs)T ≫ (1 + µˆσs+1)T . In other words, we can expect +that there is an exponential separation between the parallel and orthogonal component of UT . +Formally, we can prove the following property at the end of the spectral phase: +Lemma 5.1 (Lemma C.3, simplified version). Suppose that Assumptions 3.1 to 3.3 hold. Then +there exist positive constants Ci = Ci(X, ¯U), γi = γi(X, ¯U), i = 2, 3 independent of α such +that γ2 < γ3 and the following inequalities hold for t = T sp +α += O +� +log α−1 +log(1+µ∥X∥2) +� +when α is +sufficiently small: +∥Ut∥ ≤ ∥X∥, +σmin (UtWt) ≥ C2 · αγ2, +∥UtWt,⊥∥ ≤ C3 · αγ3, and +���V ⊤ +Xs,⊥VUtWt +��� ≤ 200δ. +5.2 +The parallel improvement phase +For small α, we have σmin (UtWt) ≫ ∥UtWt,⊥∥ by the end of the spectral phase. When (11) +no longer holds, we enter a new phase which we call the parallel improvement phase. In this +phase, the ratio σmin(UtWt) +∥UtWt,⊥∥ grows exponentially in t, until the former reaches a constant scale. +Formally, let T pi +α,s = min +� +t ⩾ 0 : σ2 +min +� +V ⊤ +XsUα,t+1 +� +> 0.3κ−1∥X∥2� +, then we can prove the +following lemma via induction. +Lemma 5.2. Suppose that Assumptions 3.1 to 3.3 hold and let c3 = 104κr +1 +2∗ δ. Then for suffi- +ciently small α, the following inequalities hold when T sp +α ≤ t < T pi +α,s: +σmin +� +V ⊤ +XsUt+1 +� +≥ σmin +� +V ⊤ +XsUt+1Wt +� +≥ +� +1 + 0.5µ +� +σ2 +s + σ2 +s+1 +�� +σmin +� +V ⊤ +XsUt +� +, +(13a) +∥Ut+1Wt+1,⊥∥ ≤ +� +1 + µ +� +0.4σ2 +s + 0.6σ2 +s+1 +�� +∥UtWt,⊥∥ , +(13b) +���V ⊤ +Xs,⊥VUt+1Wt+1 +��� ≤ c3, +(13c) +rank(V ⊤ +XsUt+1) = rank(V ⊤ +XsUt+1Wt) = s. +(13d) +We can immediately deduce from Lemmas 5.1 and 5.2 that the orthogonal term ∥UtWt,⊥∥ +remains o(1) by the end of the parallel improvement phase: +Corollary 5.1 (Lemma C.8, simplified version). Under the conditions of Lemma 5.2, when α is +sufficiently small we have +���UT pi +α,sWT pi +α,s,⊥ +��� ≤ C5 · α +1 +4κ for some constant C5 = C5(X, ¯U). +11 + +5.3 +The refinement phase +After σmin (UtWt) grows to constant scale, we enter the refinement phase for which we show +that +��XsX⊤ +s − UtU ⊤ +t +�� +F keeps decreasing until it is O +� +δκ3r∗∥X∥2� +. +Formally, let τ = +κ−1∥X∥2 and T ft +α,s = T pi +α,s − +log(10−2∥X∥−2κ−1c−1 +3 ) +log(1− 1 +2 µτ) +> T pi +α,s where c3 is defined in Lemma 5.2, +then the following lemma holds. +Lemma 5.3. Suppose that T pi +α,s ≤ t ≤ T ft +α,s and all the conditions in Lemma 5.2 hold, then we +have +���V ⊤ +Xs(XX⊤ − Ut+1U ⊤ +t+1) +��� +F +≤ +� +1 − 1 +2µτ +� ���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� +F + 20µ∥X∥4 (δ + 5c3) + 2000µ2√r∗∥X∥6. +Moreover, it holds that ∥Ut+1Wt+1,⊥∥ ≤ (1 + σ2 +s)∥UtWt,⊥∥ and +���VX⊥ +s VUtWt +��� ≤ c3. +Using Lemma 5.3, we arrive at the following result: +Corollary 5.2. For sufficiently small α, at t = T ft +α,s we have +��V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +��� +F ≤ +80δκ3r∗∥X∥2 and ∥UtWt,⊥∥ = o(1) (α → 0). +Concluding the proof of Lemma 4.1. At t = T ft +α,s, we have +���XsX⊤ +s − UtU ⊤ +t +��� +F +≤ +��� +� +XsX⊤ +s − UtU ⊤ +t +� +VXsV ⊤ +Xs +��� +F + +���UtU ⊤ +t VX⊥ +s V ⊤ +X⊥ +s +��� +F +≤ +��� +� +XsX⊤ +s − UtU ⊤ +t +� +VXsV ⊤ +Xs +��� +F + +���V ⊤ +X⊥ +s UtU ⊤ +t VX⊥ +s +��� +F +≤ +���V ⊤ +Xs +� +XsX⊤ +s − UtU ⊤ +t +���� +F + √r∗ +���V ⊤ +X⊥ +s UtWt +��� +2 ++ +√ +d +���V ⊤ +X⊥ +s UtWt,⊥ +��� +2 +(14a) +≤ +���V ⊤ +Xs +� +XsX⊤ +s − UtU ⊤ +t +���� +F +9√r∗∥X∥2 ���VX⊥ +s VUtWt +��� +2 ++ +√ +d∥UtWt,⊥∥2 +(14b) += O +� +δκ3r∗∥X∥2 + ∥X∥2c2 +3 +√r∗ +� ++ o(1) +(14c) += O +� +δκ3r∗∥X∥2� +, +(14d) +where (14a) uses ∥A∥F ≤ +� +rank(A)∥A∥, (14b) uses ∥Ut∥ ≤ 3∥X∥, (14c) follows from +Lemma 5.3 and Corollary 5.2 and the last step follows from c3 = 104κ√r∗δ and Assumption 3.1. +By Lemma 4.4, the best rank-s solution is close to the matrix factorization minimizer i.e. +��Z∗ +s − XsX⊤ +s +�� +F = O +� +δκ√r∗∥X∥2� +. We thus obtain that +��Z∗ +s − UtU ⊤ +t +�� +F = O +� +δκ3r∗∥X∥2� +. +Finally, since rank(UtWt) ≤ s (recall the decomposition (6)), we have σs+1(Ut) ≤ ∥UtWt,⊥∥ = +o(1). The conclusion follows. +12 + +(a) α = 1, 1000 measure- +ments. +(b) α = 0.1, 1000 measure- +ments. +(c) α = 0.01, 1000 measure- +ments. +(d) α = 0.001, 1000 mea- +surements. +(e) α = 0.001, 2000 mea- +surements. +(f) α = 0.001, 5000 mea- +surements. +Figure 1: The evolution of relative error against the best solution of different ranks over time. +(a) α = 1, 1000 measurements. +(b) α = 0.1, 1000 measure- +ments. +(c) α = 0.01, 1000 measure- +ments. +(d) α = 0.001, 1000 measure- +ments. +Figure 2: The evolution of the loss and relative error against best solution of different ranks in +the exact-parameterized case r = 5. +13 + +rank 1 +rank 2 +rank 3 +relative error +rank 4 +rank 5 +101 +0 +25 +50 +75 +100 +125 +150 +175 +200 +iteration100 +rank 1 +rank 2 +rank 3 +relative error +rank 4 +10-1 +rank 5 +10-2 +0 +100 +200 +300 +400 +500 +iteration100 +rank 1 +rank 2 +rank 3 +relative error +rank 4 +rank 5 +10-1 +10-2 +0 +50 +100 +150 +200 +250 +300 +iteration100 +rank 1 +rank 2 +rank 3 +relative error +rank 4 +10-1 +rank 5 +10-2 +10-3 +0 +50 +100 +150 +200 +250 +300 +iteration100 +rank 1 +rank 2 +rank 3 +10-1 +relative error +rank 4 +rank 5 +10-2 +10-3 +10-4 +0 +100 +200 +300 +400 +500 +600 +iteration100 +10-1 +error +relative +10-2 +rank 1 +rank 2 +10-3 +rank 3 +rank 4 +10-4 +rank 5 +0 +100 +200 +300 +400 +500 +600 +iteration100 +10-1 +relative error +10-2 +10-3 +rank 1 +rank 2 +10-4 +rank 3 +rank 4 +10-5 +rank 5 +0 +100 +200 +300 +400 +500 +600 +iterationrank 1 +100 +rank 2 +rank 3 +relative error +rank 4 +10-1 +rank 5 +10-2 +10-3 +10-4 +0 +100 +200 +300 +400 +500 +iteration100 +10-1 +relative error +10-2 +rank 1 +rank 2 +10-3 +rank 3 +rank 4 +rank 5 +10-4 +0 +100 +200 +300 +400 +500 +iteration100 +rank 1 +rank 2 +rank 3 +relative error +10-1 +rank 4 +rank 5 +10-2 +10-3 +0 +100 +200 +300 +400 +500 +iteration6 +Experiments +In this section, we perform some numerical experiments to illustrate our theoretical findings. +Experimental setup. We consider the matrix sensing problem (1) with d = 50, r∗ = 5, +α ∈ {1, 0.1, 0.01, 0.001}, m ∈ {1000, 2000, 5000}. We will consider different choices for ˆr in +the experiments. The ground truth Z∗ = XX⊤ is generated such that the entries of X are i.i.d. +standard Gaussian variables. We use the same ground truth throughout our experiments. +For i = 1, 2, · · · , m, all entries of the measurement Ai ∈ Rd×d are chosen i.i.d. from the +standard Gaussian N(0, 1). When m ≳ dr∗δ−2, this set of measurements satisfies the RIP with +high probability (Recht et al., 2010, Theorem 4.2). +We solve the problem (1) via running GD for T = 104 iterations starting with small initializa- +tion with scale α. Specifically, we choose U0 = α ¯U where the entries of ¯U ∈ Rd׈r are drawn +i.i.d. from N(0, 1). We consider both the over-parameterized and the exact/under-parameterized +regime. The learning rate of GD is set to be µ = 0.005. +6.1 +Implicit low-rank bias +In this subsection, we consider the over-parameterized setting with r = 50. For each iteration +t ∈ [T] and rank s ∈ [r∗], we define the relative error Es(t) = ∥UtU⊤ +t −XsX⊤ +s ∥ +2 +F +∥XsX⊤ +s ∥ +2 +F +to measure the +proximity of the GD iterates to Xs. We plot the relative error in Figure 1 for different choices of +α and m (which affects the measurement error δ). +Small initialization. The implicit low-rank bias of GD is evident when the initialization scale +α is small. Indeed, one can observe that GD first visits a small neighborhood of X1, spends a +long period of time near it, and then moves towards X2. It then proceeds to learn X3, X4, · · · in +a similar way, until it finally fits the ground truth. This is in align with Theorem 4.1. By contrast, +for large initialization we do not have this implicit bias. +The effect of measurement error. For fixed α, one can observe the relative error becomes +smaller when the number of measurements increases. This is in align with Lemma 4.1 in which +the bound depends on δ. In particular, for the case s = r∗, in the end the distance to the set of +global minima goes to zero as α → 0. +6.2 +Matrix sensing with exact parameterization +Now we study the behavior of GD in the exact parameterization regime (r = r∗). We fix +m = 1000, r = r∗ = 5 and run GD for T = 500 iterations. We plot the relative error in +Figure 2. As predicted by Theorem 4.1, we can observe that when α is small, GD exhibits an +implicit low-rank bias and takes a longer time to converge. The latter is because GD would +get into a poly(α)-small neighborhood of the saddle point Zs and take a long time to escape +the saddle. As guaranteed by Theorem 4.2, we also observe the final convergence to global +minimizers for sufficiently small α. +14 + +7 +Conclusion +In this paper, we study the matrix sensing problem with RIP measurements and show that GD +with small initialization follows an incremental learning procedure, where GD finds near-optimal +solutions with increasing ranks until it finds the ground-truth. We take a step towards under- +standing the optimization and generalization aspects of simple optimization methods, thereby +providing insights into their success in modern applications such as deep learning (Goodfellow +et al., 2016). Also, we provide a detailed landscape analysis in the under-parameterized regime, +which to the best of our knowledge is the first analysis of this kind. +Although we focus on matrix sensing in this paper, it has been revealed in a line of works that +the implicit regularization effect may vary for different models, including deep matrix factoriza- +tion (Arora et al., 2019) and nonlinear ReLU/LeakyReLU networks (Lyu et al., 2021; Timor +et al., 2022). Also, it is shown in Woodworth et al. (2020) that different initialization scales +can lead to distinct inductive bias and affect the generalization and optimization behaviors. All +these results indicate that we need further studies to comprehensively understand gradient-based +optimization methods from the generalization aspect. +References +Arora, S., Cohen, N., Hu, W., and Luo, Y. Implicit regularization in deep matrix factorization. +Advances in Neural Information Processing Systems, 32, 2019. +Belabbas, M. A. 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In 24th British machine vision conference (BMVC), pp. 1–13, 2013. +19 + +Appendix +Table of Contents +A Preliminaries +21 +A.1 +The RIP condition and its properties . . . . . . . . . . . . . . . . . . . . . . +21 +A.2 +Matrix analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +A.3 +Optimization +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +A.4 +Proof for Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +A.5 +Procrustes Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +B +Main idea for the proof of Theorem 4.1 +24 +B.1 +Heuristic explanations of the decomposition . . . . . . . . . . . . . . . . . . +25 +C Proof of Lemma 4.1 +26 +C.1 +The spectral phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +C.2 +The parallel improvement phase +. . . . . . . . . . . . . . . . . . . . . . . . +30 +C.3 +Induction +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +C.4 +The refinement phase and concluding the proof of Lemma 4.1 +. . . . . . . . +40 +D Auxiliary results for proving Lemma 4.1 +45 +D.1 +The spectral phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +D.2 +The parallel improvement phase +. . . . . . . . . . . . . . . . . . . . . . . . +46 +E +Proofs for the Landscape Results in Section 4.1 +49 +E.1 +Analysis of the matrix factorization loss . . . . . . . . . . . . . . . . . . . . +49 +E.2 +Analysis of the matrix sensing loss . . . . . . . . . . . . . . . . . . . . . . . +51 +F +Proofs for Theorems 4.1 and 4.2 +54 +G The landscape of matrix sensing with rank-1 parameterization +57 +20 + +The appendix is organized as follows: in Appendix A we present a number of results that +will be used for later proof. Appendix B sketches the main idea for proving our main results. +Appendix C is devoted to a rigorous proof of Lemma 4.1 ,with some auxiliary lemmas proved +in Appendix D. In Appendix E we analyze the landscape of low-rank matrix sensing and prove +our landscape results in Section 4.1. These results are then used in Appendix F to prove The- +orems 4.1 and 4.2. Finally, Appendix G studies the landscape of rank-1 matrix sensing, which +enjoys a strongly convex property, as we mentioned in Section 4.1 without proof. +A +Preliminaries +In this section, we present some useful results that is needed in subsequent analysis. +A.1 +The RIP condition and its properties +In this subsection, we collect a few useful properties of the RIP condition, which we recall below: +Definition A.1. We say that the measurement A satisfies the (δ, r)-RIP condition if for all ma- +trices Z ∈ Rd×d with rank(Z) ≤ r, we have +(1 − δ)∥Z∥2 +F ≤ ∥A(Z)∥2 +2 ≤ (1 − δ)∥Z∥2 +F . +The key intuition behind RIP is that A∗A ≈ I, where A∗ : v �→ +1 +√m +�m +i=1 viAi is the +adjoint of A. This intuition is made rigorous by the following proposition: +Proposition A.1. (St¨oger & Soltanolkotabi, 2021, Lemma 7.3) Suppose that A satisfies (r, δ)- +RIP with r ≥ 2, then for all symmetric Z, +(1). if rank(Z) ≤ r − 1, we have ∥(A∗A − I)Z∥2 ≤ √rδ∥Z∥. +(2). ∥(A∗A − I)Z∥2 ≤ δ∥Z∥∗, where ∥ · ∥∗ is the nuclear norm. +A.2 +Matrix analysis +The following lemma is a direct corollary of Proposition A.1 and will be frequently used in our +proof. +Lemma A.1. Suppose that the measurement A satisfies (δ, 2r∗ + 1)-RIP condition, then for all +matrices U ∈ Rd×r such that rank(U) ≤ r∗, we have +���(A∗A − I) (XX⊤ − UU ⊤) +��� ≤ δ√r∗ +� +∥X∥2 + ∥U∥2� +. +In our proof we will frequently make use of the Weyl’s inequality for singular values: +Lemma A.2 (Weyl’s inequality). Let A, ∆ ∈ Rd×d be two matrices, then for all 1 ≤ k ≤ d, we +have +|σk(A) − σk(A + ∆)| ≤ ∥∆∥. +21 + +We will also need the Wedin’s sin theorem for singular value decomposition: +Lemma A.3. (Wedin, 1972, Section 3) Define R(·) to be the column space of a matrix. Suppose +that matrices B = A+T , A1, B1 are the top-s components in the SVD of A and B respectively, +and A0 = A − A1, B0 = B − B1. If δ = σmin(B1) − σmax(A0) > 0, then we have +∥sin Θ (R(A1), R(B1))∥ ≤ ∥T ∥ +δ +where Θ(·, ·) denotes the angle between two subspaces. +Equipped with Lemma A.1, we can have the following characterization of the eigenvalues of +M (recall that M = A∗A(XX⊤)): +Lemma A.4. Let M := A∗A(XX⊤) and M = �d +k=1 ˆσ2 +kˆvkˆv⊤ +k be the eigen-decomposition of +M. For 1 ≤ i ≤ d we have +��σ2 +i − ˆσ2 +i +�� ≤ δ∥X∥2. +Proof :By Weyl’s inequality we have +��σ2 +i − ˆσ2 +i +�� ≤ +���M − XX⊤��� ≤ δ∥X∥2 +as desired. +□ +A.3 +Optimization +Lemma A.5. Suppose that a smooth function f ∈ Rm �→ R with minimum value f∗ > −∞ +satisfies the following conditions with some ϵ > 0: +(1). lim∥x∥→+∞ f(x) = +∞. +(2). There exists an open subset S ⊂ Rm such that the set S∗ of global minima of f is contained +in S, and for all stationary points x of f in Rm − S, we have f(x) − f∗ ≥ 2ϵ. Moreover, +we also have f(x) − f∗ ≥ 2ϵ on ∂S. +Then we have +{x ∈ Rm : f(x) − f∗ ≤ ϵ} ⊂ S. +Proof : Let x∗ be the minimizer of f on Rm − S. By condition (1) we can deduce that x∗ +always exists. Moreover, since any local minimizer of a function defined on a compact set must +either be a stationary point or lie on the boundary of its domain, we can see that either x∗ ∈ ∂S +or ∇f(x∗) = 0 holds. By condition (2), either cases would imply that f(x∗) − f∗ ≥ 2ϵ, as +desired. +□ +Lemma A.6. Let {xk}, {yk} ⊂ Rn be two sequences generated by xk+1 = xk −µ∇f(xk) and +yk+1 = yk − µ∇f(yk). Suppose that ∥xk∥ ≤ B and ∥yk∥ ≤ B for all k and f is L-smooth in +{x ∈ Rn : ∥x∥ ≤ B}, then we have +∥xk − yk∥ ≤ (1 + µL)k ∥x0 − y0∥ . +22 + +Proof : The update rule implies that +∥xk+1 − yk+1∥ = ∥xk − yk − µ∇f(xk) + µ∇f(yk)∥ +≤ ∥xk − yk∥ + µ ∥∇f(xk) − f(yk)∥ +≤ (1 + µL)∥xk − yk∥ +which yields the desired inequality. +□ +A.4 +Proof for Proposition 3.2 +Proposition 3.2. Suppose that all entries of ¯U ∈ Rd׈r are independently drawn from N +� +0, 1 +ˆr +� +and ρ = ϵ +√ +ˆr−√ˆr∧r∗−1 +√ +ˆr +≥ +ϵ +2r∗ , then σmin +� +V ⊤ +Xs ¯U +� +≥ ρ holds for all 1 ≤ s ≤ ˆr ∧ r∗ with +probability at least 1 − ˆr +� +Cϵ + e−cˆr� +, where c, C > 0 are universal constants. +Proposition 3.2 immediately follows from the following result: +Proposition A.2. (Restatement of Rudelson & Vershynin, 2009, Theorem 1.1) Let A be an N ×n +random matrix, N ≥ n, whose elements are independent copies of a mean zero sub-gaussian +random variable with unit variance. Then, for every ε > 0, we have P +� +sn(A) ≤ ε( +√ +N − √n − 1) +� +≤ +(Cε)N−n+1 + e−cN where C, c > 0 depend (polynomially) only on the sub-Gaussian moment. +Now we can complete the proof of Proposition 3.2. Note that the entries of U ∈ Rd׈r are +independently drawn from N(0, 1 +ˆr) and VXs ∈ Rd×s is an orthonormal matrix. We write V + +Xs ∈ +Rdˆr×sˆr as a block diagonal matrix with ˆr copies of VXs on the diagonal, and vec(U) ∈ Rdˆr be +a vector formed by the concatenation of the columns of U. Then V + +Xs is still orthonormal, and +vec(U) ∼ N(0, 1 +ˆrI). Since multivariate Gaussian distributions are invariant under orthonor- +mal transformations, we deduce that (V + +Xs)⊤vec(U) ∼ N(0, 1 +ˆrI). Equivalently, the entries of +V ⊤ +XsU are i.i.d. N(0, 1 +ˆr). +The matrix +√ +ˆrV ⊤ +XsU satisfies all the conditions in Proposition A.2. Thus, with probability +at least 1 − (Cϵ)ˆr−s+1 − e−cˆr, we have σmin( +√ +ˆrV ⊤ +XsU) ≥ ϵ( +√ +ˆr − √s − 1), or equivalently +σmin(V ⊤ +XsU) ≥ ϵ +√ +ˆr−√s−1 +√ +ˆr +. Finally, the conclusion follows from a union bound: +P +� +∃1 ≤ s ≤ ˆr ∧ r∗ s.t. σmin +� +V ⊤ +XsU +� +< ϵ +2ˆr +� +≤ +ˆr∧r∗ +� +s=1 +P +� +σmin +� +V ⊤ +XsU +� +< ϵ +√ +ˆr − √s − 1 +√r +� +≤ +ˆr∧r∗ +� +s=1 +� +e−cˆr + (Cϵ)ˆr−s+1� +≤ r +� +e−cˆr + Cϵ +� +. +(15) +A.5 +Procrustes Distance +Procrustes distance is introduced in Section 3.3. The following characterization of the optimal +R in Definition 3.3 is known in the literature (see e.g. Tu et al., 2016, Section 5.2.1) but we +provide a proof for completeness. +23 + +Lemma A.7. Let U1, U2 ∈ Rd×r where r ≤ d. Then for any orthogonal matrix R ∈ Rr×r +that minimizes ∥U1 − U2R∥F (i.e., any orthogonal R s.t. ∥U1 − U2R∥F = dist(U1, U2)), +U ⊤ +1 U2R is a symmetric positive semi-definite matrix. +Proof : We only need to consider the case when U ⊤ +2 U1 ̸= 0. Observe that +∥U1 − U2R∥2 +F = ∥U1∥2 +F + ∥U2R∥2 +F − 2 tr +� +R⊤U ⊤ +2 U1 +� += ∥U1∥2 +F + ∥U2∥2 +F − 2 tr +� +R⊤U ⊤ +2 U1 +� +. +Let AΣB⊤ be the SVD of U ⊤ +2 U1, where A⊤A = I, B⊤B = I and Σ ≻ 0. Then +tr +� +R⊤U ⊤ +2 U1 +� += tr +� +B⊤R⊤AΣ +� +≤ +���B⊤R⊤A +��� tr (Σ) = tr (Σ) , +where the final step is due to orthogonality of B⊤R⊤A ∈ Rs×s, and equality holds if and only +if B⊤R⊤A = I. Let C = R⊤A. Let bi, ci ∈ Rd be the i-th column of B and C respectively, +then B⊤C = I implies that b⊤ +i ci = 1. Note that ∥bi∥2 = ∥ci∥2 = 1, so we must have bi = ci +for all i, i.e., B = C = R⊤A. Therefore, U ⊤ +1 U2R = BΣA⊤R = BΣB⊤, which implies +that U ⊤ +1 U2R is symmetric and positive semi-definite. +□ +B +Main idea for the proof of Theorem 4.1 +In this section, we briefly introduce our main ideas for proving Theorem 4.1. Motivated by +St¨oger & Soltanolkotabi (2021), we decompose the matrix Ut into a parallel component and an +orthogonal component. Specifically, we write +Ut = +UtWtW ⊤ +t +� +�� +� +parallel component ++ UtWt,⊥W ⊤ +t,⊥ +� +�� +� +orthogonal component +, +(16) +where Wt := WV ⊤ +XsUt ∈ Rˆr×s is the matrix consisting of the right singular vectors of V ⊤ +XsUt +(Definition 3.1) and Wt,⊥ ∈ Rˆr×(ˆr−s) is an orthogonal complement of Wt. Our goal is to prove +that at some time t, we have V ⊤ +Xs +� +UtU ⊤ +t − XsX⊤ +s +� +≈ 0 and ∥UtWt,⊥∥ ≈ 0. As we will see +later, these imply that +��UtU ⊤ +t − XsX⊤ +s +�� ≈ 0. In the remaining part of this section we give a +heuristic explanation for considering (16). +Additional Notations. +Let VXs,⊥ ∈ Rd×(d−s) be an orthogonal complement of VXs ∈ Rd×s. +Let Σs = diag(σ1, . . . , σs) and Σs,⊥ = diag(σs+1, . . . , σr, 0, · · · , 0) ∈ R(d−s)×(d−s). We use +∆t := (A∗A − I)(XX⊤ − UtU ⊤ +t ) to denote the vector consisting of measurement errors for +XX⊤ − UtU ⊤ +t . +24 + +B.1 +Heuristic explanations of the decomposition +A simple and intuitive approach for showing the implicit low rank bias is to directly analyze the +growth of V ⊤ +XsUt versus V ⊤ +Xs,⊥Ut. Ideally, the former grows faster than the latter, so that GD +only learns the components in Xs. +By the update rule of GD (3), +V ⊤ +Xs,⊥Ut+1 = V ⊤ +Xs,⊥ +� +I + µA∗A(XX⊤ − UtU ⊤ +t ) +� +Ut += V ⊤ +Xs,⊥ +� +I + µXX⊤ − µUtU ⊤ +t +� +Ut +� +�� +� +=:Gt,1 ++µ V ⊤ +Xs,⊥∆tUt +� +�� +� +=:Gt,2 += Gt,1 + µGt,2. +For the first term Gt,1, we have +Gt,1 = (I + µΣ2 +s,⊥)V ⊤ +Xs,⊥Ut − µV ⊤ +Xs,⊥UtU ⊤ +t Ut += (I + µΣ2 +s,⊥)V ⊤ +Xs,⊥Ut(I − µUtU ⊤ +t ) + O(µ2), +where the last term O(µ2) is negligible when µ is sufficiently small. Since ∥Σs,⊥∥ = σs+1, the +spectral norm of Gt,1 can be bounded by +∥Gt,1∥ ≤ ∥I + µΣ2 +s,⊥∥ · ∥V ⊤ +Xs,⊥Ut∥ · ∥I − µUtU ⊤ +t ∥ + O(µ2) +≤ (1 + µσ2 +s+1)∥V ⊤ +Xs,⊥Ut∥ + O(µ2). +However, the main difference with the full-observation case (Jiang et al., 2022) is the second +term Gt,2 := V ⊤ +Xs,⊥∆tUt. Since the measurement errors ∆t are small but arbitrary, it is hard +to compare this term with V ⊤ +Xs,⊥Ut+1. As a result, we cannot directly bound the growth of +∥V ⊤ +Xs,⊥Ut∥. +However, the aforementioned problem disappears if we turn to bound the growth of ∥V ⊤ +Xs,⊥Ut+1Wt,⊥∥. +To see this, first we deduce the following by repeatedly using V ⊤ +XsUtWt,⊥ = 0 due to the defi- +nition of Wt,⊥. +Gt,1Wt,⊥ = V ⊤ +Xs,⊥ +� +I + µXX⊤ − µUtU ⊤ +t +� +UtWt,⊥ += V ⊤ +Xs,⊥(I + µXX⊤)UtWt,⊥ − µV ⊤ +Xs,⊥UtU ⊤ +t UtWt,⊥ += (I + µΣ2 +s,⊥)V ⊤ +Xs,⊥UtWt,⊥ − µV ⊤ +Xs,⊥Ut(WtW ⊤ +t + Wt,⊥W ⊤ +t,⊥)U ⊤ +t UtWt,⊥ += (I + µΣ2 +s,⊥)V ⊤ +Xs,⊥UtWt,⊥(I − µW ⊤ +t,⊥U ⊤ +t UtWt,⊥) +− µV ⊤ +Xs,⊥UtWtW ⊤ +t U ⊤ +t UtWt,⊥ + O(µ2), +Gt,2Wt,⊥ = V ⊤ +Xs,⊥∆tUtWt,⊥ = V ⊤ +Xs,⊥∆tVXs,⊥V ⊤ +Xs,⊥UtWt,⊥, +25 + +So we have the following recursion: +V ⊤ +Xs,⊥Ut+1Wt,⊥ = (I + µΣ2 +s,⊥ + µV ⊤ +Xs,⊥∆tVXs,⊥)V ⊤ +Xs,⊥UtWt,⊥(I − µW ⊤ +t,⊥U ⊤ +t UtWt,⊥) +− µV ⊤ +Xs,⊥UtWtW ⊤ +t U ⊤ +t UtWt,⊥ + O(µ2), +We further note that +V ⊤ +Xs,⊥Ut+1Wt+1,⊥ = V ⊤ +Xs,⊥Ut+1WtW ⊤ +t Wt+1,⊥ + V ⊤ +Xs,⊥Ut+1Wt,⊥W ⊤ +t,⊥Wt+1,⊥, +(17) +which establishes the relationship between V ⊤ +Xs,⊥Ut+1Wt,⊥ and V ⊤ +Xs,⊥Ut+1Wt+1,⊥. To com- +plete the proof we need to prove the following: +• The minimal eigenvalue of the parallel component UtWtW ⊤ +t +grows at a linear rate with +speed strictly faster than σs+1. +• The term +���V ⊤ +Xs,⊥VUtWt +��� ≪ 1, which implies that the first term in (17) is negligible. +C +Proof of Lemma 4.1 +In this section, we give the full proof of Lemma 4.1, with some additional technical lemmas left +to Appendix D. +Lemma 4.1. Under Assumptions 3.1 and 3.2, there exists ˆT s +α > 0 for all α > 0 and 1 ≤ s ≤ ˆr ∧ +r∗ such that limα→0 max1≤t≤ ˆT sα σs+1(Uα,t) = 0. Furthermore, it holds that +���U ˆT sαU ⊤ +ˆT sα − Z∗ +s +��� +F = +O +� +κ3r∗δ∥X∥2� +. +Appendices C.1 and C.2 are devoted to analyzing the spectral phase and parallel improvement +phase, respectively. Appendix C.3 uses induction to characterize the low-rank GD trajectory in +the parallel improvement phase. In Appendix C.4 we study the refinement phase, which allows +us to derive Lemma 4.1. +C.1 +The spectral phase +Starting from a small initialization U0 = α ¯U, α ≪ 1, we first enter the spectral phase where +GD behaves similar to power iteration. As in St¨oger & Soltanolkotabi (2021), we refer to this +phase as the spectral phase. Specifically, we have in the spectral phase that +Ut+1 = +� +I + µ (A∗A) (XX⊤ − UtU ⊤ +t ) +� +Ut ≈ +� +I + µ (A∗A) (XX⊤) +� +Ut. +The approximation holds with high accuracy as long as ∥Ut∥ ≪ 1. Moreover we have M := +(A∗A) (XX⊤) ≈ XX⊤ by the RIP condition; when δ is sufficiently small, we can still ensure +a positive eigen-gap of M. As a result, with small initialization Ut would become approximately +aligned with the top eigenvector u1 of M. Since ∥M −XX⊤∥ = O(δ√r∗) by Proposition A.1, +we have ∥u1 − v1∥ = O(δ√r∗) so that ∥V ⊤ +XsVUtWt∥ = O(δ√r∗). This proves the base case +for the induction. +26 + +Formally, we define M = A∗A(XX⊤), Kt = (I + µM)t and U sp +t += KtU0. Suppose +that M = �rank(M) +i=1 +ˆσ2 +i ˆviˆv⊤ +i is the spectral decomposition of M where {ˆσi}i≥1 is sorted in +non-increasing order. We additionally define Ms = �min{s,rank(M)} +i=1 +ˆσ2 +i ˆviˆv⊤ +i . By Lemma A.4 +and δ√r∗ ≤ 10−3κ by Assumption 3.1, we have ˆσ2 +s ≥ σ2 +s − 0.01τ and ˆσ2 +s+1 ≤ σ2 +s+1 + 0.01τ, +where we recall that τ = mins∈[r∗] +� +σ2 +s − σ2 +s+1 +� +> 0. Additionally, let Lt be the span of the +top-s left singular vectors of Ut. Recall that Assumption 3.2 is made on the initialization. Let +t⋆ := min +� +i ∈ N : +��U sp +i−1 − Ui−1 +�� > +��U sp +i−1 +��� +, +the following lemma bounds the error of approximating Ut via U sp +t : +Lemma C.1. (St¨oger & Soltanolkotabi, 2021, Lemma 8.1) Suppose that A satisfies the rank-1 +RIP with constant δ1. For all integers t such that 1 ≤ t ≤ t⋆ it holds that +∥Et∥ = +��Ut − U sp +t +�� ≤ 4ˆσ−2 +1 α3r∗ (1 + δ1) +� +1 + µˆσ2 +1 +�3t . +(18) +We can derive the following lower bound on t∗ from Lemma C.1. +Corollary C.1. We have +t∗ ≥ +log α−1 + 1 +2 log +ρˆσ2 +1 +4(1+δ1)r∗ +log +� +1 + µˆσ2 +1 +� +. +Proof : By Lemma C.1 we have +∥Et∥ ≤ 4ˆσ−2 +1 α3r∗ (1 + δ1) +� +1 + µˆσ2 +1 +�3t . +for all t ≤ t∗. On the other hand, we have +∥U sp +t ∥ = α +��(I + µM)t ¯U +�� +≥ α(1 + µˆσ2 +1)t ���ˆv1ˆv⊤ +1 ¯U +��� +≥ +� +1 + µˆσ2 +1 +�t αρ. +Thus, it follows from ∥Et∗∥ ≥ ∥U sp +t∗ ∥ that +� +1 + µˆσ2 +1 +�t∗ +≥ +� +ρˆσ2 +1 +4(1 + δ1)r∗ +· α−1 ⇒ t∗ ≥ +log α−1 + 1 +2 log +ρˆσ2 +1 +4(1+δ1)r∗ +log +� +1 + µˆσ2 +1 +� +as desired. +□ +Note that a trivial bound for the rank-1 RIP constant is δ1 ≤ δ. We can now show that for +small t, GD can be viewed as approximate power iteration. +Lemma C.2. There exists a time +t = T sp +α := +2 log α−1 + log +ρˆσ2 +1 +4r∗(1+δ) +3 log(1 + µ ˆσ2 +1) − log(1 + µˆσ2 +s+1) +≤ t∗ +27 + +such that +�����Ut − +s +� +i=1 +α(1 + µˆσ2 +i )tˆviˆv⊤ +i ¯U +����� ≤ C1 · αγ +where γ = 1 − +2 log(1+µˆσ2 +s+1) +3 log(1+µˆσ2 +1)−log(1+µˆσ2 +s+1) and C1 = C1(X, ¯U) is a constant that only depends +on X and ¯U. +Proof : It’s easy to check that T sp +α ≤ t∗ by applying Corollary C.1. +We consider the following decomposition: +�����Ut − +s +� +i=1 +α(1 + µˆσ2 +i )tˆviˆv⊤ +i ¯U +����� ≤ +��Ut − U sp +t +�� + +�����U sp +t +− +s +� +i=1 +α(1 + µˆσ2 +i )tˆviˆv⊤ +i ¯U +����� . +When t ≤ t∗, the first term can be bounded as +∥Et∥ ≤ 4ˆσ−2 +1 α3r∗ (1 + δ) +� +1 + µˆσ2 +1 +�3t . +For the second term we have +�����U sp +t +− +s +� +i=1 +α(1 + µˆσ2 +i )tˆviˆv⊤ +i U +����� ≤ +����� +r∗ +� +i=s+1 +α(1 + µˆσ2 +i )tˆviˆv⊤ +i U +����� ≤ α +� +1 + µˆσ2 +s+1 +�t . +In particular, the definition of T sp +α implies that +�����Ut − +s +� +i=1 +α(1 + µˆσ2 +i )tˆviˆv⊤ +i U +����� ≤ 2 +� +ρˆσ2 +1 +4r∗(1 + δ) +� 1−γ +2 +αγ +≤ 2 max +� +1, +ρˆσ2 +1 +4r∗(1 + δ) +� +αγ +≤ max +� +2, ρσ2 +1 +r∗ +� +� +�� +� +:=C1 +αγ. +as desired. +□ +We conclude this section with the following lemma, which states that initially the parallel +component UtWt would grow much faster than the noise term, and would become well-aligned +with Xs. +Lemma C.3 (Lemma 5.1, formal version). There exists positive constants C2 = C2(X, ¯U) and +C3 = C3(X, ¯U) such that the following inequalities hold for t = T sp +α when α ∈ +� +0, +� +ρ +10C1(X, ¯U) +�10κ� +: +28 + +∥Ut∥ ≤ ∥X∥ +(19a) +σmin (UtWt) ≥ C2 · α +1− +2 log(1+µˆσ2s) +3 log(1+µˆσ2 +1)−log(1+µˆσ2 +s+1) +(19b) +∥UtWt,⊥∥ ≤ C3 · α +1− +2 log(1+µˆσ2 +s+1) +3 log(1+µˆσ2 +1)−log(1+µˆσ2 +s+1) +(19c) +���V ⊤ +Xs,⊥VUtWt +��� ≤ 200δ +(19d) +Proof : We prove this lemma by applying Corollary D.1 to t = T sp +α +defined in the previous +lemma. +The inequality (19a) can be directly verified by using Lemma C.2: +∥Ut∥ ≤ α +� +1 + µˆσ2 +1 +�t + αγ ≤ +� +1 + +�C1(X, ¯U)ˆσ2 +1 +4r∗(1 + δ) +� 1 +3 � +· αγ/3 ≤ ˆC1(X, ¯U) · αγ/3∥X∥. +where ˆC1(X, ¯U) = 1 + +� +C1(X, ¯U)∥X∥2 +2r∗(1+δ) +� 1 +3 (the constant C1 is defined in the previous lemma). +The last inequality holds when α is sufficiently small. For the remaining inequalities, we first +verify that the assumption in Corollary D.1: +ασs(Kt) > 10 (ασs+1(Kt) + ∥Et∥) . +(20) +By definition of Kt, we can see that for α ≤ +� +ρ +10C1 +�10κ +, +ασs+1(Kt) + ∥Et∥ ≤ α +� +1 + µˆσ2 +s+1 +�t + 4ˆσ−2 +1 α3r∗(1 + δ) +� +1 + µˆσ2 +1 +�3t +≤ C1(X, ¯U) · αγ ≤ 0.1ρα +1− +2 log(1+µˆσ2s) +3 log(1+µˆσ2 +1)−log(1+µˆσ2 +s+1) +≤ 0.1ασs(Kt) +where ∥ET sp +α ∥ is bounded in the previous lemma. Hence (20) holds. Let L be the span of top-s +29 + +eigenvectors of M, then by Corollary D.1, at t = T sp +α we have +σs (UtWt) ⩾ 0.4ασs (Kt) σmin +� +V ⊤ +L ¯U +� +≥ 0.1αρ +� +1 + µˆσ2 +s +�t += 0.1ρ +� +ρˆσ2 +1 +4r∗(1 + δ) +� +log(1+µˆσ2s) +3 log(1+µˆσ2 +1)−log(1+µˆσ2 +s+1) α +1− +2 log(1+µˆσ2s) +3 log(1+µˆσ1)−log(1+µˆσ2 +s+1) +≥ 0.1ρ +�ρσ2 +1 +8r∗ +� +1 +10κ +� +�� +� +:=C2(X, ¯U) +α +1− +2 log(1+µˆσ2s) +3 log(1+µˆσ1)−log(1+µˆσ2 +s+1) +∥UtWt,⊥∥ ⩽ 2ασ2 +s+1 (Kt) + ∥Et∥ +≤ 2C1(X, ¯U) +� +�� +� +:=C3(X, ¯U) +α +1− +2 log(1+µˆσ2 +s+1) +3 log(1+µˆσ2 +1)−log(1+µˆσ2 +s+1) +���V ⊤ +Xs,⊥VUtWt +��� ⩽ 100 +� +δ + ασs+1 (Kt) + ∥Et∥ +αρσs (Kt) +� +≤ 100 +� +�δα +2 log(1+µˆσ2s)−2 log(1+µˆσ2 +s+1) +3 log(1+µˆσ2 +1)−log(1+µˆσ2 +s+1) +� +� +≤ 200δ. +(21) +The conclusion follows. +□ +C.2 +The parallel improvement phase +This subsection is devoted to proving Lemma 5.2 which we recall below. +Lemma 5.2. Suppose that Assumptions 3.1 to 3.3 hold and let c3 = 104κr +1 +2∗ δ. Then for suffi- +ciently small α, the following inequalities hold when T sp +α ≤ t < T pi +α,s: +σmin +� +V ⊤ +XsUt+1 +� +≥ σmin +� +V ⊤ +XsUt+1Wt +� +≥ +� +1 + 0.5µ +� +σ2 +s + σ2 +s+1 +�� +σmin +� +V ⊤ +XsUt +� +, +(13a) +∥Ut+1Wt+1,⊥∥ ≤ +� +1 + µ +� +0.4σ2 +s + 0.6σ2 +s+1 +�� +∥UtWt,⊥∥ , +(13b) +���V ⊤ +Xs,⊥VUt+1Wt+1 +��� ≤ c3, +(13c) +rank(V ⊤ +XsUt+1) = rank(V ⊤ +XsUt+1Wt) = s. +(13d) +30 + +C.2.1 +The parallel component +In the following we bound σmin +� +V ⊤ +XsUt+1Wt +� +. We state our main result of this section in the +lemma below. +Lemma C.4. Suppose that Assumptions 3.1 to 3.3 holds, ∥VX⊥ +s VUtWt∥ ≤ c3 < 10−2κ−1 and +∆t = (A∗A − I) (XX⊤ − UtU ⊤ +t ) satisfies ∥∆t∥ ≤ 0.2κ−1r +− 1 +2 +∗ +∥X∥2, then we have +σmin(V ⊤ +XsUt+1) ≥ σmin(V ⊤ +XsUt+1Wt) +≥ +� +1 + µ +� +σ2 +s − 5c3∥X∥2 − 2∥∆t∥ +� +− 20µ2∥X∥4� � +1 − µσ2 +min(V ⊤ +XsUt) +� +σmin(V ⊤ +XsUt). +Proof : The update rule of GD implies that +V ⊤ +XsUt+1Wt += V ⊤ +Xs +� +I + µ(XsX⊤ +s − UtU ⊤ +t ) + µ∆t +� +UtWt +(22a) += (I + µΣ2 +s)V ⊤ +XsUtWt − µV ⊤ +XsUtU ⊤ +t UtWt + µV ⊤ +Xs∆tUtWt +(22b) += (I + µΣ2 +s)V ⊤ +XsUtWt − µV ⊤ +XsUtU ⊤ +t VXsV ⊤ +XsUtWt − µV ⊤ +X UtU ⊤ +t VXs,⊥V ⊤ +Xs,⊥UtWt ++ µV ⊤ +Xs∆tUtWt += (I + µΣ2 +s)V ⊤ +XsUtWt(I − µW ⊤ +t U ⊤ +t VXsV ⊤ +XsUtWt) + µV ⊤ +Xs∆tUtWt +− µV ⊤ +XsUtU ⊤ +t VXs,⊥V ⊤ +Xs,⊥UtWt + µ2Σ2 +sV ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXsV ⊤ +XsUtWt +(22c) +where (22a) follows from V ⊤ +XsXX⊤ = V ⊤ +XsXsX⊤ +s + V ⊤ +XsXs,⊥X⊤ +s,⊥ and V ⊤ +XsXs,⊥ = +0; (22b) follows from V ⊤ +XsXsX⊤ +s += V ⊤ +XsVXsΣsV ⊤ +Xs = ΣsV ⊤ +Xs, and (22c) follows from +V ⊤ +XsUt = V ⊤ +XsUtWtW ⊤ +t ++ V ⊤ +XsUtWt,⊥W ⊤ +t,⊥ = V ⊤ +XsUtWtW ⊤ +t +by definition of Wt and +Wt,⊥. +We now relate the last three terms in (22c) to V ⊤ +XsUtWt. Since V ⊤ +XsUtWt is invertible by +Assumption 3.2, V ⊤ +XsVUtWt, ΣUtWt and WUtWt are also of full rank, thus we have +UtWt = UtWt(V ⊤ +XsUtWt)−1V ⊤ +XsUtWt += UtWt +� +V ⊤ +XsVUtWtΣUtWtW ⊤ +UtWt +�−1 +V ⊤ +XsUtWt += VUtWt +� +V ⊤ +XsVUtWt +�−1 +V ⊤ +XsUtWt. +(23) +31 + +Plugging (23) into the second and third terms of (22) and re-arranging, we deduce that +V ⊤ +XsUt+1Wt += +� +I + µ(Σ2 +s + P1 + P2) +� +V ⊤ +XsUtWt(I − µW ⊤ +t U ⊤ +t VXsV ⊤ +XsUtWt) ++ µ2 � +Σ2 +s + P1 + P2 +� +V ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXsV ⊤ +XsUtWt += +� +I + µ +� +Σ2 +s + P1 + P2 +� ++ µ2 � +Σ2 +s + P1 + P2 +� +V ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +� +I − µV ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +�−1� +· +V ⊤ +XsUtWt(I − µW ⊤ +t U ⊤ +t VXsV ⊤ +XsUtWt) +(24) +where we use the equation A = (I − µAA⊤)−1A(I − µA⊤A) with A = V ⊤ +XsUtWt (when +µ < +1 +9∥X∥2 , I − µAA⊤ is invertible by Lemma D.3), and +P1 = V ⊤ +XsUtU ⊤ +t VXs,⊥V ⊤ +Xs,⊥VUtWt +� +V ⊤ +XsVUtWt +�−1 +P2 = V ⊤ +Xs∆tVUtWt +� +V ⊤ +XsVUtWt +�−1 +(25) +By assumption we have +σmin +� +V ⊤ +XsVUtWt +� +≥ +� +1 − +���V ⊤ +Xs,⊥VUtWt +��� +2 +≥ 1 +2, +so that +∥P1∥ ≤ +���V ⊤ +XsUtU ⊤ +t VXs,⊥ +��� · +���V ⊤ +Xs,⊥VUtWt +��� · +���� +� +V ⊤ +XsVUtWt +�−1���� ≤ 5c3∥X∥2 ≤ 0.1∥X∥2 +(26) +and by our assumption we have +∥P2∥ ≤ +���� +� +V ⊤ +XsVUtWt +�−1���� · ∥∆t∥ ≤ 2∥∆t∥ ≤ 0.2κ−1r +− 1 +2 +∗ +∥X∥2. +(27) +Moreover, note that ∥Σs∥2 = σ2 +1 = ∥X∥2, and since µ < 10−4∥X∥−2 by Assumption 3.3, we +have +��� +� +I − µV ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +�−1��� < 1.1. Thus +���� +� +Σ2 +s + P1 + P2 +� +V ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +� +I − µV ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +�−1���� ≤ 20∥X∥4. +The equation (25) implies that +σmin(V ⊤ +XsUt+1Wt) +≥ σmin +� +I + µ +� +Σ2 +s + P1 + P2 +� ++ +� +Σ2 +s + P1 + P2 +� +V ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +� +I − µV ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +�−1� +· +σmin +� +V ⊤ +XsUtWt(I − µW ⊤ +t U ⊤ +t VXsV ⊤ +XsUtWt) +� +≥ +� +1 + µσ2 +min(Σs) − µ∥P1∥ − µ∥P2∥ − 20µ2∥X∥4� +σmin(V ⊤ +XsUt) +� +1 − µσ2 +min(V ⊤ +XsUt) +� += +� +1 + µσ2 +s − µ∥P1∥ − µ∥P2∥ − 20µ2∥X∥4� +σmin(V ⊤ +XsUt) +� +1 − µσ2 +min(V ⊤ +XsUt) +� +32 + +Recall that P1 and P2 are bounded in (26) and (27) respectively, so we have that +σmin(V ⊤ +XsUt+1) +≥ σmin(V ⊤ +XsUt+1Wt) +≥ +� +1 + µ +� +σ2 +s − 5c3∥X∥2 − 2∥∆t∥2� +− 20µ2∥X∥4� � +1 − µσ2 +min(V ⊤ +XsUt) +� +σmin(V ⊤ +XsUt). +The conclusion follows. +□ +The corollaries below immediately follow from Lemma C.4. +Corollary C.2. Under the conditions in Lemma C.4, if σ2 +min(V ⊤ +XsUt) < 0.3κ−1∥X∥2, then we +have +σmin(V ⊤ +XsUt+1) ≥ +� +1 + 0.5µ(σ2 +s + σ2 +s+1) +� +σmin(V ⊤ +XsUt). +Proof : By Lemma C.4 it remains to check that +� +1 + µ +� +σ2 +s − 5c3∥X∥2 − 2∥∆t∥2� +− 20µ2∥X∥4� � +1 − 0.3µκ−1∥X∥2� +≥ 1+0.5µ(σ2 +s+σ2 +s+1). +Indeed, recall from the conditions of Lemma C.4 that 5c3∥X∥2 ≤ 0.01κ−1∥X∥2 ≤ 0.01(σ2 +s − +σ2 +s+1) and similarly ∥∆t∥ ≤ 0.005(σ2 +s − σ2 +s+1) and µ2∥X∥4 ≤ 10−4κ−1µ∥X∥2 ≤ 10−4(σ2 +s − +σ2 +s+1), so that +� +1 + µ +� +σ2 +s − 5c3∥X∥2 − 2∥∆t∥2� +− 20µ2∥X∥4� � +1 − 0.3κ−1∥X∥2� +≥ +� +1 + µ(0.9σ2 +s + 0.1σ2 +s+1) +� � +1 − 0.3µ(σ2 +s − σ2 +s+1) +� += 1 + µ(0.6σ2 +s + 0.4σ2 +s+1) − µ2∥X∥2(σ2 +s − σ2 +s+1) +≥ 1 + 0.5µ(σ2 +s + σ2 +s+1) +as desired. +□ +Corollary C.3. Under the conditions in Lemma C.4, if +σ2 +min(V ⊤ +XsUt) ≤ σ2 +s − µσ4 +s − 5c3∥X∥2 − 2∥∆t∥2 − 20µ2∥X∥4, +(28) +then we have that σmin(V ⊤ +XsUt+1) ≥ σmin(V ⊤ +XsUt). +Proof : A sufficient condition for σmin(V ⊤ +XsUt+1) ≥ σmin(V ⊤ +XsUt) to hold is that +� +1 + µ +� +σ2 +s − 5c3∥X∥2 − 2∥∆t∥2� +− 20µ2∥X∥4� � +1 − µσ2 +min(V ⊤ +XsUt) +� +⇐ σ2 +s − 5c3∥X∥2 − 2∥∆t∥2 − 20µ∥X∥2 − σmin(V ⊤ +XsUt) − µσ2 +sσmin(V ⊤ +XsUt) ≥ 0. +When (28) holds, we have +σ2 +s − 5c3∥X∥2 − 2∥∆t∥2 − 20µ∥X∥2 − σmin(V ⊤ +XsUt) +≥ µσ4 +s ≥ µσ2 +sσmin(V ⊤ +XsUt) +as desired. +□ +33 + +C.2.2 +The orthogonal component +In this section we turn to analyze the noise term.The main result of this section is presented in +the following: +Lemma C.5. Suppose that Assumptions 3.1 to 3.3 hold, V ⊤ +XsUt+1Wt ∈ Rs×r is of full rank, +∥VXs,⊥VUtWt∥ ≤ c3 < 10−2κ−1 and ∥∆t∥ ≤ c3∥X∥2, then we have +∥Ut+1Wt+1,⊥∥ ≤ +� +1 + µσ2 +s+1 + 30µ∥X∥2c3 + 0.1µ2∥X∥4� +∥UtWt,⊥∥ . +Proof : By the definition of Wt,⊥, we have V ⊤ +XsUtWt,⊥ = 0, thus ∥UtWt,⊥∥ = +���V ⊤ +Xs,⊥UtWt,⊥ +���. +The latter can be decomposed as follows: +V ⊤ +Xs,⊥Ut+1Wt+1,⊥ = V ⊤ +Xs,⊥Ut+1WtW ⊤ +t Wt+1,⊥ +� +�� +� +=(a) ++ V ⊤ +Xs,⊥Ut+1Wt,⊥W ⊤ +t,⊥Wt+1,⊥ +� +�� +� +=(b) +. +In the following, we are going to show that the term (a) is bounded by c · µ where c is a small +constant, while (b) grows linearly with a slow speed. +Bounding summand (a). Since +0 = V ⊤ +XsUt+1Wt+1,⊥ = V ⊤ +XsUt+1WtW ⊤ +t Wt+1,⊥ + V ⊤ +XsUt+1Wt,⊥W ⊤ +t,⊥Wt+1,⊥ +by definition, we have +W ⊤ +t Wt+1,⊥ = − +� +V ⊤ +XsUt+1Wt +�−1 +V ⊤ +XsUt+1Wt,⊥W ⊤ +t,⊥Wt+1,⊥. +(29) +Thus the summand (a) can be rewritten as follows: +V ⊤ +Xs,⊥Ut+1WtW ⊤ +t Wt+1,⊥ += −V ⊤ +Xs,⊥Ut+1Wt +� +V ⊤ +XsUt+1Wt +�−1 +V ⊤ +XsUt+1Wt,⊥W ⊤ +t,⊥Wt+1,⊥ +(30a) += −V ⊤ +Xs,⊥Ut+1Wt +� +V ⊤ +XsVUt+1WtΣUt+1WtWUt+1Wt +�−1 +V ⊤ +XsUt+1Wt,⊥W ⊤ +t,⊥Wt+1,⊥ += −V ⊤ +Xs,⊥VUt+1Wt +� +V ⊤ +XsVUt+1Wt +�−1 +V ⊤ +XsUt+1Wt,⊥W ⊤ +t,⊥Wt+1,⊥ +(30b) += −V ⊤ +Xs,⊥VUt+1Wt +� +V ⊤ +XsVUt+1Wt +�−1 +V ⊤ +Xs +� +I + µA∗A +� +XX⊤ − UtU ⊤ +t +�� +UtWt,⊥W ⊤ +t,⊥Wt+1,⊥ += −µV ⊤ +Xs,⊥VUt+1Wt +� +V ⊤ +XsVUt+1Wt +�−1 +V ⊤ +Xs +�� +XX⊤ − UtU ⊤ +t +� ++ ∆t +� +UtWt,⊥W ⊤ +t,⊥Wt+1,⊥ +(30c) += µV ⊤ +Xs,⊥VUt+1Wt +� +V ⊤ +XsVUt+1Wt +�−1 +V ⊤ +Xs +� +UtU ⊤ +t − ∆t +� +UtWt,⊥W ⊤ +t,⊥Wt+1,⊥ += µV ⊤ +Xs,⊥VUt+1Wt +� +V ⊤ +XsVUt+1Wt +�−1 +M1V ⊤ +Xs,⊥UtWt,⊥W ⊤ +t,⊥Wt+1,⊥, +34 + +where M1 = V ⊤ +Xs +� +UtU ⊤ +t VXs,⊥ − ∆tVXs,⊥ +� +. In (30), (30a) follows from (29), (30b) holds +since ΣUt+1WtW ⊤ +Ut+1Wt ∈ Rs×s is invertible, and in (30c) we use V ⊤ +XsUtWt,⊥ = 0. It follows +that +∥(a)∥ ≤ µ +���V ⊤ +Xs,⊥VUt+1Wt +��� · +���� +� +V ⊤ +XsVUt+1Wt +�−1���� ∥M1∥ +���V ⊤ +Xs,⊥UtWt,⊥ +��� . +(31) +By Lemma D.4 we have +���V ⊤ +Xs,⊥VUt+1Wt +��� ≤ 0.01, which implies that +���� +� +V ⊤ +XsVUt+1Wt +�−1���� = σ−1 +min +� +V ⊤ +XsVUt+1Wt +� += +� +1 − +���V ⊤ +Xs,⊥VUt+1Wt +��� +2�− 1 +2 +≤ 1.1. (32) +Lastly, we bound M1 as follows: +∥M1∥ ≤ +���V ⊤ +XsUtU ⊤ +t VXs,⊥ +��� + +���(A∗A − I) +� +XX⊤ − UtU ⊤ +t +���� +≤ +���V ⊤ +XsUtWt +��� · +���V ⊤ +Xs,⊥UtWt +��� + 10−3κ−1c3∥X∥2 +≤ 10∥X∥2c3. +(33) +where the second inequality follows from our assumption on +��(A∗A − I) +� +XX⊤ − UtU ⊤ +t +���. +Combining (31), (32) and (33) yields +∥(a)∥ ≤ 20µ∥X∥2c3∥UtWt,⊥∥. +Bounding summand (b). +This is the main component in the error term. We’ll see that +although this term can grow exponentially fast, the growth speed is slower than the minimal +eigenvalue of the parallel component. +We have +V ⊤ +Xs,⊥Ut+1Wt,⊥ += V ⊤ +Xs,⊥ +� +I + µ(XX⊤ − UtU ⊤ +t ) + µ (A∗A − I) +� +XX⊤ − UtU ⊤ +t +�� +UtWt,⊥ +(34a) += +� +� +�I + µΣ2 +s,⊥ − µV ⊤ +Xs,⊥UtU ⊤ +t VXs,⊥ + µ V ⊤ +Xs,⊥∆tVXs,⊥ +� +�� +� +=:M2 +� +� +� V ⊤ +Xs,⊥UtWt,⊥ +(34b) += +� +I + µΣ2 +s,⊥ − µV ⊤ +Xs,⊥UtWtW ⊤ +t U ⊤ +t VXs,⊥ + µM2 +� +V ⊤ +Xs,⊥UtWt,⊥ +� +I − µW ⊤ +t,⊥U ⊤ +t UtWt,⊥ +� +(34c) ++ µ2 � +Σ2 +s,⊥ − V ⊤ +Xs,⊥UtWtW ⊤ +t U ⊤ +t VXs,⊥ + M2 +� +V ⊤ +Xs,⊥UtWt,⊥W ⊤ +t,⊥U ⊤ +t UtWt,⊥ +(34d) +35 + +where we recall that Σ2 +s,⊥ = diag +� +σ2 +s+1, · · · , σ2 +r, 0, · · · , 0 +� +∈ R(d−s)×(d−s). In (34), (34a) +follows from the update rule of GD, (34b) is obtained from V ⊤ +Xs,⊥XX⊤ = Σ2 +s,⊥V ⊤ +Xs,⊥ and +UtWt,⊥ = VXsV ⊤ +XsUtWt,⊥ + VXs,⊥V ⊤ +Xs,⊥UtWt,⊥ = VXs,⊥V ⊤ +Xs,⊥UtWt,⊥, and lastly in +(34d) we use +V ⊤ +Xs,⊥UtU ⊤ +t VXs,⊥V ⊤ +Xs,⊥UtWt,⊥ += V ⊤ +Xs,⊥UtWtW ⊤ +t U ⊤ +t VXs,⊥V ⊤ +Xs,⊥UtWt,⊥ + V ⊤ +Xs,⊥UtWt,⊥W ⊤ +t,⊥U ⊤ +t VXs,⊥V ⊤ +Xs,⊥UtWt,⊥ += V ⊤ +Xs,⊥UtWtW ⊤ +t U ⊤ +t VXs,⊥V ⊤ +Xs,⊥UtWt,⊥ + V ⊤ +Xs,⊥UtWt,⊥W ⊤ +t,⊥U ⊤ +t UtWt,⊥. +It follows that +���V ⊤ +Xs,⊥Ut+1Wt,⊥ +��� +≤ +����I − µV ⊤ +Xs,⊥UtWtW ⊤ +t U ⊤ +t VXs,⊥ +��� + µ∥Σs,⊥∥2 + µ∥M2∥ +� +∥V ⊤ +Xs,⊥UtWt,⊥∥ +� +I − µ∥V ⊤ +Xs,⊥UtWt,⊥∥2� ++ µ2 ∥UtWt,⊥∥3 � +σ2 +s+1 + ∥Ut∥2 + 10−3κ−1c3∥X∥2� +≤ +� +1 + µσ2 +s+1 + µ∥∆t∥ +� +∥UtWt,⊥∥ +� +1 − µ∥UtWt,⊥∥2� ++ 0.1µ2∥X∥4 ∥UtWt,⊥∥ +≤ ∥UtWt,⊥∥ +� +1 + µσ2 +s+1 + µc3∥X∥2 + 0.1µ2∥X∥4� +To summarize, we have +∥Ut+1Wt+1,⊥∥ ≤ +� +1 + µσ2 +s+1 + 30µ∥X∥2c3 + 0.1µ2∥X∥4� +∥UtWt,⊥∥ +as desired. +□ +To bound the growth speed of the orthogonal component, we need to show that the quantity +���V ⊤ +Xs,⊥VUtWt +��� remains small. The following lemma serves to complete an induction step from +t to t + 1: +Lemma C.6. Suppose V ⊤ +XsUt is of full rank, ∥VXs,⊥VUtWt∥ ≤ c3 and ∥UtWt,⊥∥ ≤ min {σmin(UtWt), c4} +with max +� +c3, c4∥X∥−1� +≤ 10−2κ−1, and ∆t = (A∗A−I)(XX⊤−UtU ⊤ +t ) satisfies ∥∆t∥ ≤ +10−3κ−1c3∥X∥2 and µ ≤ 10−4κ−1∥X∥−2c3, then we have ∥VXs,⊥VUt+1Wt+1∥ ≤ c3. +Proof : Let Mt = A∗A(XX⊤ − UtU ⊤ +t ), so the update rule of GD implies that +Ut+1Wt+1 = (I + µMt)UtWt+1 += (I + µMt) +� +UtWtW ⊤ +t Wt+1 + UtWt,⊥W ⊤ +t,⊥Wt+1 +� += (I + µMt) +� +VUtWtV ⊤ +UtW UtWtW ⊤ +t Wt+1 + UtWt,⊥W ⊤ +t,⊥Wt+1 +� += (I + µMt)(I + P )VUtWt +� +�� +� +:=H +V ⊤ +UtWtUtWtW ⊤ +t Wt+1, +where +P = UtWt,⊥W ⊤ +t,⊥Wt+1 +� +V ⊤ +UtWtUtWtW ⊤ +t Wt+1 +�−1 +V ⊤ +UtWt +36 + +and V ⊤ +UtWtUtWtW ⊤ +t Wt+1 is invertible since V ⊤ +UtWtUtWt is invertible by our assumption +that V ⊤ +XsUt is of full rank and rank(UtWt) ≥ rank(V ⊤ +XsUtWt) = rank(V ⊤ +XsUt) = s, and +W ⊤ +t Wt+1 is invertible by Lemma D.6. Indeed, Lemma D.6 implies that σmin +� +W ⊤ +t Wt+1 +� +≥ 1 +2 +by our condition on µ. +The key observation here is that because the (square) matrix V ⊤ +UtWtUtWtW ⊤ +t Wt+1 is in- +vertible, so that the column space of Ut+1Wt+1 is the same as that of H. Following the line +of proof of St¨oger & Soltanolkotabi, 2021, Lemma 9.3 (for completeness, we provide details in +Lemma D.7), we deduce that +���V ⊤ +Xs,⊥VUt+1Wt+1 +��� = +���V ⊤ +Xs,⊥VHW ⊤ +H +��� +≤ +����V ⊤ +Xs,⊥ +�� +I + B − 1 +2VUtWtV ⊤ +UtWt +� +B + B⊤�� +VUtWt − BVUtWtV ⊤ +UtWt +� +B + B⊤� +VUtWt + D +����� +≤ +����V ⊤ +Xs,⊥ +� +I + B − 1 +2VUtWtV ⊤ +UtWt +� +B + B⊤�� +VUtWt +���� + 2∥B∥2 + ∥D∥ +(35) +where B = (I + µMt)(I + P ) − I and ∥D∥ ≤ 100∥B∥2. By assumption we have +∥P ∥ ≤ +∥UtWt,⊥∥ ∥Wt,⊥Wt+1∥ +σmin(UtWt)σmin(W ⊤ +t Wt+1) +≤ 2 ∥Wt,⊥Wt+1∥ , +so that +���B − µ(XX⊤ − UtU ⊤ +t ) +��� +≤ µ∥Mt − (XX⊤ − UtU ⊤ +t )∥ + ∥P ∥ + µ∥Mt∥∥P ∥ +≤ µ ∥∆t∥ + 2 ∥Wt,⊥Wt+1∥ + 4µ∥X∥2 ∥Wt,⊥Wt+1∥ +≤ µ ∥∆t∥ + 6 ∥Wt,⊥Wt+1∥ +≤ 18µ +� +10µ∥X∥3 + c4 +� +c3∥X∥ + 7µ ∥∆t∥ +≤ 18µ +� +10µ∥X∥3 + c4 +� +c3∥X∥ + 0.01µκ−1c3∥X∥2 +(36) +where we use Lemma D.6 to bound +���W ⊤ +t,⊥Wt+1 +���. Let B1 = µ(XX⊤ − UtU ⊤ +t ) and R1 = +V ⊤ +Xs,⊥ +� +I + B1 − VUtWtV ⊤ +UtWtB1 +� +VUtWt, then we have +R1 = V ⊤ +Xs,⊥ +� +I + µ +� +I − VUtWtV ⊤ +UtWt +� � +XX⊤ − UtU ⊤ +t +�� +VUtWt += +� +I + µΣ2 +s,⊥ +� +V ⊤ +Xs,⊥VUtWt +� +I − µV ⊤ +UtWtXX⊤VUtWt +� +− µV ⊤ +Xs,⊥ +� +I − VUtWtV ⊤ +UtWt +� +UtWt,⊥W ⊤ +t,⊥U ⊤ +t VXs,⊥V ⊤ +Xs,⊥VUtWt ++ µ2Σ2 +s,⊥V ⊤ +Xs,⊥VUtWtV ⊤ +UtWtXX⊤VUtWt. +(37) +37 + +By Weyl’s inequality (cf. Lemma A.2) and our assumption on c3, +σmin +� +V ⊤ +UtWtXX⊤VUtWt +� +≥ σmin +� +V ⊤ +UtWtXsX⊤ +s VUtWt +� +− +���V ⊤ +UtWtXs,⊥X⊤ +s,⊥VUtWt +��� +2 +≥ σmin +� +V ⊤ +UtWtXsX⊤ +s VUtWt +� +− σ2 +s+1 +���V ⊤ +Xs,⊥VUtWt +��� +2 +≥ σ2 +s +���V ⊤ +UtWtVXs +��� +2 +− σ2 +s+1c2 +3 += σ2 +s − (σ2 +s + σ2 +s+1)c2 +3 > 1 +2 +� +σ2 +s + σ2 +s+1 +� +. +So we have +∥R1∥ ≤ +� +1 − µ +2 (σ2 +s − σ2 +s+1) +� ���V ⊤ +Xs,⊥VUtWt +��� + µc3c2 +4 + µ2∥X∥4. +It thus follows from (35) that +���V ⊤ +X⊥ +s VUt+1Wt+1 +��� +≤ ∥R1∥ + 2∥B − B1∥ + 102∥B∥2 +≤ +� +1 − µ +2 (σ2 +s − σ2 +s+1) +� ���V ⊤ +Xs,⊥VUtWt +��� + 40µc3c4∥X∥ + 0.02µκ−1c3∥X∥2 + 103µ2∥X∥4. +Since +���V ⊤ +Xs,⊥VUtWt +��� ≤ c3, it follows from our assumption on c3, c4 and µ that +���V ⊤ +X⊥ +s VUt+1Wt+1 +��� ≤ +c3 as well, which concludes the proof. +□ +C.3 +Induction +Let +T pi +α,s = min +� +t ⩾ 0 : σ2 +min +� +V ⊤ +XsUα,t+1 +� +> 0.3κ−1∥X∥2� +. +where pi stands for the parallel improvement phase. In this section, we show that when T sp +α ≤ +t < T pi +α,s, the parallel component grows exponentially faster than the orthogonal component. We +prove this via induction and the base case is already shown in Lemma C.3. +Lemma C.7 (Lemma 5.2, detailed version). Suppose that Assumptions 3.1 to 3.3 hold and let +c3 = 104κ√r∗δ, c4 ≤ 10−3κ−1∥X∥. Then the following holds for all T sp +α ≤ t < T pi +α,s as long +as α ≤ C4(X, ¯U) = +� +κ C2(X, ¯U)2 +C3(X, ¯U)2 +�−2κ +is sufficiently small: +σmin +� +V ⊤ +XsUt+1 +� +≥ σmin +� +V ⊤ +XsUt+1Wt +� +≥ +� +1 + 0.5µ +� +σ2 +s + σ2 +s+1 +�� +σmin +� +V ⊤ +XsUα,t +� +(38a) +∥Ut+1Wt+1,⊥∥ ≤ min +�� +1 + µ +� +0.4σ2 +s + 0.6σ2 +s+1 +�� +∥UtWt,⊥∥ , c4 +� +(38b) +���V ⊤ +Xs,⊥VUt+1Wt+1 +��� ≤ c3. +(38c) +rank(V ⊤ +XsUt+1) = rank(V ⊤ +XsUt+1Wt) = s. +(38d) +38 + +Proof : The base case t = T pi +α,s is already proved in (19). Now suppose that the lemma holds for +t, we now show that it holds for t + 1 as well. +To begin with, we bound the term ∥∆t∥ as follows: +∥∆t∥ = +���(A∗A − I)(XX⊤ − UtU ⊤ +t ) +��� +≤ +���(A∗A − I)(XX⊤ − UtWtW ⊤ +t U ⊤ +t ) +��� + +���(A∗A − I)UtWt,⊥W ⊤ +t,⊥U ⊤ +t +��� +≤ 10δ√r∗∥X∥2 + δ +���UtWt,⊥W ⊤ +t,⊥U ⊤ +t +��� +∗ +≤ 10δ√r∗∥X∥2 + δd ∥UtWt,⊥∥2 +(39) +where in the second inequality we use Proposition A.1 and Lemma A.1 and in the third inequality +we use ∥A∥∗ ≤ +√ +d∥A∥, ∀A ∈ Rd×d and the induction hypotheses. +By induction hypothesis, there exists a constant ˆC4(X, ¯U) = C2(X, ¯U) +C3(X, ¯U) (see Lemma C.3) +such that +σmin +� +V ⊤ +XsUt +� +∥UtWt,⊥∥ +≥ σmin +� +V ⊤ +XsUT sp +α +� +��UT sp +α WT sp +α ,⊥ +�� ≥ ˆC4 · α−γs +(40) +where +γs = 2 +� +log +� +1 + µˆσ2 +s +� +− log +� +1 + µˆσ2 +s+1 +�� +3 log +� +1 + µˆσ2 +1 +� +− log +� +1 + µˆσ2 +s+1 +� +≥ 1 +4κ. +Since we must have σ2 +min +� +V ⊤ +XsUt +� +≤ 0.3κ−1∥X∥2 by definition of T pi +α,s, it follows that ∥UtWt,⊥∥2 ≤ +10κ∥X∥2 ˆC2 +4α +1 +2κ , so for α ≤ ( ˆC−2 +4 κ)−2κ, ∥∆t∥ ≤ 11δ√r∗∥X∥2 holds. +The above inequality combined with our assumption on δ implies that the conditions on ∥∆t∥ +in Lemmas C.4 to C.6 hold. We now show that (38a) to (38d) hold for t + 1, which completes +the induction step. +First, since t < T pi +α,s, we have σmin +� +V ⊤ +XsUt+1 +� +≤ κ−1∥X∥2. Moreover, the induction +hypothesis implies that +���V ⊤ +Xs,⊥VUt−1Wt−1 +��� ≤ c3 and that V ⊤ +XsUα,t is of full rank. Thus the +conditions of Corollary C.2 are all satisfied, and we deduce that (38a) holds. +Second, the assumptions on c3, c4 and δ, combined with Lemma C.5, immediately implies +∥Ut+1Wt+1,⊥∥ ≤ +� +1 + µ +� +0.4σ2 +s + 0.6σ2 +s+1 +�� +∥UtWt,⊥∥ . +As a result, similar to (40) we observe that +σmin +� +V ⊤ +XsUt+1 +� +∥Ut+1Wt+1,⊥∥ ≥ σmin +� +V ⊤ +XsUT sp +α +� +��UT sp +α WT sp +α ,⊥ +�� ≥ ˆC4 · α− 1 +4κ . +Since σmin +� +V ⊤ +XsUt+1 +� +≤ ∥X∥, when α is sufficiently small we must have that ∥Ut+1Wt+1,⊥∥ ≤ +c4. +Finally, Lemma C.6 implies that (38c) is true, and (38d) follows from our application of +Lemma C.4. This concludes the proof. +□ +39 + +C.4 +The refinement phase and concluding the proof of Lemma 4.1 +We have shown that the parallel component σmin +� +V ⊤ +XsUt+1 +� +grows exponentially faster than the +orthogonal component ∥UtWt,⊥∥. In this section, we characterize the GD dynamics after T pi +α,s. +We begin with the following lemma, which is straightforward from the proof of Lemma C.7. +Lemma C.8 (Corollary 5.1, formal version). Under the conditions of Lemma 5.2, the following +inequality holds when α ≤ C4(X, ¯U): +���UT pi +α,sWT pi +α,s,⊥ +��� ≤ C5(X, ¯U) · α +1 +4κ +where C5 = +√ +10κ∥X∥C2(X, ¯U) +C3(X, ¯U). +The following lemma states that in a certain time period after T pi +α,s, the parallel and orthogonal +components still behave similarly to the second (parallel improvement) phase. +Lemma C.9. Under the conditions in Lemma C.7, there exists �tα,s ≥ +1 +log(1+µσ2s) log +� +10−4c3∥X∥2 +√ +dκC5 +α− 1 +4κ +� += +Θ +� +log α−1� +when α → 0 such that when 0 ≤ t − T pi +α,s ≤ �tα,s, we have +σmin +� +V ⊤ +XsUt +� +≥ σmin (UtWt) ≥ 0.3κ−1∥X∥2, +(41a) +∥UtWt∥ ≤ +� +1 + µ(0.4σ2 +s + 0.6σ2 +s+1) +�t−T pi +α,s ���UT pi +α,sWT pi +α,s +��� , +(41b) +∥VXs,⊥VUtWt∥ ≤ c3. +(41c) +Proof : We choose +�tα,s = min +� +t ≥ 0 : ∥Ut+1Wt+1,⊥∥2 ≤ c5 +� +(42) +where +c5 = 10−4d− 1 +2 κ−1c3∥X∥2 +(43) +We prove (41) by induction. The proof follows the idea of Lemma C.7, except that we need to +bound ∥∆t∥ in each induction step. Concretely, suppose that (41) holds at time t, then +∥∆t∥ = +���(A∗A − I)(XX⊤ − UtU ⊤ +t ) +��� +≤ +���(A∗A − I)(XX⊤ − UtWtW ⊤ +t U ⊤ +t ) +��� + +���(A∗A − I)UtWt,⊥W ⊤ +t,⊥U ⊤ +t +��� +≤ 10δ√r∗∥X∥2 + δ +���UtWt,⊥W ⊤ +t,⊥U ⊤ +t +��� +∗ +≤ 10δ√r∗∥X∥2 + δc5 +√ +d ≤ 0.02κ−1c3∥X∥2 +(44) +where we used the definition of c5 in the last step. As a result, we can apply the conclusion of +Lemmas C.4 to C.6 which implies that (41) holds for t + 1. Finally, combining Lemma C.8 and +(41b) yields �tα,s = Θ +� +log 1 +α +� +. +□ +We now present the main result of this section: +40 + +Lemma C.10. Suppose that 0 ≤ t − T π +α,s ≤ �tα,s, +���V ⊤ +Xs,⊥VUtWt +��� ≤ c3 and the conditions in +Lemma C.7 hold, then we have +���V ⊤ +Xs(XX⊤ − Ut+1U ⊤ +t+1) +��� +F +≤ +� +1 − 1 +2µτ +� ���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� +F + 20µ∥X∥4 (δ + 5c3) + 2000µ2√r∗∥X∥6. +where we recall that τ = min1≤s≤ˆr∧r∗(σ2 +s − σ2 +s+1) > 0. +Proof : +Recall that Mt = A∗A +� +XX⊤ − UtU ⊤ +t +� +. The update of GD implies that +XX⊤ − Ut+1U ⊤ +t+1 += XX⊤ − (I + µMt)UtU ⊤ +t (I + µMt) += +� +I − µUtU ⊤ +t +� � +XX⊤ − UtU ⊤ +t +� � +I − µUtU ⊤ +t +� +� +�� +� +=(i) ++µ ∆tUtU ⊤ +t +� +�� +� +=(ii) ++ µUtU ⊤ +t ∆t +� +�� +� +=(iii) ++µ2 (Et,1 + Et,2) , +where Et,1 = −UtU ⊤ +t +� +XX⊤ − UtU ⊤ +t +� +UtU ⊤ +t +and Et,2 = −MtUtU ⊤ +t Mt. Since ∥Ut∥ ≤ +3∥X∥ by Lemma D.3, and +��Mt − (XX⊤ − UtU ⊤ +t ) +�� = ∥∆t∥ ≤ ∥X∥2 which is shown in +(44), we have +���V ⊤ +XsEt,1 +��� +F = +���V ⊤ +XsUtU ⊤ +t +� +XX⊤ − UtU ⊤ +t +� +UtU ⊤ +t +��� +F +≤ √r∗ +���V ⊤ +XsUtUt +� +XX⊤ − UtU ⊤ +t +� +UtU ⊤ +t +��� +2 +≤ 103√r∗∥X∥6 +and +���V ⊤ +XsEt,2 +��� +F = +���V ⊤ +Xs +� +(A∗A) +� +XX⊤ − UtU ⊤ +t +�� +UtU ⊤ +t +� +(A∗A) +� +XX⊤ − UtU ⊤ +t +����� +F +≤ √r∗ +��� +� +(A∗A) +� +XX⊤ − UtU ⊤ +t +�� +UtU ⊤ +t +� +(A∗A) +� +XX⊤ − UtU ⊤ +t +����� +≤ 103√r∗∥X∥6. +Note that we would like to bound +��V ⊤ +Xs +� +XX⊤ − Ut+1U ⊤ +t+1 +��� +F . We deal with the above three +41 + +terms separately. For the first term, we have +���V ⊤ +Xs +� +I − µUtU ⊤ +t +� � +XX⊤ − UtU ⊤ +t +� +(I − µUtUt) +��� +F += +���V ⊤ +Xs +� +I − µUtU ⊤ +t +� +VXsV ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +� � +I − µUtU ⊤ +t +���� +F ++ +���V ⊤ +Xs +� +I − µUtU ⊤ +t +� +VXs,⊥V ⊤ +Xs,⊥ +� +XX⊤ − UtU ⊤ +t +� � +I − µUtU ⊤ +t +���� +F +≤ +���I − µV ⊤ +XsUtU ⊤ +t VXs +��� +���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� + µ +���V ⊤ +XsUtU ⊤ +t VXs,⊥V ⊤ +Xs,⊥ +� +XX⊤ − UtU ⊤ +t +���� +F +(45a) +≤ +� +1 − µσ2 +min(UtWt)σ2 +min +� +V ⊤ +XsVUtWt +�� ���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� +F + 100µ∥X∥4c3 +(45b) +≤ +� +1 − 1 +2µτ +� ���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� +F + 100µ∥X∥4c3, +(45c) +where in (45a) we use +��I − µUtU ⊤ +t +�� ≤ 1, (45b) follows from +σmin +� +V ⊤ +XsUtU ⊤ +t VXs +� += σmin +� +V ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs +� +≥ σmin +� +V ⊤ +XsUtWt +�2 +≥ σ2 +min(UtWt)σ2 +min +� +V ⊤ +XsVUtWt +� +and +���V ⊤ +XsUtU ⊤ +t VXs,⊥ +��� = +���V ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs,⊥ +��� ≤ ∥Ut∥2 ���V ⊤ +Xs,⊥UtWt +��� ≤ c3∥Ut∥2, +and lastly (45c) is obtained from +σ2 +min +� +V ⊤ +XsVUtWt +� +≥ 1 − +���V ⊤ +Xs,⊥VUtWt +��� +2 +≥ 1 − c2 +3. +For the second and the third terms, we have +���∆tUtU ⊤ +t + UtU ⊤ +t ∆t +��� ≤ 0.1κc3∥X∥4 +(46) +where we use the estimate in (44). Combining (45) and (46) yields +���V ⊤ +Xs(XX⊤ − Ut+1U ⊤ +t+1) +��� +F +≤ +� +1 − 1 +2µτ +� ���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� +F + 200µ∥X∥4c3 + 110µ2√r∗∥X∥6. +□ +To apply the result of Lemma 5.3, we need to verify that ∥VXs,⊥VUtWt∥ ≤ c3 still holds +when t ≥ T pi +α,s. In fact, this is true as long as t − T pi +α,s ≤ O +� +log 1 +α +� +. +42 + +Lemma C.11. Under the conditions in Lemma C.7, if +T pi +α,s ≤ t ≤ T pi +α,s + +γs log +c4 +C5(X, ¯U) · log 1 +α +log (1 + µσ2s) +=: T re +α,s, +then ∥Ut+1Wt+1,⊥∥ ≤ (1 + µσ2 +s) ∥UtWt,⊥∥ and +���V ⊤ +Xs,⊥VUtWt +��� ≤ c3. As a consequence, we +have ∥UtWt,⊥∥ ≤ (1 + µσ2 +s)t−T pi +α,sC5(X, ¯U) · αγs ≤ c4. +Proof : The proof is basically the same as that of Lemma C.7 and we only provide a sketch here. +We induct on t. The base case t = T pi +α,s is already proved in Lemma C.7. Suppose that the +lemma holds for t − 1 with t < T re +α,s, then the choice of T re +α,s combined with Lemma C.5 imply +that +∥UtWt,⊥∥ ≤ +� +1 + µσ2 +s +� +∥Ut−1Wt−1,⊥∥ ≤ +� +1 + µσ2 +s +�t−T pi +α,s ���UT pi +α,sWT pi +α,s,⊥ +��� . +Since we have +���UT pi +α,sWT pi +α,s,⊥ +��� ≤ C5(X, ¯U) · αγs by Lemma C.8, the choice of T re +α,s implies +that ∥UtWt,⊥∥ ≤ c4. The bound +���V ⊤ +Xs,⊥VUtWt +��� ≤ c3 then follows from Lemma C.6. +□ +We will only use a weaker version of this lemma, namely that the bounds holds for all T pi +α,s ≤ +t ≤ T ft +α,s. When α is sufficiently small, this can be directly derived from Lemmas C.10 and C.11 +since ˜tα,s, T re +α,s − T pi +α,s = Θ +� +log 1 +α +� +. Specifically, we have proven Lemma 5.3 in the main text: +Lemma 5.3. Suppose that T pi +α,s ≤ t ≤ T ft +α,s and all the conditions in Lemma 5.2 hold, then we +have +���V ⊤ +Xs(XX⊤ − Ut+1U ⊤ +t+1) +��� +F +≤ +� +1 − 1 +2µτ +� ���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� +F + 20µ∥X∥4 (δ + 5c3) + 2000µ2√r∗∥X∥6. +Moreover, it holds that ∥Ut+1Wt+1,⊥∥ ≤ (1 + σ2 +s)∥UtWt,⊥∥ and +���VX⊥ +s VUtWt +��� ≤ c3. +We are now ready to present our first main result, which states that with small initialization, +GD would visit the O(δ)-neighborhood of the rank-s minimizer of the full observation loss i.e. +XsX⊤ +s . +Theorem C.1 (Restatement of Lemma 4.1). Under Assumptions 3.1 to 3.3, if the initialization +scale α is sufficiently small, then for all 1 ≤ s ≤ ˆr ∧ r∗ there exists a time T ft +α,s ∈ Z+ (where ft +stands for fitting the ground-truth) such that +���XsX⊤ +s − UT pi +α,sU ⊤ +T pi +α,s +��� +F ≤ 107κ3r∗∥X∥2δ. +43 + +Proof : First, observe that for all t ≥ 0, +���XsX⊤ +s − UtU ⊤ +t +��� +F ≤ +��� +� +XsX⊤ +s − UtU ⊤ +t +� +VXsV ⊤ +Xs +��� +F + +���UtU ⊤ +t VX⊥ +s V ⊤ +X⊥ +s +��� +F +≤ +��� +� +XsX⊤ +s − UtU ⊤ +t +� +VXsV ⊤ +Xs +��� +F + +���V ⊤ +X⊥ +s UtU ⊤ +t VX⊥ +s +��� +F +≤ +���V ⊤ +Xs +� +XsX⊤ +s − UtU ⊤ +t +���� +F + √r∗ +���V ⊤ +X⊥ +s UtWt +��� +2 ++ +√ +d +���V ⊤ +X⊥ +s UtWt,⊥ +��� +2 +≤ +���V ⊤ +Xs +� +XsX⊤ +s − UtU ⊤ +t +���� + 9√r∗∥X∥2 ∥VXs,⊥VUtWt∥2 + +√ +d∥UtWt,⊥∥2. +(47) +We set c3 = 103κ√r∗δ and +T ft +α,s = T pi +α,s − log +� +10−2∥X∥−2τc−1 +3 +� +log +� +1 − 1 +2µτ +� +, +(48) +where we recall that τ = κ−1∥X∥2, then for small α we have T ft +α,s ≤ T pi +α,s + �tα,s (defined +in Lemma C.9). Hence for T pi +α,s ≤ t < T ft +α,s we always have ∥VXs,⊥VUtWt∥ ≤ c3. By +Lemma C.10 and the choice of c3 and δ, we have for T pi +α,s ≤ t < T ft +α,s that +���V ⊤ +Xs +� +XX⊤ − Ut+1U ⊤ +t+1 +���� +F ≤ +� +1 − 1 +2µτ +� ���V ⊤ +Xs +� +XX⊤ − UtU ⊤ +t +���� +F +30µ∥X∥4√r∗c3 +which implies that for t = T ft +α,s, +���V ⊤ +Xs +� +XX⊤ − UTsU ⊤ +Ts +���� +F ≤ 80κ∥X∥2√r∗c3. +Meanwhile, by Lemma C.9 we have ∥UtWt,⊥∥ ≤ c5 (c5 is defined in (43)) and +���V ⊤ +Xs,⊥VUtWt +��� ≤ +c3 at t = T ft +α,s. Plugging into (47) yields +���XsX⊤ +s − UT ft +α,sU ⊤ +T ft +α,s +��� +F ≤ 80κ∥X∥2√r∗c3 + 9∥X∥2c2 +3 +√r∗ + c2 +5 +√ +d. +By definition of c3 and c5 we deduce that +���XsX⊤ +s − UT ft +α,sU ⊤ +T ft +α,s +��� +F ≤ 102τ −2∥X∥6√r∗c3 ≤ +105κ3r∗∥X∥2δ, as desired. +□ +Corollary C.4. There exists a constant +C6(X, ¯U) = C5(X, ¯U) · (1 + µσ2 +s)T ft +α,s−T pi +α,s +(49) +such that +max +0≤t≤T ft +α,s +∥UtWt,⊥∥ ≤ C1 · α +1 +4κ . +Proof : The case of t ≤ T pi +α,s directly follows from Lemma C.8. For t > T pi +α,s, we know from +Lemma C.9 that +∥UtWt,⊥∥ ≤ +���UT pi +α,sWT pi +α,s,⊥ +��� · +� +1 + µσ2 +s +�T ft +α,s−T pi +α,s . +By (48), the second term is a constant independent of α, so the conclusion follows. +□ +44 + +D +Auxiliary results for proving Lemma 4.1 +This section contains a collection of auxiliary results that are used in the previous section. +D.1 +The spectral phase +In the section, we provide auxiliary results for the analysis in the spectral phase. +Recall that Kt = (I + µM)t and Ut = U sp +t ++ Et = KtU0 + Et and U0 = α ¯U +with ∥ ¯U∥ = 1. Also recall that M = �rank(M) +i=1 +ˆλiˆviˆv⊤ +i ; we additionally define Ms = +�min{s,rank(M)} +i=1 +ˆλiˆviˆv⊤ +i . Similarly, let Lt be the span of the top-s left singular vectors of Ut. +The following lemma shows that power iteration would result in large eigengap of Ut. +Lemma D.1. Let ˆρ = σmin +� +V ⊤ +Ms ¯U +� +> 0, then the following three inequalities hold, given that +the denominator of the third is positive. +σs(Ut) ≥ α +� +ˆρσs +� +ˆZt +� +− σs+1 +� +ˆZt +�� +− ∥Et∥ , +(50a) +σs+1(Ut) ≤ ασs+1 +� +ˆZt +� ++ ∥Et∥ , +(50b) +���V ⊤ +M⊥ +s VLt +��� ≤ +ασs+1 +� +ˆZt +� ++ ∥Et∥ +αˆρσs +� +ˆZt +� +− 2 +� +ασs+1 +� +ˆZt +� ++ ∥Et∥ +�. +(50c) +Proof : By Weyl’s inequality we have +σs+1(Ut) = σs+1 +� +(1 + µM)tU0 +� ++ ∥Et∥ += ασs+1 +� +(1 + µM)t ¯U +� ++ ∥Et∥ +≤ ασs+1 +� +(1 + µMs)t ¯U +� ++ α +��� +(1 + µM)t − (1 + µMs)t� ¯U +�� + ∥Et∥ +≤ α(1 + µˆλs+1)t + ∥Et∥. +Thus (50b) holds. Similarly, +σs(Ut) ≥ ασs +� +NtVMsV ⊤ +Ms ¯U +� +− α(1 + µˆλs+1)t − ∥Et∥ +≥ ασs (NtVMs) σmin +� +V ⊤ +Ms ¯U +� +− α(1 + µˆλs+1)t − ∥Et∥ +≥ αˆρ(1 + µˆλs)t − α(1 + µˆλs+1)t − ∥Et∥. +Finally, note that we can write +α(1 + µMs)t ¯U = VMs (1 + µΣMs)tV ⊤ +Ms ¯U +� +�� +� +invertible +, +so that the subspace spanned by the left singular vectors of α(1 + µMs)t ¯U coincides with the +column span of VMs. Since Lt is the span of top-s left singular vectors of Ut, we apply Wedin’s +sin theorem (Wedin, 1972) and deduce (50c). +□ +45 + +The next lemma relates the quantities studied in Lemma D.1 with those that are needed in +the induction. The proof is the same as St¨oger & Soltanolkotabi, 2021, Lemma 8.4, so we omit +it here. +Lemma D.2. Suppose that +���V ⊤ +Xs,⊥VLt +��� ≤ 0.1 for some t ≥ 1. Then it holds that +σs (UtWt) ≥ 1 +2σs (Ut) , +(51a) +���V ⊤ +Xs,⊥VUtWt +��� ≤ 10 +���V ⊤ +Xs,⊥VLt +��� , +(51b) +∥UtWt,⊥∥ ≤ 2σs+1 (Ut) . +(51c) +Combining the above two lemmas, we directly obtain the following corollary: +Corollary D.1. Suppose that ασs(Kt) > 10 (ασs+1(Kt) + ∥Et∥), then we have that +σs (UtWt) ≥ 0.4ασr⋆ (Kt) σmin +� +V ⊤ +L ¯U +� +∥UtWt,⊥∥ ≤ 2 (ασs+1 (Kt) + ∥Et∥) +���V ⊤ +Xs,⊥VUtWt +��� ≤ 100 +� +δ + ασs+1 (Kt) + ∥Et∥ +αˆρσs (Kt) +� +(52) +D.2 +The parallel improvement phase +In the section, we provide auxiliary results for the analysis in the parallel improvement phase. +Lemma D.3. (St¨oger & Soltanolkotabi, 2021, Lemma 9.4) For sufficiently small µ and δ, sup- +pose that ∥Ut∥ ≤ 3∥X∥, then we also have ∥Ut+1∥ ≤ 3∥X∥. +Lemma D.4. Under the assumptions in Lemma C.5, we have +���V ⊤ +Xs,⊥VUt+1Wt +��� ≤ 2 +� +c3 + 10µ∥X∥2� +≤ 0.01. +Proof : The proof of this lemma is essentially the same as St¨oger & Soltanolkotabi, 2021, +Lemma B.1, and we omit it here. +□ +Lemma D.5. Under the assumptions in Lemma C.6, we have +σmin +� +V ⊤ +XsUt+1 +� +≥ 1 +2σmin(UtWt). +Proof : We have +σmin +� +V ⊤ +XsUt+1 +� +≥ σmin +� +V ⊤ +XsUt+1Wt +� += σmin +� +V ⊤ +Xs (I + µMt) UtWt +� +≥ σmin +� +V ⊤ +Xs (I + µMt) VUtWt +� +· σmin +� +V ⊤ +UtWtUtWt +� +≥ +� +σmin +� +V ⊤ +XsVUtWt +� +− µ∥Mt∥ +� +· σmin(UtWt) +≥ +�� +1 − c2 +3 − 10µ∥X∥2 +� +σmin(UtWt) ≥ 1 +2σmin(UtWt) +46 + +where the last step follows from +σmin +� +V ⊤ +XsVUtWt +�2 +≥ 1 − +���V ⊤ +Xs,⊥VUtWt +��� +2 +≥ 1 − c2 +3. +The conclusion follows. +□ +Lemma D.6. Under the assumptions in Lemma C.6, we have +���W ⊤ +t,⊥Wt+1 +��� ≤ 3µ +� +10µ∥X∥2 + c4 +� +c3∥X∥ + µ +���(A∗A − I)(XX⊤ − UtU ⊤ +t ) +��� . +Proof : The proof roughly follows [St¨oger & Soltanolkotabi, 2021, Lemma B.3], but we include +it here for completeness. +Since V ⊤ +XsUt+1 = Vt+1Σt+1Wt+1 and Vt+1Σt+1 ∈ Rs×s is invertible, we have +���W ⊤ +t,⊥Wt+1 +��� = +����W ⊤ +t,⊥U ⊤ +t+1VXs +� +V ⊤ +XsUt+1U ⊤ +t+1VXs +�− 1 +2 +���� . +Since +V ⊤ +XsUt+1Wt,⊥ += V ⊤ +Xs +� +I + µA∗A(XX⊤ − UtU ⊤ +t ) +� +UtWt,⊥ += V ⊤ +Xs +� +I + µ(XX⊤ − UtU ⊤ +t ) +� +UtWt,⊥ + µV ⊤ +Xs∆tUtWt,⊥ += −µV ⊤ +XsUtU ⊤ +t UtWt,⊥ + µV ⊤ +Xs∆tUtWt,⊥ +(53a) += −µV ⊤ +XsUtWtW ⊤ +t U ⊤ +t UtWt,⊥ + µV ⊤ +Xs∆tUtWt,⊥ +(53b) += −µ V ⊤ +XsUtWtW ⊤ +t U ⊤ +t VXs,⊥V ⊤ +Xs,⊥UtWt,⊥ +� +�� +� +=:K1 ++µ V ⊤ +Xs∆tUtWt,⊥ +� +�� +� +:=K2 +(53c) +where (53a) follows from V ⊤ +XsXX⊤UtWt,⊥ = ΣsV ⊤ +XsUtWt,⊥ = 0, and in (53b) and (53c) +we use V ⊤ +XsUtWt,⊥ = 0. +For K1, note that +���� +� +V ⊤ +XsUt+1U ⊤ +t+1VXs +�− 1 +2 V ⊤ +XsUt +���� +≤ +���� +� +V ⊤ +XsUt+1U ⊤ +t+1VXs +�− 1 +2 V ⊤ +XsUt+1 +���� + µ +���� +� +V ⊤ +XsUt+1U ⊤ +t+1VXs +�− 1 +2 V ⊤ +XsA∗A(XX⊤ − UtU ⊤ +t )Ut +���� +≤ 1 + 10µ∥X∥3σ−1 +min +� +V ⊤ +XsUt+1 +� +so that +���� +� +V ⊤ +XsUt+1U ⊤ +t+1VXs +�− 1 +2 K1 +���� ≤ +� +1 + 10µ∥X∥3σ−1 +min +� +V ⊤ +XsUt+1 +�� ���V ⊤ +Xs,⊥UtWt +��� ∥UtWt,⊥∥ . +47 + +Plugging into (53), we deduce that +���W ⊤ +t,⊥Wt+1 +��� +≤ 3µ +� +1 + 10µ∥X∥3σ−1 +min +� +V ⊤ +XsUt+1 +�� ���V ⊤ +Xs,⊥VUtWt +��� ∥X∥ ∥UtWt,⊥∥ ++ µσ−1 +min +� +V ⊤ +XsUt+1 +� +∥UtWt,⊥∥ +���(A∗A − I)(XX⊤ − UtU ⊤ +t ) +��� +≤ 3µ +� +∥UtWt,⊥∥ + 10µ∥X∥3� ���V ⊤ +Xs,⊥VUtWt +��� ∥X∥ ++ µσ−1 +min +� +V ⊤ +XsUt+1 +� +∥UtWt,⊥∥ +���(A∗A − I)(XX⊤ − UtU ⊤ +t ) +��� +≤ 3µ +� +10µ∥X∥2 + c4 +� +c3∥X∥ + µ +���(A∗A − I)(XX⊤ − UtU ⊤ +t ) +��� . +where in the last step we use Lemma D.5 and the induction hypothesis which implies that +σmin(UtWt) ≥ ∥UtWt,⊥∥. +□ +Lemma D.7. The matrix H defined in the proof of Lemma C.6 satisfies the following: +H(H⊤H)− 1 +2 = VUtWt + BVUtWt − 1 +2(I + B)VUtWtV ⊤ +UtWt +� +B + B⊤� +VUtWt − D, +where ∥D∥ ≤ 30∥B∥2. +Proof : By definition of H we have +H(H⊤H)− 1 +2 += (I + µM)(I + P )VUtWt +� +V ⊤ +UtWt +� +I + P ⊤� +(I + µM)2(I + P )VUtWt +�− 1 +2 += (I + B)VUtWt +� +V ⊤ +UtWt +� +I + B⊤ + B + B⊤B +� +VUtWt +�− 1 +2 += (I + B)VUtWt +� +���I + V ⊤ +UtWt +� +B⊤ + B + B⊤B +� +VUtWt +� +�� +� +=:Θ +� +��� +− 1 +2 +. +It follows from (36) and our assumptions on c3 and c4 that +∥B∥ ≤ µ +���XX⊤ − UtU ⊤ +t +��� + 6µ +� +c3c4∥X∥ + 50∥X∥2δ +� +≤ 10µ∥X∥2 + 6µc3 (c4 + 1) ∥X∥ < 0.1 +(note that this step is independent and does not rely on earlier derivations in the proof of Lemma C.6), +so by Taylor’s formula, we have +����(I + Θ)− 1 +2 − I + 1 +2Θ +���� ≤ 3∥Θ∥2. +48 + +Hence, +����H(H⊤H)− 1 +2 − +� +VUtWt + BVUtWt − 1 +2(I + B)VUtWtV ⊤ +UtWt +� +B + B⊤� +VUtWt +����� += +����(I + B)VUtWt +� +(I + Θ)− 1 +2 − I + 1 +2Θ − 1 +2V ⊤ +UtWtB⊤BVUtWt +����� +≤ (1 + ∥B∥) +� +3∥Θ∥2 + 1 +2∥B∥2 +� +< 30∥B∥2 +as desired. +□ +E +Proofs for the Landscape Results in Section 4.1 +In this section, we study the landscape of under-parameterized matrix sensing problem +fs(U) = 1 +2 +���A(UU ⊤ − XX⊤) +��� +2 +2 , +U ∈ Rd×s +Our key result in this section is Lemma 4.3, which states a local RSI condition for the ma- +trix sensing loss. Most existing results only study the landscape of (1) in the exact- and over- +parameterized case. Zhu et al. (2021) have studied the landscape of under-parameterized matrix +factorization problem, but their main focus is the strict-saddle property of the loss. +E.1 +Analysis of the matrix factorization loss +When the measurement satisfies the RIP condition, we can expect that the landscape of fs looks +similar to that of the (under-parameterized) matrix factorization loss: +Fs(U) = 1 +2 +���UU ⊤ − XX⊤��� +2 +F , +U ∈ Rd×s +for some s < ˆr.For this reason, we first look into the landscape of Fs before analyzing fs. +Recall that XX⊤ = �r∗ +i=1 σ2 +i viv⊤ +i . The critical points of Fs(U) is characterized by the +following lemma: +Lemma E.1. U ∈ Rd×s is a critical point of Fs(U) if and only if there exists an orthogonal +matrix R ∈ Rs×s, such that all columns of UR are in {σivi : 1 ≤ i ≤ r∗}. +Proof : Assume WLOG that XX⊤ = diag(σ2 +1, σ2 +2, · · · , σ2 +r, 0, · · · , 0) =: Σ. Let U be a critical +point of Fs, then we have that +� +UU ⊤ − XX⊤� +U = 0. Let W = UU ⊤, then (Σ − W )W = +0. +Since W is symmetric, so is W 2, and we obtain that ΣW is also symmetric. It is then easy +to see that that if Σ = diag (λ1Im1, · · · , λtImt) with λ1 > λ2 > · · · > λt ≥ 0, then W is +also in block-diagonal form: W = diag (W1, W2, · · · , Wt) where Wi ∈ Rmi×mi. For each +1 ≤ i ≤ t, we then have the equation (λiImi − Wi) Wi = 0. Hence, there exists an orthogonal +matrix Ri such that R⊤ +i WiRi is a diagonal matrix where the diagonal entries are either 0 or +49 + +√λi = σi. Let R = diag (R1, R2, · · · , Rt), then R⊤W R is diagonal and its nonzero diagonal +entries form an s-subset of the multi-set {σi : 1 ≤ i ≤ r∗}. The conclusion follows. +□ +In the case of s = 1, the global minimizers of Fs are ±σ1v1, and we can show that Fs is +locally strongly convex around these minimizers. Therefore, we can deduce that f is locally +strongly-convex as well. Since our main focus is on s > 1, we put these details in Appendix G. +When s > 1, Fs(U) is not locally strongly-convex due to rotational invariance: if U is a global +minimizer, then so is UR for any orthogonal matrix R ∈ Rs×s. Instead, we establish a Re- +stricted Secant Inequality for Fs, as shown below. +Lemma E.2. For U ∈ Rd×s, let R be an orthogonal matrix that minimizes ∥U − XsR∥F . +Suppose that dist(U, Xs) ≤ 0.1∥X∥−1τ (where we recall that τ = mins∈[r∗] +� +σ2 +s − σ2 +s+1 +� +is +the eigengap of XX⊤), then we have +⟨∇Fs(U), U − XsR⟩ ≥ 0.1τ · dist2(U, Xs). +Proof : Assume WLOG that R = I. Then by Lemma A.7, U ⊤Xs is symmetric and positive +semi-definite. Let H = U − Xs, then +∇Fs(U) = (UU ⊤ − XX⊤)U += +� +(H + Xs)(H + Xs)⊤ − XX⊤� +(H + Xs). +So we have +⟨∇Fs(U), U − Xs⟩ = +�� +(H + Xs)(H + Xs)⊤ − XX⊤� +(H + Xs), H +� += tr +� +H⊤ � +(H + Xs)(H + Xs)⊤ − XX⊤� +H + H⊤ � +HH⊤ + HX⊤ +s + XsH⊤� +Xs +� +≥ − tr +� +H⊤Xs,⊥X⊤ +s,⊥H +� +− 3∥X∥∥H∥3 +F + tr +� +H⊤HX⊤ +s Xs +� +(54a) +≥ +� +σ2 +s − σ2 +s+1 +� +∥H∥2 +F − 3∥X∥∥H∥3 +F +(54b) +≥ 0.1τ∥H∥2 +F +where in (54a) we use tr +� +(H⊤Xs)2� +≥ 0 (since H⊤Xs is symmetric as noticed in the begin- +ning of the proof), and (54b) is because of +tr +� +H⊤HX⊤ +s Xs +� +≥ σmin +� +X⊤ +s Xs +� +· tr +� +H⊤H +� += σ2 +s∥H∥2 +F +and +tr +� +H⊤Xs,⊥X⊤ +s,⊥H +� += tr +� +H⊤VXs,⊥Σs,⊥V ⊤ +Xs,⊥H +� +≤ ∥Σs,⊥∥ · +���H⊤VXs,⊥ +��� +2 +F ≤ σ2 +s+1∥H∥2 +F . +□ +Corollary E.1. Under the conditions of Lemma E.2, we have ∥∇Fs(U)∥F ≥ 0.1τdist(U, Xs). +50 + +E.2 +Analysis of the matrix sensing loss +The following lemma states that the minimizer of matrix sensing loss is also near-optimal for the +matrix factorization loss. +Lemma E.3. Let Z∗ +s be a best rank-s solution as defined in Definition 1.1, then we have +���Z∗ +s − XX⊤��� +2 +F ≤ +���XsX⊤ +s − XX⊤��� +2 +F + 10δ +���XX⊤��� +2 +F . +Proof : By the RIP property Definition 3.2 we have +���XX⊤ − Z∗ +s +��� +2 +F ≤ (1 − δ)−1 ���A +� +XX⊤ − Z∗ +s +���� +2 +2 +≤ (1 − δ)−1 ���A +� +XX⊤ − XsX⊤ +s +���� +2 +2 +≤ 1 + δ +1 − δ +���XX⊤ − XsX⊤ +s +��� +2 +F +≤ +���XX⊤ − XsX⊤ +s +��� +2 +F + 10δ∥XX⊤∥2 +F , +where the second inequality holds due to Definition 1.1. +□ +We now recall Lemma 4.4. +Lemma 4.4. Under Assumption 3.1, we have dist(U ∗ +s , Xs) ≤ 40δκ∥X∥F for any global mini- +mizer U ∗ +s of fs. Moreover, +��Z∗ +s − XsX⊤ +s +�� +F ≤ 160δκ√r∗∥X∥2. +We prove the statements in this lemma separately in Lemma E.4 and Corollary E.2 below. +Lemma E.4. Suppose that Assumption 3.1 holds. Let U ∗ +s be a global minimizer of fs, then we +have +dist(U ∗ +s , Xs) ≤ 40δκ∥X∥F . +Proof : Define +S = +� +U ∈ Rd×s : dist(U, Xs) < 0.1κ−1∥X∥ +� +. +First we can show that U ∗ +s ∈ S. The main idea is to apply Lemma A.5. Indeed, it is easy to see +that +lim +∥U∥F →+∞ Fs(U) = +∞, +so the condition (1) in Lemma A.5 holds. To check condition (2), we separately analyze the two +cases U ∈ ∂S and U /∈ S. +Firstly, let U ∈ ∂S, i.e., dist2(U, Xs) = 0.1∥X∥−1τ. Assume WLOG that dist(U, Xs) = +∥U − Xs∥F , then by Lemma E.2 we have +Fs(U) − Fs(Xs) = +� 1 +0 +t ⟨∇Fs(tU + (1 − t)Xs), U − Xs⟩ dt +≥ +� 1 +0 +0.1τt2 ∥U − Xs∥2 +F dt +≥ 10−3∥X∥−2τ 3 = 10−3κ−3∥X∥2. +51 + +Secondly, let U /∈ S be a stationary point of fs. Recall that all the stationary points of Fs are +characterized in Lemma E.1, so that for all U /∈ S with ∇Fs(U) = 0, we have +Fs(U) − F ∗ +s ≥ 0.5 +� +σ4 +s − σ4 +s+1 +� +≥ 0.5τ 2. +On the other hand, we know from Lemma E.3 that +Fs(U ∗ +s ) − F ∗ +s ≤ 5δr∗∥X∥4. +(55) +By Assumption 3.1, we have 5δr∗∥X∥4 < 10−3κ−3∥X∥2 < 0.5τ 2, so Lemma A.5 implies that +U ∗ +s ∈ S. +Since ∇fs(U ∗ +s ) = 0, we have A∗A +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +U ∗ +s = 0, so that +∥∇Fs(U ∗ +s )∥F = +��� +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +U ∗ +s +��� +F += +���(A∗A − I) +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +U ∗ +s +��� +F +≤ δ +���XX⊤ − U ∗ +s (U ∗ +s )⊤��� +F ∥U ∗ +s ∥ +≤ 4δ∥X∥ · +���XX⊤��� +F . +From U ∗ +s ∈ S and Corollary E.1 we can deduce that +dist(U ∗ +s , Xs) ≤ 40δτ −1∥X∥2∥X∥F = 40δκ∥X∥F . +□ +Corollary E.2. Suppose that Assumption 3.1 holds, then we have +��Z∗ +s − XsX⊤ +s +�� +F ≤ 80δκ√r∗∥X∥2 +and σmin +� +(U ∗ +s )⊤ U ∗ +s +� +≥ σ2 +s − 80δκ√r∗∥X∥2. +Proof : We assume WLOG that ∥U ∗ +s − Xs∥F = dist(U ∗ +s , Xs) i.e. R = I in Definition 3.3. +By Lemma 4.4, we have that +���U ∗ +s (U ∗ +s )⊤ − XsX⊤ +s +��� +F ≤ 2 max {∥U ∗ +s ∥ , ∥Xs∥} · ∥U ∗ +s − Xs∥F +≤ 160δκ∥X∥∥X∥F ≤ 160δκ√r∗∥X∥2. +which proves the first inequality. Similarly, we have +���(U ∗ +s )⊤ U ∗ +s − X⊤ +s Xs +��� +F ≤ 160δκ√r∗∥X∥2. +Hence σ2 +s − σmin +� +(U ∗ +s )⊤ U ∗ +s +� +≤ +���(U ∗ +s )⊤ U ∗ +s − X⊤ +s Xs +��� ≤ 160δκ√r∗∥X∥2, as desired. +□ +Corollary 4.1. Under Assumption 3.1, we have σmin(U ∗ +s ) ≥ 1 +2σmin(Xs) = 1 +2σs ≥ 1 +2κ− 1 +2 ∥X∥. +Proof : Assumption 3.1 implies that 160δκ√r∗∥X∥2 ≤ 0.1κ−1∥X∥2 ≤ 0.1σ2 +s, so that the +conclusion immediately follows from Corollary E.2. +□ +52 + +Lemma E.5. Under Assumption 3.1, suppose that U, U ∗ +s ∈ Rd×s such that U ∗ +s is a global +minimizer of fs and ∥U − U ∗ +s ∥F = dist(U, U ∗ +s ) ≤ 10−2κ−1∥X∥ (recall that dist is defined in +Definition 3.3), then we have +⟨∇fs(U), U − U ∗ +s ⟩ ≥ 0.1κ−1∥X∥2∥U − U ∗ +s ∥2 +F . +Proof : By Lemma A.7, U ⊤U ∗ +s is symmetric and positive semi-definite. Let H = U − U ∗ +s , +then +∇fs(U) = (A∗A) (UU ⊤ − XX⊤)U += (A∗A) +� +(H + U ∗ +s )(H + U ∗ +s )⊤ − XX⊤� +(H + U ∗ +s ) += +� +(A∗A) +� +HH⊤ + U ∗ +s H⊤ + H (U ∗ +s )⊤�� +(H + U ∗ +s ) − A∗A +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +H +where we use the first-order optimality condition +A∗A +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +U ∗ +s = 0. +Since ∥U ∗ +s ∥ ≤ 2∥X∥ by Lemma 4.4, we may thus deduce that +���∇fs(U) − +�� +HH⊤ + U ∗ +s H⊤ + H (U ∗ +s )⊤� +(H + U ∗ +s ) − +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +H +���� +F +≤ +���(A∗A − I) +� +HH⊤ + U ∗ +s H⊤ + H (U ∗ +s )⊤� +(H + U ∗ +s ) +��� +F + +���(A∗A − I) +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +H +��� +≤ 50δ∥X∥2∥H∥F +Hence +⟨∇fs(U), U − U ∗ +s ⟩ +≥ +�� +HH⊤ + U ∗ +s H⊤ + H (U ∗ +s )⊤� +(H + U ∗ +s ) − +� +XX⊤ − U ∗ +s (U ∗ +s )⊤� +H, H +� +− 50δ∥X∥2∥H∥2 +F +≥ tr +� +H(H + U ∗ +s )⊤(H + U ∗ +s )H⊤ + H⊤U ∗ +s H⊤H + +� +(U ∗ +s )⊤ H +�2 +−H⊤ � +XX⊤ − U ∗ +s (U ∗ +s )⊤� +H +� +− 50δ∥X∥2∥H∥2 +F +≥ +� +σmin +� +(U ∗ +s )⊤ U ∗ +s +� +− +���XX⊤ − U ∗ +s (U ∗ +s )⊤��� − 50δ∥X∥2 − 3∥U ∗ +s ∥∥H∥ − ∥H∥2� +∥H∥2 +F . +By Corollary E.2 we have σmin +� +(U ∗ +s )⊤ U ∗ +s +� +≥ σ2 +s−80δκ∥X∥∥X∥F and +���XX⊤ − U ∗ +s (U ∗ +s )⊤��� ≤ +σ2 +s+1 + 80δκ∥X∥2 +F , so that +⟨∇fs(U), U − U ∗ +s ⟩ ≥ +� +σ2 +s − σ2 +s+1 − 160δκ∥X∥∥X∥F − 50δ∥X∥2 − 3∥U ∗ +s ∥∥H∥ − ∥H∥2� +∥H∥2 +F . +When Assumption 3.1 on δ is satisfied and ∥H∥ ≤ 10−2τ∥X∥−1, the above implies that +⟨∇fs(U), U − U ∗ +s ⟩ ≥ 0.5τ∥H∥2 +F , as desired. +□ +We now prove Lemmas 4.2 and 4.3 to conclude this section. Both results follow immediately +from Lemma E.5. +53 + +Lemma 4.2. Under Assumption 3.1, if U ∗ +s ∈ Rd×s is a global minimizer of fs, then the set of +global minimizers arg min fs is equal to +� +U ∗ +s R : R ∈ Rs×s, R⊤R = I +� +. +Proof : By Lemma E.4 we have that dist(U ∗ +s , X∗ +s ) ≤ 40δκ∥X∥F holds for any U ∗ +s ∈ arg min fs. +Suppose now that U ∗ +s , ˆU ∗ +s ∈ arg min fs such that dist(U ∗ +s , ˆU ∗ +s ) > 0. Then we also have that +dist(U ∗ +s , ˆU ∗ +s ) ≤ dist(U ∗ +s , Xs) + dist(Xs, ˆU ∗ +s ) ≤ 80δκ∥X∥F < 10−2κ−1∥X∥, where the +last inequality follows from Assumption 3.1. Without loss of generality, we can assume that +∥U ∗ +s − ˆU ∗ +s ∥F = dist(U ∗ +s , ˆU ∗ +s ), so that we can apply Lemma E.5 to obtain +� +∇fs( ˆU ∗ +s ), ˆU ∗ +s − U ∗ +s +� +≥ 0.1κ−1∥X∥2∥ ˆU ∗ +s − U ∗ +s ∥2 +F . +However, since ˆU ∗ +s is a global minimizer of fs, we have ∇fs( ˆU ∗ +s ) which is a contradiction. +Thus the global minimizer must be unique under the procrustes distance. +□ +We recall Definition 4.1 which is now guaranteed to be well-defined by Lemma 4.2. +Definition 4.1. For any U ∈ Rd×s, we use Πs(U) to denote the set of closest global minimizers +of fs to U, namely Πs(U) = arg min{∥U − U ∗ +s ∥F : U ∗ +s ∈ arg min fs}. +Equipped with this definition, Lemma E.5 directly translates into Lemma 4.3: +Lemma 4.3 (Restricted Secant Inequality). Under Assumption 3.1, if a matrix U ∈ Rd×s satis- +fies ∥U − U ∗ +s ∥F ≤ 10−2κ−1∥X∥ for some U ∗ +s ∈ Πs(U), then we have +⟨∇fs(U), U − U ∗ +s ⟩ ≥ 0.1κ−1∥X∥2∥U − U ∗ +s ∥2 +F . +(8) +F +Proofs for Theorems 4.1 and 4.2 +In this section, we prove the main theorems based on our key lemmas introduced in Section 4.1. +Based on Lemma 4.3, we first prove the following lemma, which shows that GD initialized +near global minimizers converges linearly. +Lemma F.1. Suppose that Assumptions 3.1 and 3.3 hold. Let { ˆUt}t≥0 be a trajectory of GD +that optimizes fs with step size µ, starting with ˆU0. Also let U ∗ +s be a global minimizer of fs. If +dist( ˆU0, U ∗ +s ) ≤ 10−2κ−1∥X∥, then for all t ≥ 0, +dist2( ˆUt, U ∗ +s ) ≤ (1 − 0.05τµ)t dist2( ˆUα,0, U ∗ +s ). +(56) +Proof : We prove (56) by induction. It is easy to check that (56) holds for t = 0. Now we show +that (56) holds for t + 1 assuming it holds for t. +Since dist( ˆU0, U ∗ +s ) ≤ 10−2κ−1∥X∥, we have +��� ˆU0 +��� ≤ ∥U ∗ +s ∥ + 10−2κ−1∥X∥ ≤ 2∥X∥. +(57) +54 + +Let R be the orthogonal matrix such that U ∗ +s R ∈ Π(Ut), then ∥U − U ∗ +s R∥F = dist(Ut, U ∗ +s ). +We first bound the gradient ∇f( ˆUα,t) as follows: +���∇f( ˆUα,t) +��� +F = +���A∗A +� +XX⊤ − ˆUα,t ˆU ⊤ +α,t +� +ˆUα,t +��� +F +≤ +���A∗A +� +XX⊤ − ˆUα,t ˆU ⊤ +α,t +���� +��� ˆUα,t − U ∗ +s +��� +F + +��� +� +ˆUα,t ˆU ⊤ +α,t − U ∗ +s (U ∗ +s )⊤� +U ∗ +s +��� +F +≤ 20∥X∥2 ��� ˆUα,t − U ∗ +s +��� +F +(58) +where we use (57) and the RIP property (Assumption 3.1). It follows that +dist2( ˆUt+1, U ∗ +s ) ≤ +��� ˆUt+1 − U ∗ +s R +��� +2 +F +(59a) += +��� ˆUt − µ∇f( ˆUα,t) − U ∗ +s R +��� +2 +F += +��� ˆUα,t − U ∗ +s R +��� +2 +F − µ +� +∇f( ˆUα,t), ˆUt − U ∗ +s R +� ++ µ2 ���∇f( ˆUα,t) +��� +2 +F +≤ +� +1 − 0.1τµ + 400∥X∥4µ2� ��� ˆUα,t − U ∗ +s R +��� +2 +F +(59b) +where (59a) follows from the definition of dist, and (59b) is due to Lemma 4.3 and (58). Finally, +(56) follows from Assumption 3.3. +□ +We then note the following proposition, which is straightforward from Lemma C.9 and The- +orem C.1. In the following we use Uα,t to denote the t-th iteration of GD when initialized at +U0 = α ¯U. +Proposition F.1. Suppose that Assumptions 3.1 to 3.3 hold and α is sufficiently small. Then +there exist matrices U lr +α,t for t = −T ft +α,s, −T ft +α,s + 1, · · · , 0 with rank ≤ s (where lr stands for +low rank) and a constant C6 = C6(X, ¯U) (defined in (49)) such that +max +T ft +α,s≤t≤0 +���U lr +α,t − Uα,T ft +α,s+t +��� +F = C6 · α +1 +4κ +where T ft +α,s is defined in Theorem C.1 and moreover +���U lr +α,0 +� +U lr +α,0 +�⊤ − Z∗ +s +��� +F ≤ 2 × 105κ3∥X∥2r∗δ. +where Z∗ +s = U ∗ +s (U ∗ +s )⊤ is the best rank-s solution as defined in Definition 1.1. +Proof : It follows from Corollary C.4 that max1≤t≤T ft +α,s ∥UtWt,⊥∥ ≤ C6(X, ¯U) · α +1 +4κ (recall +that C5 is defined in Lemma C.8 and T ft +α,s is defined in (48)). We choose U lr +α,t = UT ft +α,s+tWT ft +α,s+tW ⊤ +T ft +α,s+t, +then rank( ¯Ut) ≤ s and moreover by Theorem C.1 we have +���XsX⊤ +s − U lr +α,0 +� +U lr +α,0 +�⊤��� +F ≤ +105κ3∥X∥2r∗δ. On the other hand, by Lemma 4.4 we have that +��Z∗ +s − XsX⊤ +s +�� +F ≤ 80δκ√r∗∥X∥2. +Thus +���U lr +α,0 +� +U lr +α,0 +�⊤ − Z∗ +s +��� +F ≤ 2 × 105κ3∥X∥2r∗δ as desired. +□ +Let ˆUα,0 = UT ft +α,sWT ft +α,s ∈ Rd×s, then it satisfies ˆUα,0 ˆU ⊤ +α,0 = U lr +α,0 +� +U lr +α,0 +�⊤. The following +corollary shows that ˆUα,0 is close to U ∗ +s in terms of the procrustes distance. +55 + +Corollary F.1. We have dist( ˆUα,0, U ∗ +s ) ≤ 3 × 106κ4r∗∥X∥δ. +Proof : We know from Lemma 4.4 that dist(U ∗ +s , Xs) ≤ 40δκ∥X∥F , so it remains to bound +dist( ˆUα,0, Xs). +The proof idea is the same as that of Lemma 4.4, so we only provide a proof sketch here. It +has been shown in the proof of Proposition F.1 that +Fs( ˆUα,0) := 1 +2 +���XsX⊤ +s − ˆUα,0 ˆU ⊤ +α,0 +��� +2 +F ≤ r∗ +���XsX⊤ +s − ˆUα,0 ˆU ⊤ +α,0 +��� +2 +≤ 4×1010κ6r2 +∗∥X∥4δ2 ≤ 0.5τ 2. +Note that Fs is the matrix factorization loss with XsX⊤ +s being the ground-truth, so the local RSI +condition (Lemma E.2) still holds. By the same reason as (55), we deduce that dist( ˆUα,0, Xs) ≤ +0.1∥X∥−1τ, i.e., ˆUα,0 is in the local region around Xs in which the RSI condition holds. Finally, +it follows from the local RSI condition that +dist( ˆUα,0, Xs) ≤ 10τ −1 ���∇Fs( ˆUα,0) +��� +F ≤ 10τ −1∥ ˆUα,0∥ +���XsX⊤ +s − ˆUα,0 ˆU ⊤ +α,0 +��� +F ≤ 3×106κ4r∗∥X∥δ. +The conclusion follows. +□ +We are now ready to complete the proof of Theorems 4.1 and 4.2. +Theorem 4.2 (Convergence in the under-parameterized regime). Suppose that ˆr ≤ r∗, then there +exists a constant ¯α > 0 such that when α < ¯α, we have limt→+∞ Uα,tU ⊤ +α,t = Z∗ +ˆr . +Proof : When ˆr ≤ r∗, the parameterization itself ensures that Uα,t is low-rank, so that we can +choose U lr +α,t = Uα,T ft +α,ˆr+t and ˆUα,0 = Uα,T ft +α,s in Proposition F.1 and Corollary F.1 (for s = ˆr). +The proof that these choices satisfy all required conditions are identical to our proofs for these +two lemmas in the general setting, and we omit them here. +Applying Lemma F.1, we can thus deduce that limt→+∞ dist( ˆUα,t, U ∗ +ˆr ) = 0. This means +that limt→+∞ dist(Uα,t, U ∗ +ˆr ) = 0. Recall that Z∗ +ˆr = U ∗ +ˆr U ∗ +ˆr +⊤, so the conclusion immediately +follows. +□ +Theorem 4.1. Under Assumptions 3.1 to 3.3, consider GD (3) with initialization Uα,0 = α ¯U +for solving the matrix sensing problem (1). There exist universal constants c, M, constant C = +C(X, ¯U) and a sequence of time points T 1 +α < T 2 +α < · · · < T ˆr∧r∗ +α +such that for all 1 ≤ s ≤ ˆr∧r∗, +the following holds when α is sufficiently small: +���Uα,T sαU ⊤ +α,T sα − Z∗ +s +��� +F ≤ Cα +1 +Mκ2 , +(5) +where we recall that Z∗ +s is the best rank-s solution defined in Definition 1.1. Moreover, GD +follows an incremental learning procedure: we have limα→0 max1≤t≤T sα σs+1(Uα,t) = 0 for all +1 ≤ s ≤ ˆr ∧ r∗, where σi(A) denotes the i-th largest singular value of a matrix A. +Proof : Recall that +���UT ft +α,s − ¯U0 +��� +F = o(1) (α → 0) where T ft +α,s is defined in Proposition F.1; +we omit the dependence on α to simplify notations. We also note that by the update of GD, we +have ¯Ut ¯U ⊤ +t = ˆUα,t ˆU ⊤ +α,t for all t ≥ 0. +56 + +By Lemma F.1, we have that dist2( ˆUα,t, U ∗ +s ) ≤ (1 − 0.05τµ)t dist2( ˆUα,0, U ∗ +s ) and, in +particular, +��� ˆUα,t +��� ≤ 2∥X∥ for all t. Thus +�� ¯Ut +�� ≤ 2∥X∥ as well. Moreover, recall that +∥Ut∥ ≤ 3∥X∥ for all t. It’s easy to see that that the matrix sensing loss f is L-smooth in +� +U ∈ Rd×r : ∥U∥ ≤ 3∥X∥ +� +for some constant L = O(∥X∥2), so it follows from Lemma A.6 +that +���UT ft +α,s+t − ¯Ut +��� +F ≤ (1 + µL)t ���UT ft +α,s − ¯U0 +��� +F . +On the other hand, since dist2( ˆUα,t, U ∗ +s ) ≤ (1 − 0.05τµ)t dist2( ˆUα,0, U ∗ +s ), we can deduce that +���UT ft +α,s+tU ⊤ +T ft +α,s+t − Zs +��� +F ≤ +���UT ft +α,s+tU ⊤ +T ft +α,s+t − ¯Ut ¯U ⊤ +t +��� +F + +��� ¯Ut ¯U ⊤ +t − U ∗ +s (U ∗ +s )⊤��� +F += +���UT ft +α,s+tU ⊤ +T ft +α,s+t − ¯Ut ¯U ⊤ +t +��� +F + +��� ˆUα,t ˆU ⊤ +α,t − U ∗ +s (U ∗ +s )⊤��� +F +≤ 3∥X∥ +����UT ft +α,s+t − ¯Ut +��� +F + dist( ˆUα,t, U ∗ +s ) +� +≤ 3∥X∥ +� +(1 + µL)t ���UT ft +α,s − ¯U0 +��� +F + (1 − 0.05τµ) +t +2 dist2( ˆUα,0, U ∗ +s ) +� +Since when α → 0, +���UT ft +α,s − ¯U0 +��� +F = O(α +1 +4κ ), it’s easy to see that there exists a time t = ts +α so +that we have max−T ft +α,s≤t≤tsα +���UT ft +α,s+t − ¯Ut +��� +F = O +� +α +1 +M1κ2 +� +and +���UT ft +α,s+tU ⊤ +T ft +α,s+t − Zs +��� +F = +O +� +α +1 +M1κ2 +� +as well, where c1 is a universal constant. Let T s +α = T ft +α,s+ts +α, then +���UT sαU ⊤ +T sα − Zs +��� +F = +o(1) holds. Recall that rank(Ut) ≤ s, so that max0≤t≤T sα σs+1 (Ut) = o(1). Finally, for all +0 ≤ s < ˆr ∧ r∗, we need to show that T s +α < T s+1 +α +. Indeed, by Corollary E.2 and the Assump- +tion 3.1 we have σ2 +s+1 +� +UT sα +� +≥ σs+1 (Zs+1) − o(1) ≥ 0.5σ2 +s+1, so that T s+1 +α +> T s +α, as desired. +□ +G +The landscape of matrix sensing with rank-1 parameterization +In this section, we establish a local strong-convexity result Lemma G.2 for rank-1 parameterized +matrix sensing. This result is stronger than the RSI condition we established for general ranks, +though the latter is sufficient for our analysis. +Lemma G.1. Define the full-observation loss with rank-1 parameterization +g1(u) = 1 +4 +��uuT − XXT ��2 +F . +Then the global minima of g1 are u∗ = σ1v1 and −u∗. Moreover, suppose that g(u) − g(u∗) ≤ +0.5τ1 where τ1 = σ2 +1 − σ2 +2 is the eigengap, then we must have +∥u − u∗∥2 ≤ 20τ −1 +1 +(g1(u) − g1(u∗)) . +57 + +Proof : We can assume WLOG that XXT = diag +� +σ2 +1, · · · , σ2 +r∗, 0, · · · , 0 +� +. Then +g1(u) = 1 +4 +� +∥u∥4 +2 − 2 +s +� +i=1 +σ2 +i u2 +i + ∥XT X∥2 +F +� +(60a) +≥ 1 +4 +� +∥u∥4 +2 − 2σ2 +1∥u∥2 +2 + ∥XT X∥2 +F +� +(60b) +≥ 1 +4 +� +∥XT X∥2 +F − σ4 +1 +� +(60c) +where equality holds if and only if u2 = · · · = ud = 0 and ∥u∥2 = σ2 +1 i.e. u = ±σ1e1. +Moreover, suppose that g1(u) − g1(u∗) ≤ 0.5τ1, it follows from (60b) that τ1 +�d +i=2 u2 +i ≤ +2(g1(u) − g1(u∗)) which implies that �d +i=2 u2 +i ≤ 2τ −1 +1 +(g1(u) − g1(u∗)). Also (60c) yields +��∥u∥2 − σ2 +1 +�� ≤ 4 +� +g1(u) − g1(u∗). Assume WLOG that u1 > 0, then we have +∥u − σ1e1∥2 ≤ σ−2 +1 +� +u2 +1 − σ2 +1 +�2 + +d +� +i=2 +u2 +i +≤ 20τ −1 +1 +(g1(u) − g1(u∗)) . +□ +Lemma G.2. Let +f1(u) = 1 +4 +��A +� +uuT − XXT ���2 +2 , +u ∈ Rd. +Suppose that δ ≤ 10−3∥X∥−2τ1, then there exists constants a1 and ι, such that f1 is locally +ι-strongly convex in B1 = B(σ1v1, a1) ⊂ Rd. Furthermore, there is a unique global minima of +f1 inside B1. +Proof : Recall that we defined the full observation loss g1(u) = +1 +4 +��uuT − XXT ��2 +F . Let +h1 = f1 − g1, then +��∇2h1(u) +�� = 1 +2 +��(A∗A − I) (uuT − XXT ) + 2 (A∗A − I) uuT �� +≤ δ +� +2∥u∥2 + ∥X∥2� +. +When ∥u − σ1v1∥2 ≤ 0.1 min +� +σ2 +1, τ1 +� +(recall τ1 = σ2 +1 − σ2 +2), +σmin +� +∇2g1(u) +� += 1 +2σmin +� +∥u∥2I + 2uuT − XXT � +≥ 0.4τ1. +Hence we have +σmin +� +∇2f1(u) +� +≥ +� +∇2g1(u) +� +− ∥∇2h1(u)∥ ≥ 0.4τ1 − 4∥X∥2δ ≥ 0.2τ1, +i.e. strong-convexity holds for a2 +1 = 0.1 min +� +σ2 +1, τ1 +� +and ι = 0.2τ1. +58 + +Let u∗ be a global minima of f1, then we must have ∥u∗∥ ≤ 2∥X∥ (otherwise f1(u) > +f1(0)). We can thus deduce that +g1(u∗) ≤ f1(u∗) + 1 +4 +��� +uuT − XXT , (A∗A − I)(uuT − XXT ) +��� +≤ f1(u) + 10δ∥X∥2 ≤ g1(u) + 20δ∥X∥2. +It follows from Lemma G.1 and our assumption on δ that min +� +∥u∗ − σ1v1∥2 , ∥u∗ + σ1v1∥2� +≤ +1 +2a2 +1. Moreover, by strong convexity, there exists only one global minima in B1, which concludes +the proof. +□ +59 + diff --git a/d9FJT4oBgHgl3EQfSixW/content/tmp_files/load_file.txt b/d9FJT4oBgHgl3EQfSixW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dc7b171e98a0ceb398a81894e65ee18216762e9 --- /dev/null +++ b/d9FJT4oBgHgl3EQfSixW/content/tmp_files/load_file.txt @@ -0,0 +1,2731 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf,len=2730 +page_content='Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing Jikai Jin * Zhiyuan Li† Kaifeng Lyu‡ Simon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Du§ Jason D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lee¶ January 30, 2023 Abstract It is believed that Gradient Descent (GD) induces an implicit bias towards good gener- alization in training machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2020) and follows an incremental learning procedure (Gissin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2019): GD sequentially learns so- lutions with increasing ranks until it recovers the ground truth matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, we conduct numerical experiments to confirm our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 1 Introduction Understanding the optimization and generalization properties of optimization algorithms is one of the central topics in deep learning theory (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Sun, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It has long been a mystery why simple algorithms such as Gradient Descent (GD) or Stochastic Gradient Descent (SGD) can find global minima even for highly non-convex functions (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2019), and why the global minima being found can generalize well (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' One influential line of works provides theoretical analysis of the implicit bias of GD/SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' These results typically exhibit theoretical settings where the low-loss solutions found by GD/SGD attain certain optimality conditions of a particular generalization metric, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', the parameter norm (or the classifier margin) (Soudry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Gunasekar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Nacson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lyu & Li, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Ji & Telgarsky, 2020), the sharpness of local loss landscape (Blanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Damian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Peking University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Email: jkjin@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='cn †Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Email: zhiyuanli@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='edu ‡Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Email: klyu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='edu §University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Email: ssdu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='edu ¶Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Email: jasonlee@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='11500v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='LG] 27 Jan 2023 Among these works, a line of works seek to characterize the implicit bias even when the train- ing is away from convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Kalimeris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2019) empirically observed that SGD learns model from simple ones, such as linear classifiers, to more complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As a result, SGD always tries to fit the training data with minimal model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This behavior, usually re- ferred to as the simplicity bias or the incremental learning behavior of GD/SGD, can be a hidden mechanism of deep learning that prevents highly over-parameterized models from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In theory, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Frei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2021) established that GD on two-layer nets learns linear classifiers first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The goal of this paper is to demonstrate this simplicity bias/incremental learning in the matrix sensing problem, a non-convex optimization problem that arises in a wide range of real-world applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', image reconstruction (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2014), object detection (Shen & Wu, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2013) and array processing systems (Kalogerias & Petropulu, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, this problem can serve as a standard test-bed of the implicit bias of GD/SGD in deep learning theory, since it retains many of the key phenomena in deep learning while being simpler to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Formally, the matrix sensing problem asks for recovering a ground-truth matrix Z∗ ∈ Rd×d given m observations y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' , ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Each observation yi here is resulted from a linear measurement yi = ⟨Ai, Z∗⟩, where {Ai}1≤i≤m is a collection of symmetric measurement matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In this paper, we focus on the case where Z∗ is symmetric, positive semi-definite (PSD) and low-rank, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', Z∗ ⪰ 0 and rank(Z∗) = r∗ ≪ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' An intriguing approach to solve this problem is to use the Burer-Monteiro type decomposition Z∗ = UU ⊤ with U ∈ Rd׈r, and minimize the squared loss with GD: min U∈Rd׈r f(U) := 1 4m m � i=1 � yi − � Ai, UU ⊤��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (1) In the ideal case, the number of columns of U, denoted as ˆr above, should be set to r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' However, r∗ may not be known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This leads to two training regimes that are more likely to happen: the under-parameterized regime where ˆr ≤ r∗, and the over-parameterized regime where ˆr > r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The over-parameterized regime may lead to overfitting at first glance, but surprisingly, with small initialization, GD induces a good implicit bias towards solutions with the exact or ap- proximate recovery of the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It was first conjectured in Gunasekar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2017) that GD with small initialization finds the matrix with the minimum nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Gunasekar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2017) also proved this conjecture for a special case where all measurements are commutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' However, a series of works point out that this nuclear norm minimization view cannot capture the incremental learning behavior of GD, which, in the context of matrix sensing, refers to the phenomenon that GD tends to learn solutions with rank gradually increasing with training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2019) exhibited this phenomenon when there is only one observation (m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Gissin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2022) studied the full-observation case, where every entry of the ground truth is measured independently f(U) = 1 4d2 ∥Z∗ − UU ⊤∥2 F, and GD is shown to sequentially recover singular components of the ground truth from the largest singular value to the smallest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2020) provided theoretical evidence that the incremental learning 2 behavior generally occurs for matrix sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' They specifically provided a counterexample for Gunasekar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2017)’s conjecture, where GD converges to a rank-1 solution with a very large nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Razin & Cohen (2020) also pointed out a case where GD drives the norm to infinity while keeping the rank to be approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Despite these progresses, theoretical understanding of the simplicity bias of GD remains lim- ited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In fact, a vast majority of existing analyses can only show that GD is initially biased towards a rank-1 solution and cannot be generalized to higher ranks, unless additional assumptions on GD dynamics are made (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2020, Appendix H), (Belabbas, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Razin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Recently Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2022b) shows that the implicit bias of Gunasekar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2017) essentially relies on rewriting gradient flow in the space of U as continuous mirror de- scent in the space of UU ⊤, which only works a special type of reparametrized model, named “commuting parametrization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' However, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2022b) also shows that matrix sensing with general (non-commutable) measurements does not fall into this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Our Contributions In this paper, we take a step towards understanding the generalization of GD with small initial- ization by firmly demonstrating the simplicity bias/incremental learning behavior in the matrix sensing setting, assuming the Restricted Isometry Property (RIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Our main result is informally stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' See Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 for the formal version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (Best Rank-s Solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We define the best rank-s solution as the unique global minimizer Z∗ s of the following constrained optimization problem: min Z∈Rd×d 1 4m m � i=1 (yi − ⟨Ai, Z⟩)2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Z ⪰ 0, rank(Z) ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (Informal version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Consider the matrix sensing problem (1) with rank-r∗ ground-truth matrix Z∗ and measurements {Ai}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Assume that the measurements satisfy the RIP condition (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' With small learning rate µ > 0 and small initial- ization Uα,0 = αU ∈ Rd׈r, the trajectory of Uα,tU ⊤ α,t during GD training enters an o(1)- neighborhood of each of the best rank-s solutions in the order of s = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' , ˆr ∧ r∗ when α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, when ˆr ≤ r∗, we have limt→∞ Uα,tU ⊤ α,t = Z∗ ˆr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It is shown in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' St¨oger & Soltanolkotabi (2021) that GD exactly recovers the ground truth under the RIP condition, but our theorem goes beyond them in a number of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' First, in the over-parameterized regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', ˆr > r∗), it implies that GD performs incremental learning: learning solutions with increasing ranks until it finds the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Second, this result also shows that in the under-parameterized regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', ˆr ≤ r∗), GD exhibits the same implicit bias, but finally it converges to the best low-rank solution of the matrix sensing loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By contrast, to the best of our knowledge, only the over-parameterized setting is analyzed in existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 can also be considered as a generalization of previous results in Gissin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2022) which show that Uα,tU ⊤ α,t passes by the best low-rank solutions one 3 by one in the full observation case of matrix sensing f(U) = 1 4d2 ∥Z∗ − UU ⊤∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' However, our setting has two major challenges which significantly complicate our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' First, since our setting only gives partial measurements, the decomposition of signal and error terms in Gissin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2022) cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Instead, we adopt a different approach which is motivated by St¨oger & Soltanolkotabi (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Second, it is well-known that the optimal rank-s solution of matrix factorization is Xs (defined in Section 3), but little is known for Z∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 we analyze the landscape of (2), establishing the uniqueness of Z∗ s and local landscape properties under the RIP condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We find that when Uα,tU ⊤ α,t ≈ Z∗ s, GD follows an approximate low-rank trajectory, so that it behaves similarly to GD in the under-parameterized regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Using our landscape results, we can finally prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We review additional related works in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In Section 3, we provide an overview of necessary background and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We then present our main results in Section 4 with proof sketch where we also state some key lemmas that are used in the proof, including Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and some landscape results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In Section 5 we present a trajectory analysis of GD and prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Experimental results are presented in Section 6 which verify our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, in Section 7, we summarize our main contributions and discuss some promising future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Complete proofs of all results are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 2 Related work Low-rank matrix recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The goal of low-rank matrix recovery is to recover an unknown low-rank matrix from a number of (possibly noisy) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Examples include matrix sensing (Recht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2010), matrix completion (Cand`es & Recht, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Candes & Plan, 2010) and robust PCA (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Cand`es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Fornasier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Ngo & Saad (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2021) study efficient optimization algorithms with conver- gence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Interested readers can refer to Davenport & Romberg (2016) for an overview of this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Simplicity bias/incremental learning of GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Besides the works mentioned in the introduction, there are many other works studying the simplicity bias/incremental learning of GD on tensor factorization (Razin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2021, 2022), deep linear networks (Gidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2019), two-layer nets with orthogonal inputs (Boursier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Landscape analysis of non-convex low-rank problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The strict saddle property (Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016) was established for non-convex low-rank problems in a unified framework by Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2016) proved a local PL property for matrix sens- ing with exact parameterization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' the rank of parameterization and ground-truth matrix are the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The optimization geometry of general objective function with Burer-Monteiro type factorization is studied in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We provide a comprehensive analysis in this regime for matrix factorization as well as matrix sensing that improves over their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 4 3 Preliminaries In this section, we first list the notations used in this paper, and then provide details of our theoretical setup and necessary preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Notations We write min{a, b} as a ∧ b for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For any matrix A, we use ∥A∥F to denote the Frobenius norm of A, use ∥A∥ to denote the spectral norm ∥A∥2, and use σmin(A) to denote the smallest singular value of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We use the following notation for Singular Value Decomposition (SVD): Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (Singular Value Decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For any matrix A ∈ Rd1×d2 of rank r, we use A = VAΣAW ⊤ A to denote a Singular Value Decomposition (SVD) of A, where VA ∈ Rd1×r, WA ∈ Rd2×r satisfy V ⊤ A VA = I, W ⊤ AWA = I, and ΣA ∈ Rr×r is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For the matrix sensing problem (1), we write the ground-truth matrix as Z∗ = XX⊤ for some X = [v1, v2, · · · , vr∗] ∈ Rd×r∗ with orthogonal columns from an orthogonal basis {vi : i ∈ [d]} of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We denote the singular values of X as σ1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' , σr∗, then the singular values of Z∗ are σ2 1, σ2 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' , σ2 r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We set σr∗+1 := 0 for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For simplicity, we only consider the case where Z∗ has distinct singular values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', σ2 1 > σ2 2 > · · · > σ2 r∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We use κ := σ2 1 min1≤s≤r∗{σ2s−σ2 s+1} to quantify the degeneracy of the singular values of Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We also use the notation Xs = [v1, v2, · · · , vs] for the matrix consisting of the first s columns of X and X⊥ s = [vs+1, · · · , vd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Following Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we let VX⊥ s = � vs+1 ∥vs+1∥, · · · , vd ∥vd∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Note that the best rank-s solution Z∗ s (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1) does not equal XsX⊤ s in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We write the results of the measurements {Ai}m i=1 as a linear mapping A : Rd×d �→ Rm, where [A(Z)]i = 1 √m⟨Ai, Z⟩ for all 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We use A∗ : Rm → Rd×d, A∗(w) = 1 √m �m i=1 wiAi to denote the adjoint operator of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Our loss function (1) can then be written as f(U) = 1 4 ��A � Z∗ − UU ⊤���2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The gradient is given by ∇f(U) = A∗ � y − A(UU ⊤) � U = A∗A � XX⊤ − UU ⊤� U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In this paper, we consider GD with learning rate µ > 0 starting from U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The update rule is Ut+1 = Ut − µ∇f (Ut) = (I + µMt)Ut, (3) where Mt := A∗A � XXT − UtU T t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We specifically focus on GD with small initialization: letting U0 = α ¯U for some matrix ¯U ∈ Rd׈r with ∥ ¯U∥ = 1, we are interested in the trajectory of GD when α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Sometimes we write Ut as Uα,t to highlight the dependence of the trajectory on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Assumptions For our theoretical analysis of the matrix sensing problem, we make the following standard assumption in the matrix sensing literature: 5 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 (Restricted Isometry Property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We say that a measurement operator A satisfies the (δ, r)-RIP condition if (1−δ)∥Z∥2 F ≤ ∥A(Z)∥2 2 ≤ (1+δ)∥Z∥2 F for all matrices Z ∈ Rd×d with rank(Z) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The measurement operator A satisfies the (2r∗ + 1, δ)-RIP property, where r∗ = rank(Z∗) and δ ≤ 10−12κ−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5r−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The RIP condition is the key to ensure the ground truth to be recoverable with partial observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' An important consequence of RIP is that it guarantees A∗A(Z) = 1 m �m i=1 ⟨Ai, Z⟩ Ai ≈ Z when Z is low-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This is made rigorous in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (St¨oger & Soltanolkotabi, 2021, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3) Suppose that A satisfies (r, δ)- RIP with r ≥ 2, then for all symmetric Z, we have ∥(A∗A − I)Z∥2 ≤ δ∥Z∥∗, where ∥ · ∥∗ is the nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, if rank(Z) ≤ r − 1, then ∥(A∗A − I)Z∥2 ≤ √rδ∥Z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We need the following regularity condition on initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For all 1 ≤ s ≤ ˆr ∧ r∗, σmin � V ⊤ Xs ¯U � ≥ ρ for some constant ρ > 0, where VXs is defined as Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The following proposition implies that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 is satisfied with high probability with a Gaussian initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that all entries of ¯U ∈ Rd׈r are independently drawn from N � 0, 1 ˆr � and ρ = ϵ √ ˆr−√ˆr∧r∗−1 √ ˆr ≥ ϵ 2r∗ , then σmin � V ⊤ Xs ¯U � ≥ ρ holds for all 1 ≤ s ≤ ˆr ∧ r∗ with probability at least 1 − ˆr � Cϵ + e−cˆr� , where c, C > 0 are universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lastly, we make the following assumption on the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The step size µ ≤ 10−4δ∥X∥−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 Procrustes Distance Our analysis uses the notion of Procrustes distance defined as in Goodall (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 (Procrustes Distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The Procrustes distance between two matrices U1, U2 ∈ Rd×s (d, s > 0) is defined as the optimal value of the classic orthogonal Procrustes problem: dist(U1, U2) = min R∈Rs×s:R⊤R=I ∥U1 − U2R∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (4) We note that the Procrustes distance is well-defined because the set of s × s orthogonal matrices is compact and thus the continuous function ∥U1 − U2R∥F in R can attain its mini- mum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The Procrustes distance is a pseudometric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', it is symmetric and satisfies the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The following lemma is borrowed from Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2016), which connects the Procrustes dis- tance between U1 and U2 with the distance between U1U ⊤ 1 and U2U ⊤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 2016, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For any two matrices U1, U2 ∈ Rd×r, we have ��U1U ⊤ 1 − U2U ⊤ 2 �� F ≥ (2 √ 2 − 2)1/2 · σr(U1) · dist(U1, U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 6 4 Main results In this section, we present our main theorems and their proof sketches, following the theoretical setup in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Full proofs can be found in Appendices C to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, consider GD (3) with initialization Uα,0 = α ¯U for solving the matrix sensing problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' There exist universal constants c, M, constant C = C(X, ¯U) and a sequence of time points T 1 α < T 2 α < · · · < T ˆr∧r∗ α such that for all 1 ≤ s ≤ ˆr∧r∗, the following holds when α is sufficiently small: ���Uα,T sαU ⊤ α,T sα − Z∗ s ��� F ≤ Cα 1 Mκ2 , (5) where we recall that Z∗ s is the best rank-s solution defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, GD follows an incremental learning procedure: we have limα→0 max1≤t≤T sα σs+1(Uα,t) = 0 for all 1 ≤ s ≤ ˆr ∧ r∗, where σi(A) denotes the i-th largest singular value of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It is guaranteed that Z∗ s is unique for all 1 ≤ s ≤ ˆr ∧ r∗ under our assumptions (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In short, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 states that GD with small initialization discovers the best rank-s solution (s = 1, 2, · · · , ˆr ∧ r∗) sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In particular, when s = r∗, the best rank-s solution is exactly the ground truth XX⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Hence with over-parameterization (ˆr ≥ r∗), GD can discover the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' At a high level, our result characterizes the complete learning dynamics of GD and reveals an incremental learning mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', GD starts from learning simple solutions and then gradually increases the complexity of search space until it finds the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In the under-parameterized setting, we can further establish the following convergence result: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 (Convergence in the under-parameterized regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that ˆr ≤ r∗, then there exists a constant ¯α > 0 such that when α < ¯α, we have limt→+∞ Uα,tU ⊤ α,t = Z∗ ˆr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Key lemmas In this section, we present some key lemmas for proving our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' First, we can show that with small initialization, GD can get into a small neighborhood of Z∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, there exists ˆT s α > 0 for all α > 0 and 1 ≤ s ≤ ˆr ∧ r∗ such that limα→0 max1≤t≤ ˆT sα σs+1(Uα,t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Furthermore, it holds that ���U ˆT sαU ⊤ ˆT sα − Z∗ s ��� F = O � κ3r∗δ∥X∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The full proof can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Motivated by St¨oger & Soltanolkotabi (2021), we consider the following decomposition of Ut: Ut = UtWtW ⊤ t + UtWt,⊥W ⊤ t,⊥, (6) where Wt := WV ⊤ XsUt ∈ Rˆr×s is the matrix consisting of the right singular vectors of V ⊤ XsUt (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1) and Wt,⊥ ∈ Rˆr×(ˆr−s) is any orthogonal complement of Wt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', WtW ⊤ t + 7 Wt,⊥W ⊤ t,⊥ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The dependence of Wt, Wt,⊥ on s is omitted but will be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We will refer to the term UtWtW ⊤ t as the parallel component and UtWt,⊥W ⊤ t,⊥ as the orthogonal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The idea is to show that the parallel component grows quickly until it gets close to the best rank-s solution at some time ˆT s α (namely UtWtW ⊤ t U ⊤ t ≈ Z∗ s when t = ˆT s α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Meanwhile, the orthogonal term grows exponentially slower and stays o(1) before ˆT s α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' See Section 5 for a detailed proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 shows that UtU ⊤ t would enter a neighborhood of Z∗ s with constant radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' How- ever, there is still a gap between Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, since the latter states that UtU ⊤ t would actually get o(1)-close to Z∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' To proceed, we define the under-parameterized matrix sensing loss fs for every 1 ≤ s ≤ r∗: fs(U) = 1 4 ���A(Z∗ − UU ⊤) ��� 2 2 , U ∈ Rd×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (7) While the function we are minimizing is f (defined in (1)) rather than fs, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 suggests that for t ≤ ˆT s α, Ut is always approximately rank-s, so that we use a low-rank approximation for U ˆT sα and associate the dynamics locally with the GD dynamics of fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We will elaborate on how this is done in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When dist(U1, U2) = 0, it can be easily shown that fs(U1) = fs(U2) since fs is invariant to orthogonal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, we note that the global minimizer of fs is unique up to orthogonal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, if U ∗ s ∈ Rd×s is a global minimizer of fs, then the set of global minimizers arg min fs is equal to � U ∗ s R : R ∈ Rs×s, R⊤R = I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Around the global minimizers, we show that fs satisfies the Restricted Secant Inequality (RSI) which is useful for optimization analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For any U ∈ Rd×s, we use Πs(U) to denote the set of closest global minimizers of fs to U, namely Πs(U) = arg min{∥U − U ∗ s ∥F : U ∗ s ∈ arg min fs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 (Restricted Secant Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, if a matrix U ∈ Rd×s satis- fies ∥U − U ∗ s ∥F ≤ 10−2κ−1∥X∥ for some U ∗ s ∈ Πs(U), then we have ⟨∇fs(U), U − U ∗ s ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1κ−1∥X∥2∥U − U ∗ s ∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (8) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In general, a function g : Rn �→ R satisfies the RSI condition if for some µ > 0, ⟨∇g(x), x − π(x)⟩ ≥ µ∥x − π(x)∥2 holds for all x, where π(x) is a projection of x onto arg min g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This condition can be used to prove linear convergence of GD (Zhang & Yin, 2013), but it is weaker than strong convexity and stronger than Polyak-Łojasiewicz(PL) condi- tion (Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We end this subsection with the following lemma which says that all global minimizers of the fs must be close to Xs under the procrustes distance, which is used in the proof sketch of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 8 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have dist(U ∗ s , Xs) ≤ 40δκ∥X∥F for any global mini- mizer U ∗ s of fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, ��Z∗ s − XsX⊤ s �� F ≤ 160δκ√r∗∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have σmin(U ∗ s ) ≥ 1 2σmin(Xs) = 1 2σs ≥ 1 2κ− 1 2 ∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The full proofs for Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Proof outline Based on the key lemmas, here we provide the outlines of the proofs for our main theorems and defer the details to Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We first prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 which can be directly derived by combining the lemmas in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof :[Proof Sketch of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2] For any global minimizer U ∗ ˆr of (1), we have dist(U ∗ ˆr , Uα, ˆT ˆrα) ≤ (2 √ 2 − 2)−1/2σ−1 min (U ∗ ˆr ) ���Uα, ˆT ˆrαU ⊤ α, ˆT ˆrα − Z∗ ˆr ��� F ≤ O(κ 1 2 ∥X∥−1) · O(κ3r∗δ∥X∥2) = O(κ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5r∗δ∥X∥), where the first inequality is due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and the second one is due to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 then imply that Uα, ˆT ˆrα lies in the small neighborhood of the set of global minimizers of f = fˆr, in which the RSI holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Following a standard non- convex optimization analysis (Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016), we can show that GD converges linearly to arg min fˆr (in the Procrustes distance), which yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Now we turn to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' While f is not necessarily local RSI, we use a low-rank approximation for Ut and associate the dynamics in this neighborhood with the GD dynamics of fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof :[Proof sketch of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1] Recall that by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, Uα, ˆT sα is approximately rank-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' So there must exist a matrix ¯Uα,0 ∈ Rd׈r with rank( ¯Uα,0) ≤ s such that ¯Uα,0 ¯U ⊤ α,0 − Uα,T sαU ⊤ α,T sα = o(1) as α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (9) Indeed, we can let ¯Uα,t be the parallel component of Uα,T sα because the orthogonal component stays o(1) (see the discussions following (6) and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let � ¯Uα,t � t≥0 be the trajectory of GD with step size µ, initialized at ¯Uα,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since the gradient of the objective function f is locally Lipschitz, the solution obtained by the two GD trajectories { ¯Uα,t}t≥0 and {Uα, ˆT sα+t}t≥0 will remain o(1)-close for at least constantly many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Indeed we can show that they will keep o(1) close for some ¯tα = ω(1) steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', for all t ∈ [0, ¯tα], ¯Uα,t ¯U ⊤ α,t − Uα, ˆT sα+tU ⊤ α, ˆT sα+t = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (10) 9 From (3) it is evident that GD initialized at ¯Uα,0 actually lives in the space of matrices with rank ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Indeed we can identify its dynamics with another GD on fs (defined in (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Concretely, let ˆUα,0 ∈ Rd×s be a matrix so that ˆUα,0 ˆU ⊤ α,0 = ¯Uα,0 ¯U ⊤ α,0, and let { ˆUα,t}t≥0 be the trajectory of GD that optimizes fs with step size µ starting from ˆUα,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then we have ˆUα,t ˆU ⊤ α,t = ¯Uα,t ¯U ⊤ α,t for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We can now apply our landscape results for fs to analyze the GD trajectory { ˆUα,t}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By (9) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have ��� ˆUα,0 ˆU ⊤ α,0 − Z∗ s ��� F = O(κ3r∗δ∥X∥2), so using a similar argument as in the proof sketch of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 imply that the initialization ˆUα,0 is within an O � κ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5r∗δ∥X∥2� neighborhood of the set of global minimizers of fs(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' From Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 we know that that fs(U) satisfies a local RSI condition in this neighborhood, so following standard non-convex optimization analysis (Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016), we can show that { ˆUα,t}t≥0 converges linearly to its set of global minimizers in the Procrustes distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We need to choose a time t such that (10) remains true while this linear convergence process takes place for sufficiently many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This is possible since ¯tα = ω(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' indeed we can show that there always exists some t = ts α ≤ ¯tα such that both ��� ˆUα,t ˆU ⊤ α,t − Uα, ˆT sα+tU ⊤ α, ˆT sα+t ��� F and ��� ˆUα,t − U ∗ s ��� F are bounded by O(α 1 Mκ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Hence ��Uα,tU ⊤ α,t − Z∗ s �� F = O(α 1 Mκ2 ) when t = T s α := ˆT s α + ts α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For 1 ≤ s < ˆr ∧ r∗ and t ≤ ts α, since (10) holds and rank( ˆUα,t) ≤ s, we have max 1≤t≤T sα σs+1 (Uα,t) → 0 as α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 we have ��Z∗ s+1 − Xs+1X⊤ s+1 �� = O(δκ√r∗) = O(κ−1∥X∥2), so σs+1(Z∗ s+1) ≳ σ2 s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Therefore, Uα,tU ⊤ α,t cannot be close to Z∗ s+1 when t ≤ T s α, so we must have T s+1 α > T s α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This completes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ 5 Proof sketch of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 In this section, we outline the proof sketch of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We divide the GD dynamics intro three phases and characterize the dynamics separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof details for these three phases can be found in Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The spectral phase Starting from a small initialization, GD initially behaves similarly to power iteration since Ut+1 = (I + µMt)Ut ≈ (I + µM)Ut, where M := A∗A(XX⊤) is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let M = �d k=1 ˆσ2 kˆvkˆv⊤ k be the eigendecomposition of M, where ˆσ1 ≥ ˆσ2 ≥ · · · ≥ ˆσd ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Using our assumption on δ (Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1), we can show that |σi − ˆσi| , 1 ≤ i ≤ s are sufficiently 10 small so that ˆσi’s are positive and well-separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then we have UT ≈ (I + µM)T U0 = d � i=1 (1 + µˆσ2 i )T ˆviˆv⊤ i U0 ≈ s � i=1 (1 + µˆσ2 i )T ˆviˆv⊤ i U0, (11) where the last step holds because (1 + µˆσs)T ≫ (1 + µˆσs+1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In other words, we can expect that there is an exponential separation between the parallel and orthogonal component of UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Formally, we can prove the following property at the end of the spectral phase: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, simplified version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then there exist positive constants Ci = Ci(X, ¯U), γi = γi(X, ¯U), i = 2, 3 independent of α such that γ2 < γ3 and the following inequalities hold for t = T sp α = O � log α−1 log(1+µ∥X∥2) � when α is sufficiently small: ∥Ut∥ ≤ ∥X∥, σmin (UtWt) ≥ C2 · αγ2, ∥UtWt,⊥∥ ≤ C3 · αγ3, and ���V ⊤ Xs,⊥VUtWt ��� ≤ 200δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 The parallel improvement phase For small α, we have σmin (UtWt) ≫ ∥UtWt,⊥∥ by the end of the spectral phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When (11) no longer holds, we enter a new phase which we call the parallel improvement phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In this phase, the ratio σmin(UtWt) ∥UtWt,⊥∥ grows exponentially in t, until the former reaches a constant scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Formally, let T pi α,s = min � t ⩾ 0 : σ2 min � V ⊤ XsUα,t+1 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3κ−1∥X∥2� , then we can prove the following lemma via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 hold and let c3 = 104κr 1 2∗ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then for suffi- ciently small α, the following inequalities hold when T sp α ≤ t < T pi α,s: σmin � V ⊤ XsUt+1 � ≥ σmin � V ⊤ XsUt+1Wt � ≥ � 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5µ � σ2 s + σ2 s+1 �� σmin � V ⊤ XsUt � , (13a) ∥Ut+1Wt+1,⊥∥ ≤ � 1 + µ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4σ2 s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6σ2 s+1 �� ∥UtWt,⊥∥ , (13b) ���V ⊤ Xs,⊥VUt+1Wt+1 ��� ≤ c3, (13c) rank(V ⊤ XsUt+1) = rank(V ⊤ XsUt+1Wt) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (13d) We can immediately deduce from Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 that the orthogonal term ∥UtWt,⊥∥ remains o(1) by the end of the parallel improvement phase: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='8, simplified version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the conditions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, when α is sufficiently small we have ���UT pi α,sWT pi α,s,⊥ ��� ≤ C5 · α 1 4κ for some constant C5 = C5(X, ¯U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 The refinement phase After σmin (UtWt) grows to constant scale, we enter the refinement phase for which we show that ��XsX⊤ s − UtU ⊤ t �� F keeps decreasing until it is O � δκ3r∗∥X∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Formally, let τ = κ−1∥X∥2 and T ft α,s = T pi α,s − log(10−2∥X∥−2κ−1c−1 3 ) log(1− 1 2 µτ) > T pi α,s where c3 is defined in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, then the following lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that T pi α,s ≤ t ≤ T ft α,s and all the conditions in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 hold, then we have ���V ⊤ Xs(XX⊤ − Ut+1U ⊤ t+1) ��� F ≤ � 1 − 1 2µτ � ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� F + 20µ∥X∥4 (δ + 5c3) + 2000µ2√r∗∥X∥6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, it holds that ∥Ut+1Wt+1,⊥∥ ≤ (1 + σ2 s)∥UtWt,⊥∥ and ���VX⊥ s VUtWt ��� ≤ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, we arrive at the following result: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For sufficiently small α, at t = T ft α,s we have ��V ⊤ Xs � XX⊤ − UtU ⊤ t ��� F ≤ 80δκ3r∗∥X∥2 and ∥UtWt,⊥∥ = o(1) (α → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Concluding the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' At t = T ft α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' we have ���XsX⊤ s − UtU ⊤ t ��� F ≤ ��� � XsX⊤ s − UtU ⊤ t � VXsV ⊤ Xs ��� F + ���UtU ⊤ t VX⊥ s V ⊤ X⊥ s ��� F ≤ ��� � XsX⊤ s − UtU ⊤ t � VXsV ⊤ Xs ��� F + ���V ⊤ X⊥ s UtU ⊤ t VX⊥ s ��� F ≤ ���V ⊤ Xs � XsX⊤ s − UtU ⊤ t ���� F + √r∗ ���V ⊤ X⊥ s UtWt ��� 2 + √ d ���V ⊤ X⊥ s UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ ��� 2 (14a) ≤ ���V ⊤ Xs � XsX⊤ s − UtU ⊤ t ���� F +9√r∗∥X∥2 ���VX⊥ s VUtWt ��� 2 + √ d∥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥∥2 (14b) = O � δκ3r∗∥X∥2 + ∥X∥2c2 3 √r∗ � + o(1) (14c) = O � δκ3r∗∥X∥2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (14d) where (14a) uses ∥A∥F ≤ � rank(A)∥A∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (14b) uses ∥Ut∥ ≤ 3∥X∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (14c) follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 and the last step follows from c3 = 104κ√r∗δ and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4, the best rank-s solution is close to the matrix factorization minimizer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' ��Z∗ s − XsX⊤ s �� F = O � δκ√r∗∥X∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We thus obtain that ��Z∗ s − UtU ⊤ t �� F = O � δκ3r∗∥X∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, since rank(UtWt) ≤ s (recall the decomposition (6)), we have σs+1(Ut) ≤ ∥UtWt,⊥∥ = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 12 (a) α = 1, 1000 measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, 1000 measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (c) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01, 1000 measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (d) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='001, 1000 mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (e) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='001, 2000 mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (f) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='001, 5000 mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Figure 1: The evolution of relative error against the best solution of different ranks over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (a) α = 1, 1000 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, 1000 measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (c) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01, 1000 measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (d) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='001, 1000 measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Figure 2: The evolution of the loss and relative error against best solution of different ranks in the exact-parameterized case r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} 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experiments to illustrate our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We consider the matrix sensing problem (1) with d = 50, r∗ = 5, α ∈ {1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='001}, m ∈ {1000, 2000, 5000}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We will consider different choices for ˆr in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The ground truth Z∗ = XX⊤ is generated such that the entries of X are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' standard Gaussian variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We use the same ground truth throughout our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For i = 1, 2, · · · , m, all entries of the measurement Ai ∈ Rd×d are chosen i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' from the standard Gaussian N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When m ≳ dr∗δ−2, this set of measurements satisfies the RIP with high probability (Recht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2010, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We solve the problem (1) via running GD for T = 104 iterations starting with small initializa- tion with scale α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Specifically, we choose U0 = α ¯U where the entries of ¯U ∈ Rd׈r are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' from N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We consider both the over-parameterized and the exact/under-parameterized regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The learning rate of GD is set to be µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Implicit low-rank bias In this subsection, we consider the over-parameterized setting with r = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For each iteration t ∈ [T] and rank s ∈ [r∗], we define the relative error Es(t) = ∥UtU⊤ t −XsX⊤ s ∥ 2 F ∥XsX⊤ s ∥ 2 F to measure the proximity of the GD iterates to Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We plot the relative error in Figure 1 for different choices of α and m (which affects the measurement error δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Small initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The implicit low-rank bias of GD is evident when the initialization scale α is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Indeed, one can observe that GD first visits a small neighborhood of X1, spends a long period of time near it, and then moves towards X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It then proceeds to learn X3, X4, · · · in a similar way, until it finally fits the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This is in align with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By contrast, for large initialization we do not have this implicit bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The effect of measurement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For fixed α, one can observe the relative error becomes smaller when the number of measurements increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This is in align with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 in which the bound depends on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In particular, for the case s = r∗, in the end the distance to the set of global minima goes to zero as α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Matrix sensing with exact parameterization Now we study the behavior of GD in the exact parameterization regime (r = r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We fix m = 1000, r = r∗ = 5 and run GD for T = 500 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We plot the relative error in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As predicted by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we can observe that when α is small, GD exhibits an implicit low-rank bias and takes a longer time to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The latter is because GD would get into a poly(α)-small neighborhood of the saddle point Zs and take a long time to escape the saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As guaranteed by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, we also observe the final convergence to global minimizers for sufficiently small α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 14 7 Conclusion In this paper, we study the matrix sensing problem with RIP measurements and show that GD with small initialization follows an incremental learning procedure, where GD finds near-optimal solutions with increasing ranks until it finds the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We take a step towards under- standing the optimization and generalization aspects of simple optimization methods, thereby providing insights into their success in modern applications such as deep learning (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Also, we provide a detailed landscape analysis in the under-parameterized regime, which to the best of our knowledge is the first analysis of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Although we focus on matrix sensing in this paper, it has been revealed in a line of works that the implicit regularization effect may vary for different models, including deep matrix factoriza- tion (Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2019) and nonlinear ReLU/LeakyReLU networks (Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Timor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Also, it is shown in Woodworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2020) that different initialization scales can lead to distinct inductive bias and affect the generalization and optimization behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' All these results indicate that we need further studies to comprehensively understand gradient-based optimization methods from the generalization aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' References 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 1–13, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 19 Appendix Table of Contents A Preliminaries 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The RIP condition and its properties .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Matrix analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 Optimization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 Proof for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 23 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 Procrustes Distance .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 23 B Main idea for the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 24 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Heuristic explanations of the decomposition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 25 C Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 26 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The spectral phase .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 26 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 The parallel improvement phase .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 Induction .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 The refinement phase and concluding the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 40 D Auxiliary results for proving Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 45 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The spectral phase .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 45 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 The parallel improvement phase .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 46 E Proofs for the Landscape Results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 49 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Analysis of the matrix factorization loss .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 49 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Analysis of the matrix sensing loss .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 51 F Proofs for Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 54 G The landscape of matrix sensing with rank-1 parameterization 57 20 The appendix is organized as follows: in Appendix A we present a number of results that will be used for later proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Appendix B sketches the main idea for proving our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Appendix C is devoted to a rigorous proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 ,with some auxiliary lemmas proved in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In Appendix E we analyze the landscape of low-rank matrix sensing and prove our landscape results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' These results are then used in Appendix F to prove The- orems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, Appendix G studies the landscape of rank-1 matrix sensing, which enjoys a strongly convex property, as we mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' A Preliminaries In this section, we present some useful results that is needed in subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The RIP condition and its properties In this subsection, we collect a few useful properties of the RIP condition, which we recall below: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We say that the measurement A satisfies the (δ, r)-RIP condition if for all ma- trices Z ∈ Rd×d with rank(Z) ≤ r, we have (1 − δ)∥Z∥2 F ≤ ∥A(Z)∥2 2 ≤ (1 − δ)∥Z∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The key intuition behind RIP is that A∗A ≈ I, where A∗ : v �→ 1 √m �m i=1 viAi is the adjoint of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This intuition is made rigorous by the following proposition: Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (St¨oger & Soltanolkotabi, 2021, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3) Suppose that A satisfies (r, δ)- RIP with r ≥ 2, then for all symmetric Z, (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' if rank(Z) ≤ r − 1, we have ∥(A∗A − I)Z∥2 ≤ √rδ∥Z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' ∥(A∗A − I)Z∥2 ≤ δ∥Z∥∗, where ∥ · ∥∗ is the nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Matrix analysis The following lemma is a direct corollary of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and will be frequently used in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that the measurement A satisfies (δ, 2r∗ + 1)-RIP condition, then for all matrices U ∈ Rd×r such that rank(U) ≤ r∗, we have ���(A∗A − I) (XX⊤ − UU ⊤) ��� ≤ δ√r∗ � ∥X∥2 + ∥U∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In our proof we will frequently make use of the Weyl’s inequality for singular values: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 (Weyl’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let A, ∆ ∈ Rd×d be two matrices, then for all 1 ≤ k ≤ d, we have |σk(A) − σk(A + ∆)| ≤ ∥∆∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 21 We will also need the Wedin’s sin theorem for singular value decomposition: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (Wedin, 1972, Section 3) Define R(·) to be the column space of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that matrices B = A+T , A1, B1 are the top-s components in the SVD of A and B respectively, and A0 = A − A1, B0 = B − B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' If δ = σmin(B1) − σmax(A0) > 0, then we have ∥sin Θ (R(A1), R(B1))∥ ≤ ∥T ∥ δ where Θ(·, ·) denotes the angle between two subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Equipped with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we can have the following characterization of the eigenvalues of M (recall that M = A∗A(XX⊤)): Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let M := A∗A(XX⊤) and M = �d k=1 ˆσ2 kˆvkˆv⊤ k be the eigen-decomposition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For 1 ≤ i ≤ d we have ��σ2 i − ˆσ2 i �� ≤ δ∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof :By Weyl’s inequality we have ��σ2 i − ˆσ2 i �� ≤ ���M − XX⊤��� ≤ δ∥X∥2 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 Optimization Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that a smooth function f ∈ Rm �→ R with minimum value f∗ > −∞ satisfies the following conditions with some ϵ > 0: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' lim∥x∥→+∞ f(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' There exists an open subset S ⊂ Rm such that the set S∗ of global minima of f is contained in S, and for all stationary points x of f in Rm − S, we have f(x) − f∗ ≥ 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, we also have f(x) − f∗ ≥ 2ϵ on ∂S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then we have {x ∈ Rm : f(x) − f∗ ≤ ϵ} ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Let x∗ be the minimizer of f on Rm − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By condition (1) we can deduce that x∗ always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, since any local minimizer of a function defined on a compact set must either be a stationary point or lie on the boundary of its domain, we can see that either x∗ ∈ ∂S or ∇f(x∗) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By condition (2), either cases would imply that f(x∗) − f∗ ≥ 2ϵ, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let {xk}, {yk} ⊂ Rn be two sequences generated by xk+1 = xk −µ∇f(xk) and yk+1 = yk − µ∇f(yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that ∥xk∥ ≤ B and ∥yk∥ ≤ B for all k and f is L-smooth in {x ∈ Rn : ∥x∥ ≤ B}, then we have ∥xk − yk∥ ≤ (1 + µL)k ∥x0 − y0∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 22 Proof : The update rule implies that ∥xk+1 − yk+1∥ = ∥xk − yk − µ∇f(xk) + µ∇f(yk)∥ ≤ ∥xk − yk∥ + µ ∥∇f(xk) − f(yk)∥ ≤ (1 + µL)∥xk − yk∥ which yields the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 Proof for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that all entries of ¯U ∈ Rd׈r are independently drawn from N � 0, 1 ˆr � and ρ = ϵ √ ˆr−√ˆr∧r∗−1 √ ˆr ≥ ϵ 2r∗ , then σmin � V ⊤ Xs ¯U � ≥ ρ holds for all 1 ≤ s ≤ ˆr ∧ r∗ with probability at least 1 − ˆr � Cϵ + e−cˆr� , where c, C > 0 are universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 immediately follows from the following result: Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (Restatement of Rudelson & Vershynin, 2009, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1) Let A be an N ×n random matrix, N ≥ n, whose elements are independent copies of a mean zero sub-gaussian random variable with unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then, for every ε > 0, we have P � sn(A) ≤ ε( √ N − √n − 1) � ≤ (Cε)N−n+1 + e−cN where C, c > 0 depend (polynomially) only on the sub-Gaussian moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Now we can complete the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Note that the entries of U ∈ Rd׈r are independently drawn from N(0, 1 ˆr) and VXs ∈ Rd×s is an orthonormal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We write V + Xs ∈ Rdˆr×sˆr as a block diagonal matrix with ˆr copies of VXs on the diagonal, and vec(U) ∈ Rdˆr be a vector formed by the concatenation of the columns of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then V + Xs is still orthonormal, and vec(U) ∼ N(0, 1 ˆrI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since multivariate Gaussian distributions are invariant under orthonor- mal transformations, we deduce that (V + Xs)⊤vec(U) ∼ N(0, 1 ˆrI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Equivalently, the entries of V ⊤ XsU are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' N(0, 1 ˆr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The matrix √ ˆrV ⊤ XsU satisfies all the conditions in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus, with probability at least 1 − (Cϵ)ˆr−s+1 − e−cˆr, we have σmin( √ ˆrV ⊤ XsU) ≥ ϵ( √ ˆr − √s − 1), or equivalently σmin(V ⊤ XsU) ≥ ϵ √ ˆr−√s−1 √ ˆr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, the conclusion follows from a union bound: P � ∃1 ≤ s ≤ ˆr ∧ r∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' σmin � V ⊤ XsU � < ϵ 2ˆr � ≤ ˆr∧r∗ � s=1 P � σmin � V ⊤ XsU � < ϵ √ ˆr − √s − 1 √r � ≤ ˆr∧r∗ � s=1 � e−cˆr + (Cϵ)ˆr−s+1� ≤ r � e−cˆr + Cϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (15) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 Procrustes Distance Procrustes distance is introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The following characterization of the optimal R in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 is known in the literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2016, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1) but we provide a proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 23 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let U1, U2 ∈ Rd×r where r ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then for any orthogonal matrix R ∈ Rr×r that minimizes ∥U1 − U2R∥F (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', any orthogonal R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' ∥U1 − U2R∥F = dist(U1, U2)), U ⊤ 1 U2R is a symmetric positive semi-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : We only need to consider the case when U ⊤ 2 U1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Observe that ∥U1 − U2R∥2 F = ∥U1∥2 F + ∥U2R∥2 F − 2 tr � R⊤U ⊤ 2 U1 � = ∥U1∥2 F + ∥U2∥2 F − 2 tr � R⊤U ⊤ 2 U1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let AΣB⊤ be the SVD of U ⊤ 2 U1, where A⊤A = I, B⊤B = I and Σ ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then tr � R⊤U ⊤ 2 U1 � = tr � B⊤R⊤AΣ � ≤ ���B⊤R⊤A ��� tr (Σ) = tr (Σ) , where the final step is due to orthogonality of B⊤R⊤A ∈ Rs×s, and equality holds if and only if B⊤R⊤A = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let C = R⊤A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let bi, ci ∈ Rd be the i-th column of B and C respectively, then B⊤C = I implies that b⊤ i ci = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Note that ∥bi∥2 = ∥ci∥2 = 1, so we must have bi = ci for all i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', B = C = R⊤A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Therefore, U ⊤ 1 U2R = BΣA⊤R = BΣB⊤, which implies that U ⊤ 1 U2R is symmetric and positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ B Main idea for the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 In this section, we briefly introduce our main ideas for proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Motivated by St¨oger & Soltanolkotabi (2021), we decompose the matrix Ut into a parallel component and an orthogonal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Specifically, we write Ut = UtWtW ⊤ t � �� � parallel component + UtWt,⊥W ⊤ t,⊥ � �� � orthogonal component , (16) where Wt := WV ⊤ XsUt ∈ Rˆr×s is the matrix consisting of the right singular vectors of V ⊤ XsUt (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1) and Wt,⊥ ∈ Rˆr×(ˆr−s) is an orthogonal complement of Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Our goal is to prove that at some time t, we have V ⊤ Xs � UtU ⊤ t − XsX⊤ s � ≈ 0 and ∥UtWt,⊥∥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As we will see later, these imply that ��UtU ⊤ t − XsX⊤ s �� ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In the remaining part of this section we give a heuristic explanation for considering (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Additional Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let VXs,⊥ ∈ Rd×(d−s) be an orthogonal complement of VXs ∈ Rd×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let Σs = diag(σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' , σs) and Σs,⊥ = diag(σs+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' , σr, 0, · · · , 0) ∈ R(d−s)×(d−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We use ∆t := (A∗A − I)(XX⊤ − UtU ⊤ t ) to denote the vector consisting of measurement errors for XX⊤ − UtU ⊤ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 24 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Heuristic explanations of the decomposition A simple and intuitive approach for showing the implicit low rank bias is to directly analyze the growth of V ⊤ XsUt versus V ⊤ Xs,⊥Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Ideally, the former grows faster than the latter, so that GD only learns the components in Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By the update rule of GD (3), V ⊤ Xs,⊥Ut+1 = V ⊤ Xs,⊥ � I + µA∗A(XX⊤ − UtU ⊤ t ) � Ut = V ⊤ Xs,⊥ � I + µXX⊤ − µUtU ⊤ t � Ut � �� � =:Gt,1 +µ V ⊤ Xs,⊥∆tUt � �� � =:Gt,2 = Gt,1 + µGt,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For the first term Gt,1, we have Gt,1 = (I + µΣ2 s,⊥)V ⊤ Xs,⊥Ut − µV ⊤ Xs,⊥UtU ⊤ t Ut = (I + µΣ2 s,⊥)V ⊤ Xs,⊥Ut(I − µUtU ⊤ t ) + O(µ2), where the last term O(µ2) is negligible when µ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since ∥Σs,⊥∥ = σs+1, the spectral norm of Gt,1 can be bounded by ∥Gt,1∥ ≤ ∥I + µΣ2 s,⊥∥ · ∥V ⊤ Xs,⊥Ut∥ · ∥I − µUtU ⊤ t ∥ + O(µ2) ≤ (1 + µσ2 s+1)∥V ⊤ Xs,⊥Ut∥ + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' However, the main difference with the full-observation case (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', 2022) is the second term Gt,2 := V ⊤ Xs,⊥∆tUt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since the measurement errors ∆t are small but arbitrary, it is hard to compare this term with V ⊤ Xs,⊥Ut+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As a result, we cannot directly bound the growth of ∥V ⊤ Xs,⊥Ut∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' However, the aforementioned problem disappears if we turn to bound the growth of ∥V ⊤ Xs,⊥Ut+1Wt,⊥∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' To see this, first we deduce the following by repeatedly using V ⊤ XsUtWt,⊥ = 0 due to the defi- nition of Wt,⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � I + µXX⊤ − µUtU ⊤ t � UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥(I + µXX⊤)UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ − µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtU ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = (I + µΣ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥)V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ − µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut(WtW ⊤ t + Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥)U ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = (I + µΣ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥)V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥(I − µW ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥U ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥) − µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWtW ⊤ t U ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ + O(µ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥∆tUtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥∆tVXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 25 So we have the following recursion: V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = (I + µΣ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ + µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥∆tVXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥)V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥(I − µW ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥U ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥) − µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWtW ⊤ t U ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ + O(µ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We further note that V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1WtW ⊤ t Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ + V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (17) which establishes the relationship between V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ and V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' To com- plete the proof we need to prove the following: The minimal eigenvalue of the parallel component UtWtW ⊤ t grows at a linear rate with speed strictly faster than σs+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The term ���V ⊤ Xs,⊥VUtWt ��� ≪ 1, which implies that the first term in (17) is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' C Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 In this section, we give the full proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, with some additional technical lemmas left to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, there exists ˆT s α > 0 for all α > 0 and 1 ≤ s ≤ ˆr ∧ r∗ such that limα→0 max1≤t≤ ˆT sα σs+1(Uα,t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Furthermore, it holds that ���U ˆT sαU ⊤ ˆT sα − Z∗ s ��� F = O � κ3r∗δ∥X∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 are devoted to analyzing the spectral phase and parallel improvement phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 uses induction to characterize the low-rank GD trajectory in the parallel improvement phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 we study the refinement phase, which allows us to derive Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The spectral phase Starting from a small initialization U0 = α ¯U, α ≪ 1, we first enter the spectral phase where GD behaves similar to power iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As in St¨oger & Soltanolkotabi (2021), we refer to this phase as the spectral phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Specifically, we have in the spectral phase that Ut+1 = � I + µ (A∗A) (XX⊤ − UtU ⊤ t ) � Ut ≈ � I + µ (A∗A) (XX⊤) � Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The approximation holds with high accuracy as long as ∥Ut∥ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover we have M := (A∗A) (XX⊤) ≈ XX⊤ by the RIP condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' when δ is sufficiently small, we can still ensure a positive eigen-gap of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As a result, with small initialization Ut would become approximately aligned with the top eigenvector u1 of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since ∥M −XX⊤∥ = O(δ√r∗) by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have ∥u1 − v1∥ = O(δ√r∗) so that ∥V ⊤ XsVUtWt∥ = O(δ√r∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This proves the base case for the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 26 Formally, we define M = A∗A(XX⊤), Kt = (I + µM)t and U sp t = KtU0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that M = �rank(M) i=1 ˆσ2 i ˆviˆv⊤ i is the spectral decomposition of M where {ˆσi}i≥1 is sorted in non-increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We additionally define Ms = �min{s,rank(M)} i=1 ˆσ2 i ˆviˆv⊤ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 and δ√r∗ ≤ 10−3κ by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have ˆσ2 s ≥ σ2 s − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01τ and ˆσ2 s+1 ≤ σ2 s+1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01τ, where we recall that τ = mins∈[r∗] � σ2 s − σ2 s+1 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Additionally, let Lt be the span of the top-s left singular vectors of Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Recall that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 is made on the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let t⋆ := min � i ∈ N : ��U sp i−1 − Ui−1 �� > ��U sp i−1 ��� , the following lemma bounds the error of approximating Ut via U sp t : Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (St¨oger & Soltanolkotabi, 2021, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1) Suppose that A satisfies the rank-1 RIP with constant δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For all integers t such that 1 ≤ t ≤ t⋆ it holds that ∥Et∥ = ��Ut − U sp t �� ≤ 4ˆσ−2 1 α3r∗ (1 + δ1) � 1 + µˆσ2 1 �3t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (18) We can derive the following lower bound on t∗ from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We have t∗ ≥ log α−1 + 1 2 log ρˆσ2 1 4(1+δ1)r∗ log � 1 + µˆσ2 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 we have ∥Et∥ ≤ 4ˆσ−2 1 α3r∗ (1 + δ1) � 1 + µˆσ2 1 �3t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' for all t ≤ t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' On the other hand, we have ∥U sp t ∥ = α ��(I + µM)t ¯U �� ≥ α(1 + µˆσ2 1)t ���ˆv1ˆv⊤ 1 ¯U ��� ≥ � 1 + µˆσ2 1 �t αρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus, it follows from ∥Et∗∥ ≥ ∥U sp t∗ ∥ that � 1 + µˆσ2 1 �t∗ ≥ � ρˆσ2 1 4(1 + δ1)r∗ α−1 ⇒ t∗ ≥ log α−1 + 1 2 log ρˆσ2 1 4(1+δ1)r∗ log � 1 + µˆσ2 1 � as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Note that a trivial bound for the rank-1 RIP constant is δ1 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We can now show that for small t, GD can be viewed as approximate power iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' There exists a time t = T sp α := 2 log α−1 + log ρˆσ2 1 4r∗(1+δ) 3 log(1 + µ ˆσ2 1) − log(1 + µˆσ2 s+1) ≤ t∗ 27 such that �����Ut − s � i=1 α(1 + µˆσ2 i )tˆviˆv⊤ i ¯U ����� ≤ C1 · αγ where γ = 1 − 2 log(1+µˆσ2 s+1) 3 log(1+µˆσ2 1)−log(1+µˆσ2 s+1) and C1 = C1(X, ¯U) is a constant that only depends on X and ¯U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : It’s easy to check that T sp α ≤ t∗ by applying Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We consider the following decomposition: �����Ut − s � i=1 α(1 + µˆσ2 i )tˆviˆv⊤ i ¯U ����� ≤ ��Ut − U sp t �� + �����U sp t − s � i=1 α(1 + µˆσ2 i )tˆviˆv⊤ i ¯U ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When t ≤ t∗, the first term can be bounded as ∥Et∥ ≤ 4ˆσ−2 1 α3r∗ (1 + δ) � 1 + µˆσ2 1 �3t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For the second term we have �����U sp t − s � i=1 α(1 + µˆσ2 i )tˆviˆv⊤ i U ����� ≤ ����� r∗ � i=s+1 α(1 + µˆσ2 i )tˆviˆv⊤ i U ����� ≤ α � 1 + µˆσ2 s+1 �t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In particular, the definition of T sp α implies that �����Ut − s � i=1 α(1 + µˆσ2 i )tˆviˆv⊤ i U ����� ≤ 2 � ρˆσ2 1 4r∗(1 + δ) � 1−γ 2 αγ ≤ 2 max � 1, ρˆσ2 1 4r∗(1 + δ) � αγ ≤ max � 2, ρσ2 1 r∗ � � �� � :=C1 αγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We conclude this section with the following lemma, which states that initially the parallel component UtWt would grow much faster than the noise term, and would become well-aligned with Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, formal version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' There exists positive constants C2 = C2(X, ¯U) and C3 = C3(X, ¯U) such that the following inequalities hold for t = T sp α when α ∈ � 0, � ρ 10C1(X, ¯U) �10κ� : 28 ∥Ut∥ ≤ ∥X∥ (19a) σmin (UtWt) ≥ C2 · α 1− 2 log(1+µˆσ2s) 3 log(1+µˆσ2 1)−log(1+µˆσ2 s+1) (19b) ∥UtWt,⊥∥ ≤ C3 · α 1− 2 log(1+µˆσ2 s+1) 3 log(1+µˆσ2 1)−log(1+µˆσ2 s+1) (19c) ���V ⊤ Xs,⊥VUtWt ��� ≤ 200δ (19d) Proof : We prove this lemma by applying Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to t = T sp α defined in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The inequality (19a) can be directly verified by using Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2: ∥Ut∥ ≤ α � 1 + µˆσ2 1 �t + αγ ≤ � 1 + �C1(X, ¯U)ˆσ2 1 4r∗(1 + δ) � 1 3 � αγ/3 ≤ ˆC1(X, ¯U) · αγ/3∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' where ˆC1(X, ¯U) = 1 + � C1(X, ¯U)∥X∥2 2r∗(1+δ) � 1 3 (the constant C1 is defined in the previous lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The last inequality holds when α is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For the remaining inequalities, we first verify that the assumption in Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1: ασs(Kt) > 10 (ασs+1(Kt) + ∥Et∥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (20) By definition of Kt, we can see that for α ≤ � ρ 10C1 �10κ , ασs+1(Kt) + ∥Et∥ ≤ α � 1 + µˆσ2 s+1 �t + 4ˆσ−2 1 α3r∗(1 + δ) � 1 + µˆσ2 1 �3t ≤ C1(X, ¯U) · αγ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1ρα 1− 2 log(1+µˆσ2s) 3 log(1+µˆσ2 1)−log(1+µˆσ2 s+1) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1ασs(Kt) where ∥ET sp α ∥ is bounded in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Hence (20) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let L be the span of top-s 29 eigenvectors of M, then by Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, at t = T sp α we have σs (UtWt) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4ασs (Kt) σmin � V ⊤ L ¯U � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1αρ � 1 + µˆσ2 s �t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1ρ � ρˆσ2 1 4r∗(1 + δ) � log(1+µˆσ2s) 3 log(1+µˆσ2 1)−log(1+µˆσ2 s+1) α 1− 2 log(1+µˆσ2s) 3 log(1+µˆσ1)−log(1+µˆσ2 s+1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1ρ �ρσ2 1 8r∗ � 1 10κ � �� � :=C2(X, ¯U) α 1− 2 log(1+µˆσ2s) 3 log(1+µˆσ1)−log(1+µˆσ2 s+1) ∥UtWt,⊥∥ ⩽ 2ασ2 s+1 (Kt) + ∥Et∥ ≤ 2C1(X, ¯U) � �� � :=C3(X, ¯U) α 1− 2 log(1+µˆσ2 s+1) 3 log(1+µˆσ2 1)−log(1+µˆσ2 s+1) ���V ⊤ Xs,⊥VUtWt ��� ⩽ 100 � δ + ασs+1 (Kt) + ∥Et∥ αρσs (Kt) � ≤ 100 � �δα 2 log(1+µˆσ2s)−2 log(1+µˆσ2 s+1) 3 log(1+µˆσ2 1)−log(1+µˆσ2 s+1) � � ≤ 200δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (21) The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 The parallel improvement phase This subsection is devoted to proving Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 which we recall below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 hold and let c3 = 104κr 1 2∗ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then for suffi- ciently small α, the following inequalities hold when T sp α ≤ t < T pi α,s: σmin � V ⊤ XsUt+1 � ≥ σmin � V ⊤ XsUt+1Wt � ≥ � 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5µ � σ2 s + σ2 s+1 �� σmin � V ⊤ XsUt � , (13a) ∥Ut+1Wt+1,⊥∥ ≤ � 1 + µ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4σ2 s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6σ2 s+1 �� ∥UtWt,⊥∥ , (13b) ���V ⊤ Xs,⊥VUt+1Wt+1 ��� ≤ c3, (13c) rank(V ⊤ XsUt+1) = rank(V ⊤ XsUt+1Wt) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (13d) 30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The parallel component In the following we bound σmin � V ⊤ XsUt+1Wt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We state our main result of this section in the lemma below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 holds, ∥VX⊥ s VUtWt∥ ≤ c3 < 10−2κ−1 and ∆t = (A∗A − I) (XX⊤ − UtU ⊤ t ) satisfies ∥∆t∥ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2κ−1r − 1 2 ∗ ∥X∥2, then we have σmin(V ⊤ XsUt+1) ≥ σmin(V ⊤ XsUt+1Wt) ≥ � 1 + µ � σ2 s − 5c3∥X∥2 − 2∥∆t∥ � − 20µ2∥X∥4� � 1 − µσ2 min(V ⊤ XsUt) � σmin(V ⊤ XsUt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : The update rule of GD implies that V ⊤ XsUt+1Wt = V ⊤ Xs � I + µ(XsX⊤ s − UtU ⊤ t ) + µ∆t � UtWt (22a) = (I + µΣ2 s)V ⊤ XsUtWt − µV ⊤ XsUtU ⊤ t UtWt + µV ⊤ Xs∆tUtWt (22b) = (I + µΣ2 s)V ⊤ XsUtWt − µV ⊤ XsUtU ⊤ t VXsV ⊤ XsUtWt − µV ⊤ X UtU ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt + µV ⊤ Xs∆tUtWt = (I + µΣ2 s)V ⊤ XsUtWt(I − µW ⊤ t U ⊤ t VXsV ⊤ XsUtWt) + µV ⊤ Xs∆tUtWt − µV ⊤ XsUtU ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt + µ2Σ2 sV ⊤ XsUtWtW ⊤ t U ⊤ t VXsV ⊤ XsUtWt (22c) where (22a) follows from V ⊤ XsXX⊤ = V ⊤ XsXsX⊤ s + V ⊤ XsXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥X⊤ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ and V ⊤ XsXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (22b) follows from V ⊤ XsXsX⊤ s = V ⊤ XsVXsΣsV ⊤ Xs = ΣsV ⊤ Xs, and (22c) follows from V ⊤ XsUt = V ⊤ XsUtWtW ⊤ t + V ⊤ XsUtWt,⊥W ⊤ t,⊥ = V ⊤ XsUtWtW ⊤ t by definition of Wt and Wt,⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We now relate the last three terms in (22c) to V ⊤ XsUtWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since V ⊤ XsUtWt is invertible by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, V ⊤ XsVUtWt, ΣUtWt and WUtWt are also of full rank, thus we have UtWt = UtWt(V ⊤ XsUtWt)−1V ⊤ XsUtWt = UtWt � V ⊤ XsVUtWtΣUtWtW ⊤ UtWt �−1 V ⊤ XsUtWt = VUtWt � V ⊤ XsVUtWt �−1 V ⊤ XsUtWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (23) 31 Plugging (23) into the second and third terms of (22) and re-arranging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' we deduce that V ⊤ XsUt+1Wt = � I + µ(Σ2 s + P1 + P2) � V ⊤ XsUtWt(I − µW ⊤ t U ⊤ t VXsV ⊤ XsUtWt) + µ2 � Σ2 s + P1 + P2 � V ⊤ XsUtWtW ⊤ t U ⊤ t VXsV ⊤ XsUtWt = � I + µ � Σ2 s + P1 + P2 � + µ2 � Σ2 s + P1 + P2 � V ⊤ XsUtWtW ⊤ t U ⊤ t VXs � I − µV ⊤ XsUtWtW ⊤ t U ⊤ t VXs �−1� V ⊤ XsUtWt(I − µW ⊤ t U ⊤ t VXsV ⊤ XsUtWt) (24) where we use the equation A = (I − µAA⊤)−1A(I − µA⊤A) with A = V ⊤ XsUtWt (when µ < 1 9∥X∥2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' I − µAA⊤ is invertible by Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3), and P1 = V ⊤ XsUtU ⊤ t VXs,⊥V ⊤ Xs,⊥VUtWt � V ⊤ XsVUtWt �−1 P2 = V ⊤ Xs∆tVUtWt � V ⊤ XsVUtWt �−1 (25) By assumption we have σmin � V ⊤ XsVUtWt � ≥ � 1 − ���V ⊤ Xs,⊥VUtWt ��� 2 ≥ 1 2, so that ∥P1∥ ≤ ���V ⊤ XsUtU ⊤ t VXs,⊥ ��� · ���V ⊤ Xs,⊥VUtWt ��� · ���� � V ⊤ XsVUtWt �−1���� ≤ 5c3∥X∥2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1∥X∥2 (26) and by our assumption we have ∥P2∥ ≤ ���� � V ⊤ XsVUtWt �−1���� · ∥∆t∥ ≤ 2∥∆t∥ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2κ−1r − 1 2 ∗ ∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (27) Moreover, note that ∥Σs∥2 = σ2 1 = ∥X∥2, and since µ < 10−4∥X∥−2 by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, we have ��� � I − µV ⊤ XsUtWtW ⊤ t U ⊤ t VXs �−1��� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus ���� � Σ2 s + P1 + P2 � V ⊤ XsUtWtW ⊤ t U ⊤ t VXs � I − µV ⊤ XsUtWtW ⊤ t U ⊤ t VXs �−1���� ≤ 20∥X∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='The equation (25) implies that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='σmin(V ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUt+1Wt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='≥ σmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='I + µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='Σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='s + P1 + P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='Σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='s + P1 + P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='V ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUtWtW ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t U ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t VXs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='I − µV ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUtWtW ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t U ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t VXs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='�−1� σmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='V ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUtWt(I − µW ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t U ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='t VXsV ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUtWt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 + µσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='min(Σs) − µ∥P1∥ − µ∥P2∥ − 20µ2∥X∥4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='σmin(V ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 − µσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='min(V ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 + µσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='s − µ∥P1∥ − µ∥P2∥ − 20µ2∥X∥4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='σmin(V ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 − µσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='min(V ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='XsUt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='Recall that P1 and P2 are bounded in (26) and (27) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' so we have that σmin(V ⊤ XsUt+1) ≥ σmin(V ⊤ XsUt+1Wt) ≥ � 1 + µ � σ2 s − 5c3∥X∥2 − 2∥∆t∥2� − 20µ2∥X∥4� � 1 − µσ2 min(V ⊤ XsUt) � σmin(V ⊤ XsUt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ The corollaries below immediately follow from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the conditions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4, if σ2 min(V ⊤ XsUt) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3κ−1∥X∥2, then we have σmin(V ⊤ XsUt+1) ≥ � 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5µ(σ2 s + σ2 s+1) � σmin(V ⊤ XsUt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 it remains to check that � 1 + µ � σ2 s − 5c3∥X∥2 − 2∥∆t∥2� − 20µ2∥X∥4� � 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3µκ−1∥X∥2� ≥ 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5µ(σ2 s+σ2 s+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Indeed, recall from the conditions of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 that 5c3∥X∥2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01κ−1∥X∥2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01(σ2 s − σ2 s+1) and similarly ∥∆t∥ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='005(σ2 s − σ2 s+1) and µ2∥X∥4 ≤ 10−4κ−1µ∥X∥2 ≤ 10−4(σ2 s − σ2 s+1), so that � 1 + µ � σ2 s − 5c3∥X∥2 − 2∥∆t∥2� − 20µ2∥X∥4� � 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3κ−1∥X∥2� ≥ � 1 + µ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='9σ2 s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1σ2 s+1) � � 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3µ(σ2 s − σ2 s+1) � = 1 + µ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6σ2 s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4σ2 s+1) − µ2∥X∥2(σ2 s − σ2 s+1) ≥ 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5µ(σ2 s + σ2 s+1) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the conditions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4, if σ2 min(V ⊤ XsUt) ≤ σ2 s − µσ4 s − 5c3∥X∥2 − 2∥∆t∥2 − 20µ2∥X∥4, (28) then we have that σmin(V ⊤ XsUt+1) ≥ σmin(V ⊤ XsUt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : A sufficient condition for σmin(V ⊤ XsUt+1) ≥ σmin(V ⊤ XsUt) to hold is that � 1 + µ � σ2 s − 5c3∥X∥2 − 2∥∆t∥2� − 20µ2∥X∥4� � 1 − µσ2 min(V ⊤ XsUt) � ⇐ σ2 s − 5c3∥X∥2 − 2∥∆t∥2 − 20µ∥X∥2 − σmin(V ⊤ XsUt) − µσ2 sσmin(V ⊤ XsUt) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When (28) holds, we have σ2 s − 5c3∥X∥2 − 2∥∆t∥2 − 20µ∥X∥2 − σmin(V ⊤ XsUt) ≥ µσ4 s ≥ µσ2 sσmin(V ⊤ XsUt) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ 33 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 The orthogonal component In this section we turn to analyze the noise term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='The main result of this section is presented in the following: Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 hold, V ⊤ XsUt+1Wt ∈ Rs×r is of full rank, ∥VXs,⊥VUtWt∥ ≤ c3 < 10−2κ−1 and ∥∆t∥ ≤ c3∥X∥2, then we have ∥Ut+1Wt+1,⊥∥ ≤ � 1 + µσ2 s+1 + 30µ∥X∥2c3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1µ2∥X∥4� ∥UtWt,⊥∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : By the definition of Wt,⊥, we have V ⊤ XsUtWt,⊥ = 0, thus ∥UtWt,⊥∥ = ���V ⊤ Xs,⊥UtWt,⊥ ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The latter can be decomposed as follows: V ⊤ Xs,⊥Ut+1Wt+1,⊥ = V ⊤ Xs,⊥Ut+1WtW ⊤ t Wt+1,⊥ � �� � =(a) + V ⊤ Xs,⊥Ut+1Wt,⊥W ⊤ t,⊥Wt+1,⊥ � �� � =(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In the following, we are going to show that the term (a) is bounded by c · µ where c is a small constant, while (b) grows linearly with a slow speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Bounding summand (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since 0 = V ⊤ XsUt+1Wt+1,⊥ = V ⊤ XsUt+1WtW ⊤ t Wt+1,⊥ + V ⊤ XsUt+1Wt,⊥W ⊤ t,⊥Wt+1,⊥ by definition, we have W ⊤ t Wt+1,⊥ = − � V ⊤ XsUt+1Wt �−1 V ⊤ XsUt+1Wt,⊥W ⊤ t,⊥Wt+1,⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (29) Thus the summand (a) can be rewritten as follows: V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1WtW ⊤ t Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = −V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt � V ⊤ XsUt+1Wt �−1 V ⊤ XsUt+1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ (30a) = −V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt � V ⊤ XsVUt+1WtΣUt+1WtWUt+1Wt �−1 V ⊤ XsUt+1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = −V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥VUt+1Wt � V ⊤ XsVUt+1Wt �−1 V ⊤ XsUt+1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ (30b) = −V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥VUt+1Wt � V ⊤ XsVUt+1Wt �−1 V ⊤ Xs � I + µA∗A � XX⊤ − UtU ⊤ t �� UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = −µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥VUt+1Wt � V ⊤ XsVUt+1Wt �−1 V ⊤ Xs �� XX⊤ − UtU ⊤ t � + ∆t � UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ (30c) = µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥VUt+1Wt � V ⊤ XsVUt+1Wt �−1 V ⊤ Xs � UtU ⊤ t − ∆t � UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥VUt+1Wt � V ⊤ XsVUt+1Wt �−1 M1V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 34 where M1 = V ⊤ Xs � UtU ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ − ∆tVXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In (30), (30a) follows from (29), (30b) holds since ΣUt+1WtW ⊤ Ut+1Wt ∈ Rs×s is invertible, and in (30c) we use V ⊤ XsUtWt,⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It follows that ∥(a)∥ ≤ µ ���V ⊤ Xs,⊥VUt+1Wt ��� · ���� � V ⊤ XsVUt+1Wt �−1���� ∥M1∥ ���V ⊤ Xs,⊥UtWt,⊥ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (31) By Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 we have ���V ⊤ Xs,⊥VUt+1Wt ��� ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01, which implies that ���� � V ⊤ XsVUt+1Wt �−1���� = σ−1 min � V ⊤ XsVUt+1Wt � = � 1 − ���V ⊤ Xs,⊥VUt+1Wt ��� 2�− 1 2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (32) Lastly, we bound M1 as follows: ∥M1∥ ≤ ���V ⊤ XsUtU ⊤ t VXs,⊥ ��� + ���(A∗A − I) � XX⊤ − UtU ⊤ t ���� ≤ ���V ⊤ XsUtWt ��� · ���V ⊤ Xs,⊥UtWt ��� + 10−3κ−1c3∥X∥2 ≤ 10∥X∥2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (33) where the second inequality follows from our assumption on ��(A∗A − I) � XX⊤ − UtU ⊤ t ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Combining (31), (32) and (33) yields ∥(a)∥ ≤ 20µ∥X∥2c3∥UtWt,⊥∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Bounding summand (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This is the main component in the error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We’ll see that although this term can grow exponentially fast, the growth speed is slower than the minimal eigenvalue of the parallel component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We have V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Ut+1Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � I + µ(XX⊤ − UtU ⊤ t ) + µ (A∗A − I) � XX⊤ − UtU ⊤ t �� UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ (34a) = � � �I + µΣ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ − µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtU ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ + µ V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥∆tVXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � �� � =:M2 � � � V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ (34b) = � I + µΣ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ − µV ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWtW ⊤ t U ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ + µM2 � V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � I − µW ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥U ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � (34c) + µ2 � Σ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ − V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWtW ⊤ t U ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ + M2 � V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥U ⊤ t UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ (34d) 35 where we recall that Σ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ = diag � σ2 s+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' σ2 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 0 � ∈ R(d−s)×(d−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In (34), (34a) follows from the update rule of GD, (34b) is obtained from V ⊤ Xs,⊥XX⊤ = Σ2 s,⊥V ⊤ Xs,⊥ and UtWt,⊥ = VXsV ⊤ XsUtWt,⊥ + VXs,⊥V ⊤ Xs,⊥UtWt,⊥ = VXs,⊥V ⊤ Xs,⊥UtWt,⊥, and lastly in (34d) we use V ⊤ Xs,⊥UtU ⊤ t VXs,⊥V ⊤ Xs,⊥UtWt,⊥ = V ⊤ Xs,⊥UtWtW ⊤ t U ⊤ t VXs,⊥V ⊤ Xs,⊥UtWt,⊥ + V ⊤ Xs,⊥UtWt,⊥W ⊤ t,⊥U ⊤ t VXs,⊥V ⊤ Xs,⊥UtWt,⊥ = V ⊤ Xs,⊥UtWtW ⊤ t U ⊤ t VXs,⊥V ⊤ Xs,⊥UtWt,⊥ + V ⊤ Xs,⊥UtWt,⊥W ⊤ t,⊥U ⊤ t UtWt,⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It follows that ���V ⊤ Xs,⊥Ut+1Wt,⊥ ��� ≤ ����I − µV ⊤ Xs,⊥UtWtW ⊤ t U ⊤ t VXs,⊥ ��� + µ∥Σs,⊥∥2 + µ∥M2∥ � ∥V ⊤ Xs,⊥UtWt,⊥∥ � I − µ∥V ⊤ Xs,⊥UtWt,⊥∥2� + µ2 ∥UtWt,⊥∥3 � σ2 s+1 + ∥Ut∥2 + 10−3κ−1c3∥X∥2� ≤ � 1 + µσ2 s+1 + µ∥∆t∥ � ∥UtWt,⊥∥ � 1 − µ∥UtWt,⊥∥2� + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1µ2∥X∥4 ∥UtWt,⊥∥ ≤ ∥UtWt,⊥∥ � 1 + µσ2 s+1 + µc3∥X∥2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1µ2∥X∥4� To summarize, we have ∥Ut+1Wt+1,⊥∥ ≤ � 1 + µσ2 s+1 + 30µ∥X∥2c3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1µ2∥X∥4� ∥UtWt,⊥∥ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ To bound the growth speed of the orthogonal component, we need to show that the quantity ���V ⊤ Xs,⊥VUtWt ��� remains small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The following lemma serves to complete an induction step from t to t + 1: Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose V ⊤ XsUt is of full rank, ∥VXs,⊥VUtWt∥ ≤ c3 and ∥UtWt,⊥∥ ≤ min {σmin(UtWt), c4} with max � c3, c4∥X∥−1� ≤ 10−2κ−1, and ∆t = (A∗A−I)(XX⊤−UtU ⊤ t ) satisfies ∥∆t∥ ≤ 10−3κ−1c3∥X∥2 and µ ≤ 10−4κ−1∥X∥−2c3, then we have ∥VXs,⊥VUt+1Wt+1∥ ≤ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Let Mt = A∗A(XX⊤ − UtU ⊤ t ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' so the update rule of GD implies that Ut+1Wt+1 = (I + µMt)UtWt+1 = (I + µMt) � UtWtW ⊤ t Wt+1 + UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1 � = (I + µMt) � VUtWtV ⊤ UtW UtWtW ⊤ t Wt+1 + UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1 � = (I + µMt)(I + P )VUtWt � �� � :=H V ⊤ UtWtUtWtW ⊤ t Wt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' where P = UtWt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥W ⊤ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥Wt+1 � V ⊤ UtWtUtWtW ⊤ t Wt+1 �−1 V ⊤ UtWt 36 and V ⊤ UtWtUtWtW ⊤ t Wt+1 is invertible since V ⊤ UtWtUtWt is invertible by our assumption that V ⊤ XsUt is of full rank and rank(UtWt) ≥ rank(V ⊤ XsUtWt) = rank(V ⊤ XsUt) = s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' and W ⊤ t Wt+1 is invertible by Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Indeed, Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6 implies that σmin � W ⊤ t Wt+1 � ≥ 1 2 by our condition on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The key observation here is that because the (square) matrix V ⊤ UtWtUtWtW ⊤ t Wt+1 is in- vertible, so that the column space of Ut+1Wt+1 is the same as that of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Following the line of proof of St¨oger & Soltanolkotabi, 2021, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 (for completeness, we provide details in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7), we deduce that ���V ⊤ Xs,⊥VUt+1Wt+1 ��� = ���V ⊤ Xs,⊥VHW ⊤ H ��� ≤ ����V ⊤ Xs,⊥ �� I + B − 1 2VUtWtV ⊤ UtWt � B + B⊤�� VUtWt − BVUtWtV ⊤ UtWt � B + B⊤� VUtWt + D ����� ≤ ����V ⊤ Xs,⊥ � I + B − 1 2VUtWtV ⊤ UtWt � B + B⊤�� VUtWt ���� + 2∥B∥2 + ∥D∥ (35) where B = (I + µMt)(I + P ) − I and ∥D∥ ≤ 100∥B∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By assumption we have ∥P ∥ ≤ ∥UtWt,⊥∥ ∥Wt,⊥Wt+1∥ σmin(UtWt)σmin(W ⊤ t Wt+1) ≤ 2 ∥Wt,⊥Wt+1∥ , so that ���B − µ(XX⊤ − UtU ⊤ t ) ��� ≤ µ∥Mt − (XX⊤ − UtU ⊤ t )∥ + ∥P ∥ + µ∥Mt∥∥P ∥ ≤ µ ∥∆t∥ + 2 ∥Wt,⊥Wt+1∥ + 4µ∥X∥2 ∥Wt,⊥Wt+1∥ ≤ µ ∥∆t∥ + 6 ∥Wt,⊥Wt+1∥ ≤ 18µ � 10µ∥X∥3 + c4 � c3∥X∥ + 7µ ∥∆t∥ ≤ 18µ � 10µ∥X∥3 + c4 � c3∥X∥ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01µκ−1c3∥X∥2 (36) where we use Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6 to bound ���W ⊤ t,⊥Wt+1 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let B1 = µ(XX⊤ − UtU ⊤ t ) and R1 = V ⊤ Xs,⊥ � I + B1 − VUtWtV ⊤ UtWtB1 � VUtWt, then we have R1 = V ⊤ Xs,⊥ � I + µ � I − VUtWtV ⊤ UtWt � � XX⊤ − UtU ⊤ t �� VUtWt = � I + µΣ2 s,⊥ � V ⊤ Xs,⊥VUtWt � I − µV ⊤ UtWtXX⊤VUtWt � − µV ⊤ Xs,⊥ � I − VUtWtV ⊤ UtWt � UtWt,⊥W ⊤ t,⊥U ⊤ t VXs,⊥V ⊤ Xs,⊥VUtWt + µ2Σ2 s,⊥V ⊤ Xs,⊥VUtWtV ⊤ UtWtXX⊤VUtWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (37) 37 By Weyl’s inequality (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2) and our assumption on c3, σmin � V ⊤ UtWtXX⊤VUtWt � ≥ σmin � V ⊤ UtWtXsX⊤ s VUtWt � − ���V ⊤ UtWtXs,⊥X⊤ s,⊥VUtWt ��� 2 ≥ σmin � V ⊤ UtWtXsX⊤ s VUtWt � − σ2 s+1 ���V ⊤ Xs,⊥VUtWt ��� 2 ≥ σ2 s ���V ⊤ UtWtVXs ��� 2 − σ2 s+1c2 3 = σ2 s − (σ2 s + σ2 s+1)c2 3 > 1 2 � σ2 s + σ2 s+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' So we have ∥R1∥ ≤ � 1 − µ 2 (σ2 s − σ2 s+1) � ���V ⊤ Xs,⊥VUtWt ��� + µc3c2 4 + µ2∥X∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It thus follows from (35) that ���V ⊤ X⊥ s VUt+1Wt+1 ��� ≤ ∥R1∥ + 2∥B − B1∥ + 102∥B∥2 ≤ � 1 − µ 2 (σ2 s − σ2 s+1) � ���V ⊤ Xs,⊥VUtWt ��� + 40µc3c4∥X∥ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='02µκ−1c3∥X∥2 + 103µ2∥X∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since ���V ⊤ Xs,⊥VUtWt ��� ≤ c3, it follows from our assumption on c3, c4 and µ that ���V ⊤ X⊥ s VUt+1Wt+1 ��� ≤ c3 as well, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 Induction Let T pi α,s = min � t ⩾ 0 : σ2 min � V ⊤ XsUα,t+1 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3κ−1∥X∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' where pi stands for the parallel improvement phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In this section, we show that when T sp α ≤ t < T pi α,s, the parallel component grows exponentially faster than the orthogonal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We prove this via induction and the base case is already shown in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, detailed version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 hold and let c3 = 104κ√r∗δ, c4 ≤ 10−3κ−1∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then the following holds for all T sp α ≤ t < T pi α,s as long as α ≤ C4(X, ¯U) = � κ C2(X, ¯U)2 C3(X, ¯U)2 �−2κ is sufficiently small: σmin � V ⊤ XsUt+1 � ≥ σmin � V ⊤ XsUt+1Wt � ≥ � 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5µ � σ2 s + σ2 s+1 �� σmin � V ⊤ XsUα,t � (38a) ∥Ut+1Wt+1,⊥∥ ≤ min �� 1 + µ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4σ2 s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6σ2 s+1 �� ∥UtWt,⊥∥ , c4 � (38b) ���V ⊤ Xs,⊥VUt+1Wt+1 ��� ≤ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (38c) rank(V ⊤ XsUt+1) = rank(V ⊤ XsUt+1Wt) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (38d) 38 Proof : The base case t = T pi α,s is already proved in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Now suppose that the lemma holds for t, we now show that it holds for t + 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' To begin with, we bound the term ∥∆t∥ as follows: ∥∆t∥ = ���(A∗A − I)(XX⊤ − UtU ⊤ t ) ��� ≤ ���(A∗A − I)(XX⊤ − UtWtW ⊤ t U ⊤ t ) ��� + ���(A∗A − I)UtWt,⊥W ⊤ t,⊥U ⊤ t ��� ≤ 10δ√r∗∥X∥2 + δ ���UtWt,⊥W ⊤ t,⊥U ⊤ t ��� ∗ ≤ 10δ√r∗∥X∥2 + δd ∥UtWt,⊥∥2 (39) where in the second inequality we use Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and in the third inequality we use ∥A∥∗ ≤ √ d∥A∥, ∀A ∈ Rd×d and the induction hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By induction hypothesis, there exists a constant ˆC4(X, ¯U) = C2(X, ¯U) C3(X, ¯U) (see Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3) such that σmin � V ⊤ XsUt � ∥UtWt,⊥∥ ≥ σmin � V ⊤ XsUT sp α � ��UT sp α WT sp α ,⊥ �� ≥ ˆC4 · α−γs (40) where γs = 2 � log � 1 + µˆσ2 s � − log � 1 + µˆσ2 s+1 �� 3 log � 1 + µˆσ2 1 � − log � 1 + µˆσ2 s+1 � ≥ 1 4κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since we must have σ2 min � V ⊤ XsUt � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3κ−1∥X∥2 by definition of T pi α,s, it follows that ∥UtWt,⊥∥2 ≤ 10κ∥X∥2 ˆC2 4α 1 2κ , so for α ≤ ( ˆC−2 4 κ)−2κ, ∥∆t∥ ≤ 11δ√r∗∥X∥2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The above inequality combined with our assumption on δ implies that the conditions on ∥∆t∥ in Lemmas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We now show that (38a) to (38d) hold for t + 1, which completes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' First, since t < T pi α,s, we have σmin � V ⊤ XsUt+1 � ≤ κ−1∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, the induction hypothesis implies that ���V ⊤ Xs,⊥VUt−1Wt−1 ��� ≤ c3 and that V ⊤ XsUα,t is of full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus the conditions of Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 are all satisfied, and we deduce that (38a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Second, the assumptions on c3, c4 and δ, combined with Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5, immediately implies ∥Ut+1Wt+1,⊥∥ ≤ � 1 + µ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4σ2 s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6σ2 s+1 �� ∥UtWt,⊥∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As a result, similar to (40) we observe that σmin � V ⊤ XsUt+1 � ∥Ut+1Wt+1,⊥∥ ≥ σmin � V ⊤ XsUT sp α � ��UT sp α WT sp α ,⊥ �� ≥ ˆC4 · α− 1 4κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since σmin � V ⊤ XsUt+1 � ≤ ∥X∥, when α is sufficiently small we must have that ∥Ut+1Wt+1,⊥∥ ≤ c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6 implies that (38c) is true, and (38d) follows from our application of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ 39 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 The refinement phase and concluding the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 We have shown that the parallel component σmin � V ⊤ XsUt+1 � grows exponentially faster than the orthogonal component ∥UtWt,⊥∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In this section, we characterize the GD dynamics after T pi α,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We begin with the following lemma, which is straightforward from the proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='8 (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, formal version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the conditions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, the following inequality holds when α ≤ C4(X, ¯U): ���UT pi α,sWT pi α,s,⊥ ��� ≤ C5(X, ¯U) · α 1 4κ where C5 = √ 10κ∥X∥C2(X, ¯U) C3(X, ¯U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The following lemma states that in a certain time period after T pi α,s, the parallel and orthogonal components still behave similarly to the second (parallel improvement) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the conditions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7, there exists �tα,s ≥ 1 log(1+µσ2s) log � 10−4c3∥X∥2 √ dκC5 α− 1 4κ � = Θ � log α−1� when α → 0 such that when 0 ≤ t − T pi α,s ≤ �tα,s, we have σmin � V ⊤ XsUt � ≥ σmin (UtWt) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3κ−1∥X∥2, (41a) ∥UtWt∥ ≤ � 1 + µ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4σ2 s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6σ2 s+1) �t−T pi α,s ���UT pi α,sWT pi α,s ��� , (41b) ∥VXs,⊥VUtWt∥ ≤ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (41c) Proof : We choose �tα,s = min � t ≥ 0 : ∥Ut+1Wt+1,⊥∥2 ≤ c5 � (42) where c5 = 10−4d− 1 2 κ−1c3∥X∥2 (43) We prove (41) by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The proof follows the idea of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7, except that we need to bound ∥∆t∥ in each induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Concretely, suppose that (41) holds at time t, then ∥∆t∥ = ���(A∗A − I)(XX⊤ − UtU ⊤ t ) ��� ≤ ���(A∗A − I)(XX⊤ − UtWtW ⊤ t U ⊤ t ) ��� + ���(A∗A − I)UtWt,⊥W ⊤ t,⊥U ⊤ t ��� ≤ 10δ√r∗∥X∥2 + δ ���UtWt,⊥W ⊤ t,⊥U ⊤ t ��� ∗ ≤ 10δ√r∗∥X∥2 + δc5 √ d ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='02κ−1c3∥X∥2 (44) where we used the definition of c5 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As a result, we can apply the conclusion of Lemmas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6 which implies that (41) holds for t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, combining Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='8 and (41b) yields �tα,s = Θ � log 1 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We now present the main result of this section: 40 Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that 0 ≤ t − T π α,s ≤ �tα,s, ���V ⊤ Xs,⊥VUtWt ��� ≤ c3 and the conditions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7 hold, then we have ���V ⊤ Xs(XX⊤ − Ut+1U ⊤ t+1) ��� F ≤ � 1 − 1 2µτ � ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� F + 20µ∥X∥4 (δ + 5c3) + 2000µ2√r∗∥X∥6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' where we recall that τ = min1≤s≤ˆr∧r∗(σ2 s − σ2 s+1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Recall that Mt = A∗A � XX⊤ − UtU ⊤ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The update of GD implies that XX⊤ − Ut+1U ⊤ t+1 = XX⊤ − (I + µMt)UtU ⊤ t (I + µMt) = � I − µUtU ⊤ t � � XX⊤ − UtU ⊤ t � � I − µUtU ⊤ t � � �� � =(i) +µ ∆tUtU ⊤ t � �� � =(ii) + µUtU ⊤ t ∆t � �� � =(iii) +µ2 (Et,1 + Et,2) , where Et,1 = −UtU ⊤ t � XX⊤ − UtU ⊤ t � UtU ⊤ t and Et,2 = −MtUtU ⊤ t Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since ∥Ut∥ ≤ 3∥X∥ by Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, and ��Mt − (XX⊤ − UtU ⊤ t ) �� = ∥∆t∥ ≤ ∥X∥2 which is shown in (44), we have ���V ⊤ XsEt,1 ��� F = ���V ⊤ XsUtU ⊤ t � XX⊤ − UtU ⊤ t � UtU ⊤ t ��� F ≤ √r∗ ���V ⊤ XsUtUt � XX⊤ − UtU ⊤ t � UtU ⊤ t ��� 2 ≤ 103√r∗∥X∥6 and ���V ⊤ XsEt,2 ��� F = ���V ⊤ Xs � (A∗A) � XX⊤ − UtU ⊤ t �� UtU ⊤ t � (A∗A) � XX⊤ − UtU ⊤ t ����� F ≤ √r∗ ��� � (A∗A) � XX⊤ − UtU ⊤ t �� UtU ⊤ t � (A∗A) � XX⊤ − UtU ⊤ t ����� ≤ 103√r∗∥X∥6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Note that we would like to bound ��V ⊤ Xs � XX⊤ − Ut+1U ⊤ t+1 ��� F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We deal with the above three 41 terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' we have ���V ⊤ Xs � I − µUtU ⊤ t � � XX⊤ − UtU ⊤ t � (I − µUtUt) ��� F = ���V ⊤ Xs � I − µUtU ⊤ t � VXsV ⊤ Xs � XX⊤ − UtU ⊤ t � � I − µUtU ⊤ t ���� F + ���V ⊤ Xs � I − µUtU ⊤ t � VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � XX⊤ − UtU ⊤ t � � I − µUtU ⊤ t ���� F ≤ ���I − µV ⊤ XsUtU ⊤ t VXs ��� ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� + µ ���V ⊤ XsUtU ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ � XX⊤ − UtU ⊤ t ���� F (45a) ≤ � 1 − µσ2 min(UtWt)σ2 min � V ⊤ XsVUtWt �� ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� F + 100µ∥X∥4c3 (45b) ≤ � 1 − 1 2µτ � ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� F + 100µ∥X∥4c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (45c) where in (45a) we use ��I − µUtU ⊤ t �� ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (45b) follows from σmin � V ⊤ XsUtU ⊤ t VXs � = σmin � V ⊤ XsUtWtW ⊤ t U ⊤ t VXs � ≥ σmin � V ⊤ XsUtWt �2 ≥ σ2 min(UtWt)σ2 min � V ⊤ XsVUtWt � and ���V ⊤ XsUtU ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ ��� = ���V ⊤ XsUtWtW ⊤ t U ⊤ t VXs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥ ��� ≤ ∥Ut∥2 ���V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥UtWt ��� ≤ c3∥Ut∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' and lastly (45c) is obtained from σ2 min � V ⊤ XsVUtWt � ≥ 1 − ���V ⊤ Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='⊥VUtWt ��� 2 ≥ 1 − c2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For the second and the third terms, we have ���∆tUtU ⊤ t + UtU ⊤ t ∆t ��� ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1κc3∥X∥4 (46) where we use the estimate in (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Combining (45) and (46) yields ���V ⊤ Xs(XX⊤ − Ut+1U ⊤ t+1) ��� F ≤ � 1 − 1 2µτ � ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� F + 200µ∥X∥4c3 + 110µ2√r∗∥X∥6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ To apply the result of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, we need to verify that ∥VXs,⊥VUtWt∥ ≤ c3 still holds when t ≥ T pi α,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In fact, this is true as long as t − T pi α,s ≤ O � log 1 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 42 Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the conditions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7, if T pi α,s ≤ t ≤ T pi α,s + γs log c4 C5(X, ¯U) · log 1 α log (1 + µσ2s) =: T re α,s, then ∥Ut+1Wt+1,⊥∥ ≤ (1 + µσ2 s) ∥UtWt,⊥∥ and ���V ⊤ Xs,⊥VUtWt ��� ≤ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' As a consequence, we have ∥UtWt,⊥∥ ≤ (1 + µσ2 s)t−T pi α,sC5(X, ¯U) · αγs ≤ c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : The proof is basically the same as that of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7 and we only provide a sketch here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We induct on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The base case t = T pi α,s is already proved in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that the lemma holds for t − 1 with t < T re α,s, then the choice of T re α,s combined with Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 imply that ∥UtWt,⊥∥ ≤ � 1 + µσ2 s � ∥Ut−1Wt−1,⊥∥ ≤ � 1 + µσ2 s �t−T pi α,s ���UT pi α,sWT pi α,s,⊥ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since we have ���UT pi α,sWT pi α,s,⊥ ��� ≤ C5(X, ¯U) · αγs by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='8, the choice of T re α,s implies that ∥UtWt,⊥∥ ≤ c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The bound ���V ⊤ Xs,⊥VUtWt ��� ≤ c3 then follows from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We will only use a weaker version of this lemma, namely that the bounds holds for all T pi α,s ≤ t ≤ T ft α,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When α is sufficiently small, this can be directly derived from Lemmas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='10 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='11 since ˜tα,s, T re α,s − T pi α,s = Θ � log 1 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Specifically, we have proven Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 in the main text: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that T pi α,s ≤ t ≤ T ft α,s and all the conditions in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 hold, then we have ���V ⊤ Xs(XX⊤ − Ut+1U ⊤ t+1) ��� F ≤ � 1 − 1 2µτ � ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� F + 20µ∥X∥4 (δ + 5c3) + 2000µ2√r∗∥X∥6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, it holds that ∥Ut+1Wt+1,⊥∥ ≤ (1 + σ2 s)∥UtWt,⊥∥ and ���VX⊥ s VUtWt ��� ≤ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We are now ready to present our first main result, which states that with small initialization, GD would visit the O(δ)-neighborhood of the rank-s minimizer of the full observation loss i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' XsX⊤ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (Restatement of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, if the initialization scale α is sufficiently small, then for all 1 ≤ s ≤ ˆr ∧ r∗ there exists a time T ft α,s ∈ Z+ (where ft stands for fitting the ground-truth) such that ���XsX⊤ s − UT pi α,sU ⊤ T pi α,s ��� F ≤ 107κ3r∗∥X∥2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 43 Proof : First, observe that for all t ≥ 0, ���XsX⊤ s − UtU ⊤ t ��� F ≤ ��� � XsX⊤ s − UtU ⊤ t � VXsV ⊤ Xs ��� F + ���UtU ⊤ t VX⊥ s V ⊤ X⊥ s ��� F ≤ ��� � XsX⊤ s − UtU ⊤ t � VXsV ⊤ Xs ��� F + ���V ⊤ X⊥ s UtU ⊤ t VX⊥ s ��� F ≤ ���V ⊤ Xs � XsX⊤ s − UtU ⊤ t ���� F + √r∗ ���V ⊤ X⊥ s UtWt ��� 2 + √ d ���V ⊤ X⊥ s UtWt,⊥ ��� 2 ≤ ���V ⊤ Xs � XsX⊤ s − UtU ⊤ t ���� + 9√r∗∥X∥2 ∥VXs,⊥VUtWt∥2 + √ d∥UtWt,⊥∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (47) We set c3 = 103κ√r∗δ and T ft α,s = T pi α,s − log � 10−2∥X∥−2τc−1 3 � log � 1 − 1 2µτ � , (48) where we recall that τ = κ−1∥X∥2, then for small α we have T ft α,s ≤ T pi α,s + �tα,s (defined in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Hence for T pi α,s ≤ t < T ft α,s we always have ∥VXs,⊥VUtWt∥ ≤ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='10 and the choice of c3 and δ, we have for T pi α,s ≤ t < T ft α,s that ���V ⊤ Xs � XX⊤ − Ut+1U ⊤ t+1 ���� F ≤ � 1 − 1 2µτ � ���V ⊤ Xs � XX⊤ − UtU ⊤ t ���� F +30µ∥X∥4√r∗c3 which implies that for t = T ft α,s, ���V ⊤ Xs � XX⊤ − UTsU ⊤ Ts ���� F ≤ 80κ∥X∥2√r∗c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Meanwhile, by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='9 we have ∥UtWt,⊥∥ ≤ c5 (c5 is defined in (43)) and ���V ⊤ Xs,⊥VUtWt ��� ≤ c3 at t = T ft α,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Plugging into (47) yields ���XsX⊤ s − UT ft α,sU ⊤ T ft α,s ��� F ≤ 80κ∥X∥2√r∗c3 + 9∥X∥2c2 3 √r∗ + c2 5 √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By definition of c3 and c5 we deduce that ���XsX⊤ s − UT ft α,sU ⊤ T ft α,s ��� F ≤ 102τ −2∥X∥6√r∗c3 ≤ 105κ3r∗∥X∥2δ, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' There exists a constant C6(X, ¯U) = C5(X, ¯U) · (1 + µσ2 s)T ft α,s−T pi α,s (49) such that max 0≤t≤T ft α,s ∥UtWt,⊥∥ ≤ C1 · α 1 4κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : The case of t ≤ T pi α,s directly follows from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For t > T pi α,s, we know from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='9 that ∥UtWt,⊥∥ ≤ ���UT pi α,sWT pi α,s,⊥ ��� · � 1 + µσ2 s �T ft α,s−T pi α,s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By (48), the second term is a constant independent of α, so the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ 44 D Auxiliary results for proving Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 This section contains a collection of auxiliary results that are used in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 The spectral phase In the section, we provide auxiliary results for the analysis in the spectral phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Recall that Kt = (I + µM)t and Ut = U sp t + Et = KtU0 + Et and U0 = α ¯U with ∥ ¯U∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Also recall that M = �rank(M) i=1 ˆλiˆviˆv⊤ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' we additionally define Ms = �min{s,rank(M)} i=1 ˆλiˆviˆv⊤ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Similarly, let Lt be the span of the top-s left singular vectors of Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The following lemma shows that power iteration would result in large eigengap of Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let ˆρ = σmin � V ⊤ Ms ¯U � > 0, then the following three inequalities hold, given that the denominator of the third is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' σs(Ut) ≥ α � ˆρσs � ˆZt � − σs+1 � ˆZt �� − ∥Et∥ , (50a) σs+1(Ut) ≤ ασs+1 � ˆZt � + ∥Et∥ , (50b) ���V ⊤ M⊥ s VLt ��� ≤ ασs+1 � ˆZt � + ∥Et∥ αˆρσs � ˆZt � − 2 � ασs+1 � ˆZt � + ∥Et∥ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (50c) Proof : By Weyl’s inequality we have σs+1(Ut) = σs+1 � (1 + µM)tU0 � + ∥Et∥ = ασs+1 � (1 + µM)t ¯U � + ∥Et∥ ≤ ασs+1 � (1 + µMs)t ¯U � + α ��� (1 + µM)t − (1 + µMs)t� ¯U �� + ∥Et∥ ≤ α(1 + µˆλs+1)t + ∥Et∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus (50b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Similarly, σs(Ut) ≥ ασs � NtVMsV ⊤ Ms ¯U � − α(1 + µˆλs+1)t − ∥Et∥ ≥ ασs (NtVMs) σmin � V ⊤ Ms ¯U � − α(1 + µˆλs+1)t − ∥Et∥ ≥ αˆρ(1 + µˆλs)t − α(1 + µˆλs+1)t − ∥Et∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, note that we can write α(1 + µMs)t ¯U = VMs (1 + µΣMs)tV ⊤ Ms ¯U � �� � invertible , so that the subspace spanned by the left singular vectors of α(1 + µMs)t ¯U coincides with the column span of VMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since Lt is the span of top-s left singular vectors of Ut, we apply Wedin’s sin theorem (Wedin, 1972) and deduce (50c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ 45 The next lemma relates the quantities studied in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 with those that are needed in the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The proof is the same as St¨oger & Soltanolkotabi, 2021, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4, so we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that ���V ⊤ Xs,⊥VLt ��� ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 for some t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then it holds that σs (UtWt) ≥ 1 2σs (Ut) , (51a) ���V ⊤ Xs,⊥VUtWt ��� ≤ 10 ���V ⊤ Xs,⊥VLt ��� , (51b) ∥UtWt,⊥∥ ≤ 2σs+1 (Ut) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (51c) Combining the above two lemmas, we directly obtain the following corollary: Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that ασs(Kt) > 10 (ασs+1(Kt) + ∥Et∥), then we have that σs (UtWt) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4ασr⋆ (Kt) σmin � V ⊤ L ¯U � ∥UtWt,⊥∥ ≤ 2 (ασs+1 (Kt) + ∥Et∥) ���V ⊤ Xs,⊥VUtWt ��� ≤ 100 � δ + ασs+1 (Kt) + ∥Et∥ αˆρσs (Kt) � (52) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 The parallel improvement phase In the section, we provide auxiliary results for the analysis in the parallel improvement phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (St¨oger & Soltanolkotabi, 2021, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4) For sufficiently small µ and δ, sup- pose that ∥Ut∥ ≤ 3∥X∥, then we also have ∥Ut+1∥ ≤ 3∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the assumptions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5, we have ���V ⊤ Xs,⊥VUt+1Wt ��� ≤ 2 � c3 + 10µ∥X∥2� ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : The proof of this lemma is essentially the same as St¨oger & Soltanolkotabi, 2021, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, and we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the assumptions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6, we have σmin � V ⊤ XsUt+1 � ≥ 1 2σmin(UtWt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : We have σmin � V ⊤ XsUt+1 � ≥ σmin � V ⊤ XsUt+1Wt � = σmin � V ⊤ Xs (I + µMt) UtWt � ≥ σmin � V ⊤ Xs (I + µMt) VUtWt � σmin � V ⊤ UtWtUtWt � ≥ � σmin � V ⊤ XsVUtWt � − µ∥Mt∥ � σmin(UtWt) ≥ �� 1 − c2 3 − 10µ∥X∥2 � σmin(UtWt) ≥ 1 2σmin(UtWt) 46 where the last step follows from σmin � V ⊤ XsVUtWt �2 ≥ 1 − ���V ⊤ Xs,⊥VUtWt ��� 2 ≥ 1 − c2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the assumptions in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6, we have ���W ⊤ t,⊥Wt+1 ��� ≤ 3µ � 10µ∥X∥2 + c4 � c3∥X∥ + µ ���(A∗A − I)(XX⊤ − UtU ⊤ t ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : The proof roughly follows [St¨oger & Soltanolkotabi, 2021, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3], but we include it here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since V ⊤ XsUt+1 = Vt+1Σt+1Wt+1 and Vt+1Σt+1 ∈ Rs×s is invertible, we have ���W ⊤ t,⊥Wt+1 ��� = ����W ⊤ t,⊥U ⊤ t+1VXs � V ⊤ XsUt+1U ⊤ t+1VXs �− 1 2 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since V ⊤ XsUt+1Wt,⊥ = V ⊤ Xs � I + µA∗A(XX⊤ − UtU ⊤ t ) � UtWt,⊥ = V ⊤ Xs � I + µ(XX⊤ − UtU ⊤ t ) � UtWt,⊥ + µV ⊤ Xs∆tUtWt,⊥ = −µV ⊤ XsUtU ⊤ t UtWt,⊥ + µV ⊤ Xs∆tUtWt,⊥ (53a) = −µV ⊤ XsUtWtW ⊤ t U ⊤ t UtWt,⊥ + µV ⊤ Xs∆tUtWt,⊥ (53b) = −µ V ⊤ XsUtWtW ⊤ t U ⊤ t VXs,⊥V ⊤ Xs,⊥UtWt,⊥ � �� � =:K1 +µ V ⊤ Xs∆tUtWt,⊥ � �� � :=K2 (53c) where (53a) follows from V ⊤ XsXX⊤UtWt,⊥ = ΣsV ⊤ XsUtWt,⊥ = 0, and in (53b) and (53c) we use V ⊤ XsUtWt,⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For K1, note that ���� � V ⊤ XsUt+1U ⊤ t+1VXs �− 1 2 V ⊤ XsUt ���� ≤ ���� � V ⊤ XsUt+1U ⊤ t+1VXs �− 1 2 V ⊤ XsUt+1 ���� + µ ���� � V ⊤ XsUt+1U ⊤ t+1VXs �− 1 2 V ⊤ XsA∗A(XX⊤ − UtU ⊤ t )Ut ���� ≤ 1 + 10µ∥X∥3σ−1 min � V ⊤ XsUt+1 � so that ���� � V ⊤ XsUt+1U ⊤ t+1VXs �− 1 2 K1 ���� ≤ � 1 + 10µ∥X∥3σ−1 min � V ⊤ XsUt+1 �� ���V ⊤ Xs,⊥UtWt ��� ∥UtWt,⊥∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 47 Plugging into (53), we deduce that ���W ⊤ t,⊥Wt+1 ��� ≤ 3µ � 1 + 10µ∥X∥3σ−1 min � V ⊤ XsUt+1 �� ���V ⊤ Xs,⊥VUtWt ��� ∥X∥ ∥UtWt,⊥∥ + µσ−1 min � V ⊤ XsUt+1 � ∥UtWt,⊥∥ ���(A∗A − I)(XX⊤ − UtU ⊤ t ) ��� ≤ 3µ � ∥UtWt,⊥∥ + 10µ∥X∥3� ���V ⊤ Xs,⊥VUtWt ��� ∥X∥ + µσ−1 min � V ⊤ XsUt+1 � ∥UtWt,⊥∥ ���(A∗A − I)(XX⊤ − UtU ⊤ t ) ��� ≤ 3µ � 10µ∥X∥2 + c4 � c3∥X∥ + µ ���(A∗A − I)(XX⊤ − UtU ⊤ t ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' where in the last step we use Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 and the induction hypothesis which implies that σmin(UtWt) ≥ ∥UtWt,⊥∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The matrix H defined in the proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6 satisfies the following: H(H⊤H)− 1 2 = VUtWt + BVUtWt − 1 2(I + B)VUtWtV ⊤ UtWt � B + B⊤� VUtWt − D, where ∥D∥ ≤ 30∥B∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : By definition of H we have H(H⊤H)− 1 2 = (I + µM)(I + P )VUtWt � V ⊤ UtWt � I + P ⊤� (I + µM)2(I + P )VUtWt �− 1 2 = (I + B)VUtWt � V ⊤ UtWt � I + B⊤ + B + B⊤B � VUtWt �− 1 2 = (I + B)VUtWt � ���I + V ⊤ UtWt � B⊤ + B + B⊤B � VUtWt � �� � =:Θ � ��� − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It follows from (36) and our assumptions on c3 and c4 that ∥B∥ ≤ µ ���XX⊤ − UtU ⊤ t ��� + 6µ � c3c4∥X∥ + 50∥X∥2δ � ≤ 10µ∥X∥2 + 6µc3 (c4 + 1) ∥X∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (note that this step is independent and does not rely on earlier derivations in the proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6), so by Taylor’s formula, we have ����(I + Θ)− 1 2 − I + 1 2Θ ���� ≤ 3∥Θ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 48 Hence, ����H(H⊤H)− 1 2 − � VUtWt + BVUtWt − 1 2(I + B)VUtWtV ⊤ UtWt � B + B⊤� VUtWt ����� = ����(I + B)VUtWt � (I + Θ)− 1 2 − I + 1 2Θ − 1 2V ⊤ UtWtB⊤BVUtWt ����� ≤ (1 + ∥B∥) � 3∥Θ∥2 + 1 2∥B∥2 � < 30∥B∥2 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ E Proofs for the Landscape Results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 In this section, we study the landscape of under-parameterized matrix sensing problem fs(U) = 1 2 ���A(UU ⊤ − XX⊤) ��� 2 2 , U ∈ Rd×s Our key result in this section is Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, which states a local RSI condition for the ma- trix sensing loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Most existing results only study the landscape of (1) in the exact- and over- parameterized case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (2021) have studied the landscape of under-parameterized matrix factorization problem, but their main focus is the strict-saddle property of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 Analysis of the matrix factorization loss When the measurement satisfies the RIP condition, we can expect that the landscape of fs looks similar to that of the (under-parameterized) matrix factorization loss: Fs(U) = 1 2 ���UU ⊤ − XX⊤��� 2 F , U ∈ Rd×s for some s < ˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='For this reason, we first look into the landscape of Fs before analyzing fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Recall that XX⊤ = �r∗ i=1 σ2 i viv⊤ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The critical points of Fs(U) is characterized by the following lemma: Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' U ∈ Rd×s is a critical point of Fs(U) if and only if there exists an orthogonal matrix R ∈ Rs×s, such that all columns of UR are in {σivi : 1 ≤ i ≤ r∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Assume WLOG that XX⊤ = diag(σ2 1, σ2 2, · · · , σ2 r, 0, · · · , 0) =: Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let U be a critical point of Fs, then we have that � UU ⊤ − XX⊤� U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let W = UU ⊤, then (Σ − W )W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since W is symmetric, so is W 2, and we obtain that ΣW is also symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It is then easy to see that that if Σ = diag (λ1Im1, · · · , λtImt) with λ1 > λ2 > · · · > λt ≥ 0, then W is also in block-diagonal form: W = diag (W1, W2, · · · , Wt) where Wi ∈ Rmi×mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For each 1 ≤ i ≤ t, we then have the equation (λiImi − Wi) Wi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Hence, there exists an orthogonal matrix Ri such that R⊤ i WiRi is a diagonal matrix where the diagonal entries are either 0 or 49 √λi = σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let R = diag (R1, R2, · · · , Rt), then R⊤W R is diagonal and its nonzero diagonal entries form an s-subset of the multi-set {σi : 1 ≤ i ≤ r∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ In the case of s = 1, the global minimizers of Fs are ±σ1v1, and we can show that Fs is locally strongly convex around these minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Therefore, we can deduce that f is locally strongly-convex as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since our main focus is on s > 1, we put these details in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When s > 1, Fs(U) is not locally strongly-convex due to rotational invariance: if U is a global minimizer, then so is UR for any orthogonal matrix R ∈ Rs×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Instead, we establish a Re- stricted Secant Inequality for Fs, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For U ∈ Rd×s, let R be an orthogonal matrix that minimizes ∥U − XsR∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that dist(U, Xs) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1∥X∥−1τ (where we recall that τ = mins∈[r∗] � σ2 s − σ2 s+1 � is the eigengap of XX⊤), then we have ⟨∇Fs(U), U − XsR⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1τ · dist2(U, Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Assume WLOG that R = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7, U ⊤Xs is symmetric and positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let H = U − Xs, then ∇Fs(U) = (UU ⊤ − XX⊤)U = � (H + Xs)(H + Xs)⊤ − XX⊤� (H + Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' So we have ⟨∇Fs(U), U − Xs⟩ = �� (H + Xs)(H + Xs)⊤ − XX⊤� (H + Xs), H � = tr � H⊤ � (H + Xs)(H + Xs)⊤ − XX⊤� H + H⊤ � HH⊤ + HX⊤ s + XsH⊤� Xs � ≥ − tr � H⊤Xs,⊥X⊤ s,⊥H � − 3∥X∥∥H∥3 F + tr � H⊤HX⊤ s Xs � (54a) ≥ � σ2 s − σ2 s+1 � ∥H∥2 F − 3∥X∥∥H∥3 F (54b) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1τ∥H∥2 F where in (54a) we use tr � (H⊤Xs)2� ≥ 0 (since H⊤Xs is symmetric as noticed in the begin- ning of the proof), and (54b) is because of tr � H⊤HX⊤ s Xs � ≥ σmin � X⊤ s Xs � tr � H⊤H � = σ2 s∥H∥2 F and tr � H⊤Xs,⊥X⊤ s,⊥H � = tr � H⊤VXs,⊥Σs,⊥V ⊤ Xs,⊥H � ≤ ∥Σs,⊥∥ · ���H⊤VXs,⊥ ��� 2 F ≤ σ2 s+1∥H∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under the conditions of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2, we have ∥∇Fs(U)∥F ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1τdist(U, Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 50 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 Analysis of the matrix sensing loss The following lemma states that the minimizer of matrix sensing loss is also near-optimal for the matrix factorization loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let Z∗ s be a best rank-s solution as defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, then we have ���Z∗ s − XX⊤��� 2 F ≤ ���XsX⊤ s − XX⊤��� 2 F + 10δ ���XX⊤��� 2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : By the RIP property Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 we have ���XX⊤ − Z∗ s ��� 2 F ≤ (1 − δ)−1 ���A � XX⊤ − Z∗ s ���� 2 2 ≤ (1 − δ)−1 ���A � XX⊤ − XsX⊤ s ���� 2 2 ≤ 1 + δ 1 − δ ���XX⊤ − XsX⊤ s ��� 2 F ≤ ���XX⊤ − XsX⊤ s ��� 2 F + 10δ∥XX⊤∥2 F , where the second inequality holds due to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We now recall Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have dist(U ∗ s , Xs) ≤ 40δκ∥X∥F for any global mini- mizer U ∗ s of fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, ��Z∗ s − XsX⊤ s �� F ≤ 160δκ√r∗∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We prove the statements in this lemma separately in Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 and Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let U ∗ s be a global minimizer of fs, then we have dist(U ∗ s , Xs) ≤ 40δκ∥X∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Define S = � U ∈ Rd×s : dist(U, Xs) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1κ−1∥X∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' First we can show that U ∗ s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The main idea is to apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Indeed, it is easy to see that lim ∥U∥F →+∞ Fs(U) = +∞, so the condition (1) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' To check condition (2), we separately analyze the two cases U ∈ ∂S and U /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Firstly, let U ∈ ∂S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', dist2(U, Xs) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1∥X∥−1τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Assume WLOG that dist(U, Xs) = ∥U − Xs∥F , then by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 we have Fs(U) − Fs(Xs) = � 1 0 t ⟨∇Fs(tU + (1 − t)Xs), U − Xs⟩ dt ≥ � 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1τt2 ∥U − Xs∥2 F dt ≥ 10−3∥X∥−2τ 3 = 10−3κ−3∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 51 Secondly, let U /∈ S be a stationary point of fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Recall that all the stationary points of Fs are characterized in Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, so that for all U /∈ S with ∇Fs(U) = 0, we have Fs(U) − F ∗ s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 � σ4 s − σ4 s+1 � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' On the other hand, we know from Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 that Fs(U ∗ s ) − F ∗ s ≤ 5δr∗∥X∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (55) By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have 5δr∗∥X∥4 < 10−3κ−3∥X∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5τ 2, so Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 implies that U ∗ s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since ∇fs(U ∗ s ) = 0, we have A∗A � XX⊤ − U ∗ s (U ∗ s )⊤� U ∗ s = 0, so that ∥∇Fs(U ∗ s )∥F = ��� � XX⊤ − U ∗ s (U ∗ s )⊤� U ∗ s ��� F = ���(A∗A − I) � XX⊤ − U ∗ s (U ∗ s )⊤� U ∗ s ��� F ≤ δ ���XX⊤ − U ∗ s (U ∗ s )⊤��� F ∥U ∗ s ∥ ≤ 4δ∥X∥ · ���XX⊤��� F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' From U ∗ s ∈ S and Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 we can deduce that dist(U ∗ s , Xs) ≤ 40δτ −1∥X∥2∥X∥F = 40δκ∥X∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 holds, then we have ��Z∗ s − XsX⊤ s �� F ≤ 80δκ√r∗∥X∥2 and σmin � (U ∗ s )⊤ U ∗ s � ≥ σ2 s − 80δκ√r∗∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : We assume WLOG that ∥U ∗ s − Xs∥F = dist(U ∗ s , Xs) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' R = I in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4, we have that ���U ∗ s (U ∗ s )⊤ − XsX⊤ s ��� F ≤ 2 max {∥U ∗ s ∥ , ∥Xs∥} · ∥U ∗ s − Xs∥F ≤ 160δκ∥X∥∥X∥F ≤ 160δκ√r∗∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' which proves the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Similarly, we have ���(U ∗ s )⊤ U ∗ s − X⊤ s Xs ��� F ≤ 160δκ√r∗∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Hence σ2 s − σmin � (U ∗ s )⊤ U ∗ s � ≤ ���(U ∗ s )⊤ U ∗ s − X⊤ s Xs ��� ≤ 160δκ√r∗∥X∥2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have σmin(U ∗ s ) ≥ 1 2σmin(Xs) = 1 2σs ≥ 1 2κ− 1 2 ∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 implies that 160δκ√r∗∥X∥2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1κ−1∥X∥2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1σ2 s, so that the conclusion immediately follows from Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ 52 Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, suppose that U, U ∗ s ∈ Rd×s such that U ∗ s is a global minimizer of fs and ∥U − U ∗ s ∥F = dist(U, U ∗ s ) ≤ 10−2κ−1∥X∥ (recall that dist is defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3), then we have ⟨∇fs(U), U − U ∗ s ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1κ−1∥X∥2∥U − U ∗ s ∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='7, U ⊤U ∗ s is symmetric and positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let H = U − U ∗ s , then ∇fs(U) = (A∗A) (UU ⊤ − XX⊤)U = (A∗A) � (H + U ∗ s )(H + U ∗ s )⊤ − XX⊤� (H + U ∗ s ) = � (A∗A) � HH⊤ + U ∗ s H⊤ + H (U ∗ s )⊤�� (H + U ∗ s ) − A∗A � XX⊤ − U ∗ s (U ∗ s )⊤� H where we use the first-order optimality condition A∗A � XX⊤ − U ∗ s (U ∗ s )⊤� U ∗ s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since ∥U ∗ s ∥ ≤ 2∥X∥ by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' we may thus deduce that ���∇fs(U) − �� HH⊤ + U ∗ s H⊤ + H (U ∗ s )⊤� (H + U ∗ s ) − � XX⊤ − U ∗ s (U ∗ s )⊤� H ���� F ≤ ���(A∗A − I) � HH⊤ + U ∗ s H⊤ + H (U ∗ s )⊤� (H + U ∗ s ) ��� F + ���(A∗A − I) � XX⊤ − U ∗ s (U ∗ s )⊤� H ��� ≤ 50δ∥X∥2∥H∥F Hence ⟨∇fs(U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' U − U ∗ s ⟩ ≥ �� HH⊤ + U ∗ s H⊤ + H (U ∗ s )⊤� (H + U ∗ s ) − � XX⊤ − U ∗ s (U ∗ s )⊤� H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' H � − 50δ∥X∥2∥H∥2 F ≥ tr � H(H + U ∗ s )⊤(H + U ∗ s )H⊤ + H⊤U ∗ s H⊤H + � (U ∗ s )⊤ H �2 −H⊤ � XX⊤ − U ∗ s (U ∗ s )⊤� H � − 50δ∥X∥2∥H∥2 F ≥ � σmin � (U ∗ s )⊤ U ∗ s � − ���XX⊤ − U ∗ s (U ∗ s )⊤��� − 50δ∥X∥2 − 3∥U ∗ s ∥∥H∥ − ∥H∥2� ∥H∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 we have σmin � (U ∗ s )⊤ U ∗ s � ≥ σ2 s−80δκ∥X∥∥X∥F and ���XX⊤ − U ∗ s (U ∗ s )⊤��� ≤ σ2 s+1 + 80δκ∥X∥2 F , so that ⟨∇fs(U), U − U ∗ s ⟩ ≥ � σ2 s − σ2 s+1 − 160δκ∥X∥∥X∥F − 50δ∥X∥2 − 3∥U ∗ s ∥∥H∥ − ∥H∥2� ∥H∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 on δ is satisfied and ∥H∥ ≤ 10−2τ∥X∥−1, the above implies that ⟨∇fs(U), U − U ∗ s ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5τ∥H∥2 F , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We now prove Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 to conclude this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Both results follow immediately from Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 53 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, if U ∗ s ∈ Rd×s is a global minimizer of fs, then the set of global minimizers arg min fs is equal to � U ∗ s R : R ∈ Rs×s, R⊤R = I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : By Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 we have that dist(U ∗ s , X∗ s ) ≤ 40δκ∥X∥F holds for any U ∗ s ∈ arg min fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose now that U ∗ s , ˆU ∗ s ∈ arg min fs such that dist(U ∗ s , ˆU ∗ s ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then we also have that dist(U ∗ s , ˆU ∗ s ) ≤ dist(U ∗ s , Xs) + dist(Xs, ˆU ∗ s ) ≤ 80δκ∥X∥F < 10−2κ−1∥X∥, where the last inequality follows from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Without loss of generality, we can assume that ∥U ∗ s − ˆU ∗ s ∥F = dist(U ∗ s , ˆU ∗ s ), so that we can apply Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 to obtain � ∇fs( ˆU ∗ s ), ˆU ∗ s − U ∗ s � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1κ−1∥X∥2∥ ˆU ∗ s − U ∗ s ∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' However, since ˆU ∗ s is a global minimizer of fs, we have ∇fs( ˆU ∗ s ) which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus the global minimizer must be unique under the procrustes distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We recall Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 which is now guaranteed to be well-defined by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' For any U ∈ Rd×s, we use Πs(U) to denote the set of closest global minimizers of fs to U, namely Πs(U) = arg min{∥U − U ∗ s ∥F : U ∗ s ∈ arg min fs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Equipped with this definition, Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5 directly translates into Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 (Restricted Secant Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, if a matrix U ∈ Rd×s satis- fies ∥U − U ∗ s ∥F ≤ 10−2κ−1∥X∥ for some U ∗ s ∈ Πs(U), then we have ⟨∇fs(U), U − U ∗ s ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1κ−1∥X∥2∥U − U ∗ s ∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (8) F Proofs for Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 In this section, we prove the main theorems based on our key lemmas introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Based on Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, we first prove the following lemma, which shows that GD initialized near global minimizers converges linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let { ˆUt}t≥0 be a trajectory of GD that optimizes fs with step size µ, starting with ˆU0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Also let U ∗ s be a global minimizer of fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' If dist( ˆU0, U ∗ s ) ≤ 10−2κ−1∥X∥, then for all t ≥ 0, dist2( ˆUt, U ∗ s ) ≤ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='05τµ)t dist2( ˆUα,0, U ∗ s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (56) Proof : We prove (56) by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It is easy to check that (56) holds for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Now we show that (56) holds for t + 1 assuming it holds for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Since dist( ˆU0, U ∗ s ) ≤ 10−2κ−1∥X∥, we have ��� ˆU0 ��� ≤ ∥U ∗ s ∥ + 10−2κ−1∥X∥ ≤ 2∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' (57) 54 Let R be the orthogonal matrix such that U ∗ s R ∈ Π(Ut), then ∥U − U ∗ s R∥F = dist(Ut, U ∗ s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We first bound the gradient ∇f( ˆUα,t) as follows: ���∇f( ˆUα,t) ��� F = ���A∗A � XX⊤ − ˆUα,t ˆU ⊤ α,t � ˆUα,t ��� F ≤ ���A∗A � XX⊤ − ˆUα,t ˆU ⊤ α,t ���� ��� ˆUα,t − U ∗ s ��� F + ��� � ˆUα,t ˆU ⊤ α,t − U ∗ s (U ∗ s )⊤� U ∗ s ��� F ≤ 20∥X∥2 ��� ˆUα,t − U ∗ s ��� F (58) where we use (57) and the RIP property (Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It follows that dist2( ˆUt+1, U ∗ s ) ≤ ��� ˆUt+1 − U ∗ s R ��� 2 F (59a) = ��� ˆUt − µ∇f( ˆUα,t) − U ∗ s R ��� 2 F = ��� ˆUα,t − U ∗ s R ��� 2 F − µ � ∇f( ˆUα,t), ˆUt − U ∗ s R � + µ2 ���∇f( ˆUα,t) ��� 2 F ≤ � 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1τµ + 400∥X∥4µ2� ��� ˆUα,t − U ∗ s R ��� 2 F (59b) where (59a) follows from the definition of dist, and (59b) is due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 and (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, (56) follows from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We then note the following proposition, which is straightforward from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='9 and The- orem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' In the following we use Uα,t to denote the t-th iteration of GD when initialized at U0 = α ¯U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3 hold and α is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then there exist matrices U lr α,t for t = −T ft α,s, −T ft α,s + 1, · · · , 0 with rank ≤ s (where lr stands for low rank) and a constant C6 = C6(X, ¯U) (defined in (49)) such that max T ft α,s≤t≤0 ���U lr α,t − Uα,T ft α,s+t ��� F = C6 · α 1 4κ where T ft α,s is defined in Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and moreover ���U lr α,0 � U lr α,0 �⊤ − Z∗ s ��� F ≤ 2 × 105κ3∥X∥2r∗δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' where Z∗ s = U ∗ s (U ∗ s )⊤ is the best rank-s solution as defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : It follows from Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 that max1≤t≤T ft α,s ∥UtWt,⊥∥ ≤ C6(X, ¯U) · α 1 4κ (recall that C5 is defined in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='8 and T ft α,s is defined in (48)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We choose U lr α,t = UT ft α,s+tWT ft α,s+tW ⊤ T ft α,s+t, then rank( ¯Ut) ≤ s and moreover by Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 we have ���XsX⊤ s − U lr α,0 � U lr α,0 �⊤��� F ≤ 105κ3∥X∥2r∗δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' On the other hand, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 we have that ��Z∗ s − XsX⊤ s �� F ≤ 80δκ√r∗∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus ���U lr α,0 � U lr α,0 �⊤ − Z∗ s ��� F ≤ 2 × 105κ3∥X∥2r∗δ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Let ˆUα,0 = UT ft α,sWT ft α,s ∈ Rd×s, then it satisfies ˆUα,0 ˆU ⊤ α,0 = U lr α,0 � U lr α,0 �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The following corollary shows that ˆUα,0 is close to U ∗ s in terms of the procrustes distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 55 Corollary F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We have dist( ˆUα,0, U ∗ s ) ≤ 3 × 106κ4r∗∥X∥δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : We know from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4 that dist(U ∗ s , Xs) ≤ 40δκ∥X∥F , so it remains to bound dist( ˆUα,0, Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The proof idea is the same as that of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4, so we only provide a proof sketch here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It has been shown in the proof of Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 that Fs( ˆUα,0) := 1 2 ���XsX⊤ s − ˆUα,0 ˆU ⊤ α,0 ��� 2 F ≤ r∗ ���XsX⊤ s − ˆUα,0 ˆU ⊤ α,0 ��� 2 ≤ 4×1010κ6r2 ∗∥X∥4δ2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Note that Fs is the matrix factorization loss with XsX⊤ s being the ground-truth, so the local RSI condition (Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2) still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' By the same reason as (55), we deduce that dist( ˆUα,0, Xs) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1∥X∥−1τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=', ˆUα,0 is in the local region around Xs in which the RSI condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, it follows from the local RSI condition that dist( ˆUα,0, Xs) ≤ 10τ −1 ���∇Fs( ˆUα,0) ��� F ≤ 10τ −1∥ ˆUα,0∥ ���XsX⊤ s − ˆUα,0 ˆU ⊤ α,0 ��� F ≤ 3×106κ4r∗∥X∥δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ We are now ready to complete the proof of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 (Convergence in the under-parameterized regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that ˆr ≤ r∗, then there exists a constant ¯α > 0 such that when α < ¯α, we have limt→+∞ Uα,tU ⊤ α,t = Z∗ ˆr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : When ˆr ≤ r∗, the parameterization itself ensures that Uα,t is low-rank, so that we can choose U lr α,t = Uα,T ft α,ˆr+t and ˆUα,0 = Uα,T ft α,s in Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and Corollary F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 (for s = ˆr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' The proof that these choices satisfy all required conditions are identical to our proofs for these two lemmas in the general setting, and we omit them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Applying Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we can thus deduce that limt→+∞ dist( ˆUα,t, U ∗ ˆr ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This means that limt→+∞ dist(Uα,t, U ∗ ˆr ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Recall that Z∗ ˆr = U ∗ ˆr U ∗ ˆr ⊤, so the conclusion immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='3, consider GD (3) with initialization Uα,0 = α ¯U for solving the matrix sensing problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' There exist universal constants c, M, constant C = C(X, ¯U) and a sequence of time points T 1 α < T 2 α < · · · < T ˆr∧r∗ α such that for all 1 ≤ s ≤ ˆr∧r∗, the following holds when α is sufficiently small: ���Uα,T sαU ⊤ α,T sα − Z∗ s ��� F ≤ Cα 1 Mκ2 , (5) where we recall that Z∗ s is the best rank-s solution defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, GD follows an incremental learning procedure: we have limα→0 max1≤t≤T sα σs+1(Uα,t) = 0 for all 1 ≤ s ≤ ˆr ∧ r∗, where σi(A) denotes the i-th largest singular value of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Recall that ���UT ft α,s − ¯U0 ��� F = o(1) (α → 0) where T ft α,s is defined in Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' we omit the dependence on α to simplify notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We also note that by the update of GD, we have ¯Ut ¯U ⊤ t = ˆUα,t ˆU ⊤ α,t for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 56 By Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1, we have that dist2( ˆUα,t, U ∗ s ) ≤ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='05τµ)t dist2( ˆUα,0, U ∗ s ) and, in particular, ��� ˆUα,t ��� ≤ 2∥X∥ for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Thus �� ¯Ut �� ≤ 2∥X∥ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, recall that ∥Ut∥ ≤ 3∥X∥ for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It’s easy to see that that the matrix sensing loss f is L-smooth in � U ∈ Rd×r : ∥U∥ ≤ 3∥X∥ � for some constant L = O(∥X∥2), so it follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='6 that ���UT ft α,s+t − ¯Ut ��� F ≤ (1 + µL)t ���UT ft α,s − ¯U0 ��� F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' On the other hand, since dist2( ˆUα,t, U ∗ s ) ≤ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='05τµ)t dist2( ˆUα,0, U ∗ s ), we can deduce that ���UT ft α,s+tU ⊤ T ft α,s+t − Zs ��� F ≤ ���UT ft α,s+tU ⊤ T ft α,s+t − ¯Ut ¯U ⊤ t ��� F + ��� ¯Ut ¯U ⊤ t − U ∗ s (U ∗ s )⊤��� F = ���UT ft α,s+tU ⊤ T ft α,s+t − ¯Ut ¯U ⊤ t ��� F + ��� ˆUα,t ˆU ⊤ α,t − U ∗ s (U ∗ s )⊤��� F ≤ 3∥X∥ ����UT ft α,s+t − ¯Ut ��� F + dist( ˆUα,t, U ∗ s ) � ≤ 3∥X∥ � (1 + µL)t ���UT ft α,s − ¯U0 ��� F + (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='05τµ) t 2 dist2( ˆUα,0, U ∗ s ) � Since when α → 0, ���UT ft α,s − ¯U0 ��� F = O(α 1 4κ ), it’s easy to see that there exists a time t = ts α so that we have max−T ft α,s≤t≤tsα ���UT ft α,s+t − ¯Ut ��� F = O � α 1 M1κ2 � and ���UT ft α,s+tU ⊤ T ft α,s+t − Zs ��� F = O � α 1 M1κ2 � as well, where c1 is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let T s α = T ft α,s+ts α, then ���UT sαU ⊤ T sα − Zs ��� F = o(1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Recall that rank(Ut) ≤ s, so that max0≤t≤T sα σs+1 (Ut) = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Finally, for all 0 ≤ s < ˆr ∧ r∗, we need to show that T s α < T s+1 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Indeed, by Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 and the Assump- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 we have σ2 s+1 � UT sα � ≥ σs+1 (Zs+1) − o(1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5σ2 s+1, so that T s+1 α > T s α, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ G The landscape of matrix sensing with rank-1 parameterization In this section, we establish a local strong-convexity result Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2 for rank-1 parameterized matrix sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' This result is stronger than the RSI condition we established for general ranks, though the latter is sufficient for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Define the full-observation loss with rank-1 parameterization g1(u) = 1 4 ��uuT − XXT ��2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then the global minima of g1 are u∗ = σ1v1 and −u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, suppose that g(u) − g(u∗) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5τ1 where τ1 = σ2 1 − σ2 2 is the eigengap, then we must have ∥u − u∗∥2 ≤ 20τ −1 1 (g1(u) − g1(u∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 57 Proof : We can assume WLOG that XXT = diag � σ2 1, · · · , σ2 r∗, 0, · · · , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Then g1(u) = 1 4 � ∥u∥4 2 − 2 s � i=1 σ2 i u2 i + ∥XT X∥2 F � (60a) ≥ 1 4 � ∥u∥4 2 − 2σ2 1∥u∥2 2 + ∥XT X∥2 F � (60b) ≥ 1 4 � ∥XT X∥2 F − σ4 1 � (60c) where equality holds if and only if u2 = · · · = ud = 0 and ∥u∥2 = σ2 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' u = ±σ1e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, suppose that g1(u) − g1(u∗) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='5τ1, it follows from (60b) that τ1 �d i=2 u2 i ≤ 2(g1(u) − g1(u∗)) which implies that �d i=2 u2 i ≤ 2τ −1 1 (g1(u) − g1(u∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Also (60c) yields ��∥u∥2 − σ2 1 �� ≤ 4 � g1(u) − g1(u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Assume WLOG that u1 > 0, then we have ∥u − σ1e1∥2 ≤ σ−2 1 � u2 1 − σ2 1 �2 + d � i=2 u2 i ≤ 20τ −1 1 (g1(u) − g1(u∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let f1(u) = 1 4 ��A � uuT − XXT ���2 2 , u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Suppose that δ ≤ 10−3∥X∥−2τ1, then there exists constants a1 and ι, such that f1 is locally ι-strongly convex in B1 = B(σ1v1, a1) ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Furthermore, there is a unique global minima of f1 inside B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Proof : Recall that we defined the full observation loss g1(u) = 1 4 ��uuT − XXT ��2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Let h1 = f1 − g1, then ��∇2h1(u) �� = 1 2 ��(A∗A − I) (uuT − XXT ) + 2 (A∗A − I) uuT �� ≤ δ � 2∥u∥2 + ∥X∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' When ∥u − σ1v1∥2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 min � σ2 1, τ1 � (recall τ1 = σ2 1 − σ2 2), σmin � ∇2g1(u) � = 1 2σmin � ∥u∥2I + 2uuT − XXT � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Hence we have σmin � ∇2f1(u) � ≥ � ∇2g1(u) � − ∥∇2h1(u)∥ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='4τ1 − 4∥X∥2δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2τ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' strong-convexity holds for a2 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 min � σ2 1, τ1 � and ι = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='2τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' 58 Let u∗ be a global minima of f1, then we must have ∥u∗∥ ≤ 2∥X∥ (otherwise f1(u) > f1(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' We can thus deduce that g1(u∗) ≤ f1(u∗) + 1 4 ��� uuT − XXT , (A∗A − I)(uuT − XXT ) ��� ≤ f1(u) + 10δ∥X∥2 ≤ g1(u) + 20δ∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' It follows from Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content='1 and our assumption on δ that min � ∥u∗ − σ1v1∥2 , ∥u∗ + σ1v1∥2� ≤ 1 2a2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' Moreover, by strong convexity, there exists only one global minima in B1, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} +page_content=' □ 59' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQfSixW/content/2301.11500v1.pdf'} diff --git a/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf b/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ee42d2ce9db34749ab2d1cd91094c8079b7e1ddd --- /dev/null +++ b/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:19dc2bc3fee5968f597d8e4ca7934f8a3508e27533e498896eccc3805bfa3131 +size 1524858 diff --git a/ddE3T4oBgHgl3EQfeQr_/content/tmp_files/2301.04543v1.pdf.txt b/ddE3T4oBgHgl3EQfeQr_/content/tmp_files/2301.04543v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b4664d95ac80e3bd95b166ecc39bad9935410f9 --- /dev/null +++ b/ddE3T4oBgHgl3EQfeQr_/content/tmp_files/2301.04543v1.pdf.txt @@ -0,0 +1,2508 @@ +Draft version January 12, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +A Characterization of the ALMA Phasing System at 345 GHz +G. B. Crew,1 C. Goddi,2, 3, 4, 5 L. D. Matthews,1 H. Rottmann,6 A. Saez,7 and I. Mart´ı-Vidal8, 9 +1Massachusetts Institute of Technology Haystack Observatory, 99 Millstone Road, Westford, MA 01886 USA +2 Universidade de S˜ao Paulo, Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, Departamento de Astronomia, S˜ao Paulo, SP +05508-090, Brazil +3Dipartimento di Fisica, Universit´a degli Studi di Cagliari, SP Monserrato-Sestu km 0.7, I-09042 Monserrato, Italy +4INAF - Osservatorio Astronomico di Cagliari, via della Scienza 5, I-09047 Selargius (CA), Italy +5 INFN, Sezione di Cagliari, Cittadella Univ., I-09042 Monserrato (CA), Italy +6Max Planck Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, 53121 Bonn, Germany +7ALMA Observatory, Av. Alonso de C´ordova 3107, Vitacura, Regi´on Metropolitana, Chile +8Departament d’Astronomia i Astrof´ısica, Universitat de Val`encia, C. Dr. Moliner 50, E-46100 Burjassot, Val`encia, Spain +9Observatori Astron`omic, Universitat de Val`encia, C. Catedr´atico Jos´e Beltr´an 2, E-46980 Paterna, Val`encia, Spain +ABSTRACT +The development of the Atacama Large Millimeter/submillimeter Array (ALMA) phasing system +(APS) has allowed ALMA to function as an extraordinarily sensitive station for very long baseline +interferometry (VLBI) at frequencies of up to 230 GHz (λ ≈1.3 mm). Efforts are now underway to ex- +tend use of the APS to 345 GHz (λ ≈0.87 mm). Here we report a characterization of APS performance +at 345 GHz based on a series of tests carried out between 2015–2021, including a successful global VLBI +test campaign conducted in 2018 October in collaboration with the Event Horizon Telescope (EHT). +Keywords: instrumentation: interferometers – methods: observational - techniques: high angular res- +olution +1. INTRODUCTION +In addition to operating as a connected element inter- +ferometer, the Atacama Large Millimeter/submillimeter +Array (ALMA) can function as the equivalent of a single +very large aperture antenna if the data from its individ- +ual antennas are phase-corrected and coherently added. +The development of a phased-array capability for ALMA +(Doeleman et al. 2009; Matthews et al. 2018) has allowed +ALMA to play a transformational role in the technique +of very long baseline interferometry (VLBI) at millime- +ter (mm) wavelengths. By boosting the sensitivity of +VLBI baselines in previously existing arrays operating at +230 GHz (λ ≈1.3 mm) by up to an order of magnitude, +phased ALMA was crucial to the achievement of the +first horizon scale images of the supermassive black hole +at the center of the M87 Galaxy, M87* (Event Horizon +Telescope Collaboration et al. 2019a,b,c,d,e,f) and the +one at the center of our own Milky Way, Sgr A* (Event +Horizon Telescope Collaboration et al. 2022a,b,c,d,e,f). +Corresponding author: Ciriaco Goddi +cgoddi@usp.br +At 86 GHz (λ ≈3 mm), phased ALMA was also key to +achieving the first scatter-corrected images of SgrA*, the +supermassive black hole candidate at the Galactic Cen- +ter (Issaoun et al. 2019) and the first ALMA detection +of pulsed emission from radio pulsars (Liu et al. 2019, +2021). +The ALMA phasing system (APS) has been offered to +the community for VLBI science observations in ALMA +Bands 3 (λ ≈3 mm; ν ≈ 86 GHz) and 6 (λ ≈1.3 +mm; ν ≈ 230 GHz), where the Global mm-VLBI Ar- +ray (GMVA) and the Event Horizon Telescope (EHT), +respectively, serve as partner networks. +The first sci- +ence observations that included the APS in this capacity +were conducted in 2017 April as part of ALMA Cycle 4 +(Goddi et al. 2019). To further expand scientific possi- +bilities, there is now growing motivation to push VLBI +techniques to still shorter wavelengths, i.e., λ ≈0.87 mm +or ν ≈345 GHz (e.g., Falcke et al. 2001; Miyoshi & +Kameno 2002; Weintroub 2008; Krichbaum et al. 2008; +Doeleman et al. 2009; Event Horizon Telescope Collab- +oration et al. 2019a,b). Not only will this enable even +higher angular resolution (≲ 20µas for Earth-sized base- +lines), but it will help to improve uv coverage by en- +arXiv:2301.04543v1 [astro-ph.IM] 11 Jan 2023 + +2 +Crew, Goddi, Matthews, et al. +abling the combination of 230 GHz and 345 GHz obser- +vations (thus enabling higher fidelity imaging), and will +minimize the effects of interstellar scattering on achiev- +able image quality. The latter is particularly important +for imaging SgrA*, where interstellar ionized gas along +the line-of-sight causes significant blurring of images at +longer radio wavelengths (Johnson & Gwinn 2015; John- +son 2016; Event Horizon Telescope Collaboration et al. +2022g). +A key component of the effort to extend VLBI capabil- +ities into the sub-mm regime is the extension of ALMA’s +phased array capabilities to 345 GHz. While this had +been an envisioned application of the APS since its con- +ception (e.g., Doeleman 2010; Fish et al. 2013), com- +missioning of ALMA’s phasing capabilities was initially +limited to the 86 GHz and 230 GHz bands (ALMA Band +3 and 6, respectively) owing to time constraints and to +the limited availability of suitably equipped VLBI part- +ner sites (see Matthews et al. 2018). +While the APS itself is agnostic to observing fre- +quency, there are practical considerations that impact +the use of the APS at ν >∼345 GHz and the optimization +of phasing efficiency at higher frequencies. One of the +most important is the shorter coherence timescales at +higher frequencies owing to the effects of tropospheric +water vapor, which become increasingly significant with +increasing baseline length. +This in turn will impact +choices such as the maximum baseline length to include +in the phased array and whether or not to apply “fast” +phasing corrections—derived from water vapor radiome- +ter (WVR) data at each ALMA antenna—in addition +to the nominal “slow” phasing corrections derived by +the phasing engine within the (TelCal) telescope cali- +bration software (Matthews et al. 2018). Additionally, +effects such as pointing errors and wind speeds will have +increasingly important impacts on phased array perfor- +mance at higher frequency owing to the smaller beam +size of the antennas (see, e.g., Smith et al. 2000). +Here we present a characterization of the performance +and phasing efficiency of the APS at 345 GHz based on +test sessions conducted between 2015 March and 2021 +September. The data sets include observations obtained +in 2018 as part of a multi-day global VLBI test cam- +paign that produced for the first time VLBI fringes in +the 345 GHz band (Event Horizon Telescope Collabora- +tion et al., in preparation; hereafter Paper II). +2. OBSERVATIONS +Testing and characterization of the performance of the +APS for use in Band 7 were done using a combination of +ALMA standalone tests and a global VLBI campaign. +These tests are described in detail in the next two sub- +sections. In the discussions of phased ALMA that fol- +low, we adopt the following nomenclature: the reference +antenna is a designated antenna relative to which the +phasing corrections are computed for all other anten- +nas; the sum antenna is a virtual antenna containing +the phased signals of all of the phased-array antennas +summed together; a comparison antenna is an ALMA +antenna that is participating in the observations, but is +not being phased and is not included in the phased sum. +2.1. Initial Testing: ALMA Standalone Observations +Initial testing of the APS at 345 GHz began in 2015 +and continued throughout 2021 with a series of short +ALMA standalone tests. These observations were con- +ducted during ALMA Extension and Optimization of +Capabilities (EOC) time or Engineering time, and typ- +ically lasted from a few minutes up to 40 minutes. A +summary of these tests is provided in Table 1, including +array and weather parameters. +Selected observing targets comprised bright ( >∼1 Jy +at 345 GHz) quasars and other compact extragalactic +sources that are unresolved on intra-ALMA baselines. +Most of the target quasars are routinely observed at +ALMA as part of the flux-density monitoring program +with the ALMA Compact Array (ACA). This program +includes measurements, mostly in Band 3 and Band 7, +of bright reference sources, referred to as ”Grid Sources” +(Remijan et al. 2019)). +Data from these standalone APS tests allowed us to +demonstrate the feasibility of phased ALMA operations +at sub-mm wavelengths. Although in several instances +the test data were taken in conditions that were sub- +optimal for Band 7 observing, such data enable explo- +ration of the impacts of weather conditions on phasing +performance and help to establish guidelines on the pa- +rameter space for scientifically useful phasing operations +at higher frequencies (e.g. maximum baseline length in +the phased array, maximum wind speed). More details +on the analysis of these test datasets are given in Sec- +tion 3. +2.2. The 2018 Global VLBI Test Campaign +2018 October marked the first time that ALMA’s +345 GHz phasing capability was tested over a sustained +observing session, as well as during a global VLBI cam- +paign, with the goal of obtaining 345 GHz VLBI fringes +on global baselines (see Paper II). During this campaign, +APS operations at 345 GHz were characterized during +a series of four observing windows from October 17–21. +A total of six ALMA scheduling blocks were built and +executed, including four blocks in Band 7 (each span- +ning ∼90 min) and two blocks in Band 6 (each spanning + +Phasing ALMA at 345 GHz +3 +∼35 min) for comparison purposes. A summary of the +observations is reported in Table 2. +2.2.1. Observed Targets +As for the standalone phasing tests in Table 1, se- +lected observing targets comprised bright quasars and +other compact extragalactic sources. An effort was made +to select sources that would be point-like on the angu- +lar scales sampled by intra-ALMA baselines (to max- +imize phasing efficiency), while still having sufficient +correlated flux density to allow high signal-to-noise ra- +tio (SNR) fringe detections with short integration times +on VLBI baselines. +This is particularly important in +Band 7, where coherence timescales are expected to be +only ∼10 s (e.g., Doeleman et al. 2011). +A list of observed sources and their calibration intent +is given in Table 3; only VLBI targets observed on Octo- +ber 18/19 and 21 are listed (targets observed on previous +days were not included in the analysis owing to poor +weather conditions; see Sect. 2.2.3 and Appendix A). +In most cases, recent flux density measurements at 230 +and 345 GHz were available from the ALMA Calibra- +tor Source Catalogue.1 +These can be used for cross- +comparison and validation of the APS performance and +calibration in Band 7 (see Section 5.1). +2.2.2. Observational Setup +During the 2018 October test campaign the ALMA +antennas were in transition from configuration C43-6 +(maximum baseline 2500 m) to C43-5 (maximum base- +line 1400 m). There were 23–29 12 m antennas included +in the ALMA phased array (depending on the session), +with a phasing radius limited to 300 m. An additional +16–22 outlying antennas (with maximum baselines be- +tween 1400 m and 2500 m, depending on the day) were +withheld for comparison purposes. +Antenna locations +are plotted in Figure 1. The observing array was more +extended than during previous science observations in +VLBI mode, where only antennas within a radius of +180 m were phased (e.g., Goddi et al. 2019). Only 12 m +antennas were included in the array; ALMA’s 7 m CM +antennas can also be used with the APS, but are typi- +cally excluded from phased-array operations for a vari- +ety of practical reasons. +The spectral setup included four spectral windows +(SPWs), each with a bandwidth of 1875 MHz, that +were processed by the ALMA Baseline Correlator. Two +SPWs were in the lower sideband and two in the up- +per sideband. +In Band 7 the data outputted by the +1 https://almascience.eso.org/sc/ +1000 +500 +0 +500 +1000 +X (m) +2000 +1500 +1000 +500 +0 +Y (m) +Figure 1. ALMA antenna locations for the phased array +(orange points) and the unphased comparison antennas (blue +points) during the Band 7 phasing tests in 2018 October. +Positions are plotted with positive values of X toward local +east and positive values of Y toward local north. +ALMA Baseline Correlator were averaged in frequency +to produce 120 channels per SPW (corresponding to a +channel spacing of 15.625 MHz). In Band 6 there were +240 channels per SPW (resulting in a channel spacing +of 7.8125 MHz). +The frequency setup is summarized +in Table 4. The spectral data from the ALMA Baseline +Correlator are available with a time resolution of 4.032 s. +In parallel, VLBI recordings of all four basebands, each +corresponding to one of the four SPWs, were recorded in +dual linear polarizations, thus exercising the full 64 Gb +s−1 VLBI recording capability at ALMA (see Paper II). +2.2.3. Weather Conditions +During the 2018 October campaign there were no +significant technical issues at ALMA and all aspects +of its phasing and VLBI systems appeared to be per- +forming nominally. However, with the exception of the +last observing night where conditions were exceptional, +weather conditions did not meet the usual requirements +for Band 7 observing at ALMA. As discussed below, +these weather issues significantly impacted the quality +of the phased array data, but at the same time pro- +vided valuable insights into the range of weather condi- +tions where scientifically useful phased array operations +in Band 7 are likely to be possible. +In the first two sessions of the campaign there was high +and variable precipitable water vapor (PWV≳2 mm) + +4 +Crew, Goddi, Matthews, et al. +Table 1. Standalone ALMA Band 7 Phasing Tests +UTC Starta +UTC Enda +Archive UIDb +Nphased +Baselinesc +PWVd +Wind Speedd +ηve +Qual.f +(YYYY MMM DD/hh:mm:ss.s) +(YYYY MMM DD/hh:mm:ss.s) +(m) +(mm) +(m s−1) +2015 Mar 30/02:49:26.4 +2015 Mar 30/02:51:46.9 +uid +A002 X9cdda2 X42c +9 +15–193 +0.53±0.03 +8.5±2.5 +0.86 +0.91 +2015 Aug 02/14:37:58.7 +2015 Aug 02/14:45:01.4 +uid +A002 Xa73e10 X28dc +35 +15–1492 +0.50±0.04 +7.5±4.5 +0.46 +0.94 +2016 Jul 10/08:51:25.1 +2016 Jul 10/09:38:54.0 +uid +A002 Xb53e10 Xa7a +9 +19-396 +2.0±0.5 +7±5 +0.15 +0.66 +2017 Jan 30/21:47:33.5 +2017 Jan 30/21:51:58.4 +uid +A002 Xbd3836 X4ba +37 +15–260 +5.0±1.5 +12±5 +0.07 +0.36 +2017 Jan 30/21:56:56.8 +2017 Jan 30/22:08:59.2 +uid +A002 Xbd3836 X579 +37 +15–260 +5.0±1.5 +13±6 +0.10 +0.50 +2017 Jan 30/22:18:57.0 +2017 Jan 30/22:34:29.2 +uid +A002 Xbd3836 X739 +37 +15–260 +4.5±1.2 +14±6 +0.16 +0.66 +2017 Jan 30/22:39:27.4 +2017 Jan 30/22:51:29.2 +uid +A002 Xbd3836 X87c +37 +15–260 +4.5±1.5 +13±4 +0.11 +0.55 +2017 Feb 01/03:19:27.0 +2017 Feb 01/04:02:59.7 +uid +A002 Xbd3836 X4363g +41 +15–331 +1.6±0.6 +9±5 +0.75 +0.93 +2019 Mar 08/04:12:43.8 +2019 Mar 08/04:19:14.4 +uid +A002 Xd9435e X2859 +45 +15–314 +1.0±0.1 +3.5±3.5 +0.74 +0.95 +2021 Mar 23/23:13:24.1 +2021 Mar 23/23:21:11.3 +uid +A002 Xea64a8 X321 +33 +15–1232 +2.7±0.15 +10±4 +0.49 +0.88 +2021 Mar 24/21:37:20.6 +2021 Mar 24/21:44:11.3 +uid +A002 Xea6cf9 X1c1 +33 +15–1214 +2.15±0.25 +12±5 +0.21 +0.84 +2021 Mar 25/01:07:24.7 +2021 Mar 25/01:15:11.3 +uid +A002 Xea6cf9 Xb5a +35 +15–1231 +2.55±0.15 +4±3 +0.38 +0.90 +2021 Mar 25/19:12:42.6 +2021 Mar 25/19:20:11.3 +uid +A002 Xea6cf9 X1d9e +25 +22–969 +2.0±0.2 +13±5 +0.11 +0.69 +2021 Aug 26/19:55:06.2 +2021 Aug 26/20:02:18.0 +uid +A002 Xefb0d3 X7c2 +31 +92–6855 +1.2±0.2 +12±8 +0.21 +0.72 +2021 Sep 02/19:37:24.1 +2021 Sep 02/19:45:11.6 +uid +A002 Xf02179 X1ea +25 +237–6855 +0.45±0.15 +10±6 +0.26 +0.87 +2021 Sep 03/02:07:24.7 +2021 Sep 03/02:15:29.9 +uid +A002 Xf02179 X10a0 +29 +237–6855 +0.4±0.1 +4±3 +0.65 +0.91 +a Start times and end times include observations in APS-mode only (i.e. standard ALMA-mode calibration scans are excluded). +b Unique identifier (UID) of the data set in the ALMA Archive. +c Approximate range of baseline lengths in the phased array (excluding unphased comparison antennas). +d Weather data (including precipitable water vapor or PWV and wind speed) reported by meteorological stations on the Chajnantor plateau. The ± values refer to the +range of values reported by different stations. +e Phasing efficiency, averaged over polarizations and basebands, computed according to Eq. E3. +f Phasing quality, averaged over polarizations and basebands, (see Section 3). +g This block also included Band 6 observations. +Table 2. Observations Log for 2018 October Band 7 Test Campaign +UTC Start +UTC End +Archive UID +Nphased +PWV +Wind Speed +ηv +Qual. +Band +(YYYY MMM DD/hh:mm:ss.s) +(YYYY MMM DD/hh:mm:ss.s) +(mm) +(m s−1) +2018 Oct 16/23:42:52.4 +2018 Oct 17/01:00:26.0 +uid +A002 Xd3607d X6f14 +23 +2.0 ± 0.3 +9 ± 5 +0.13 +0.42 +7 +2018 Oct 17/01:05:24.6 +2018 Oct 17/01:40:54.3 +uid +A002 Xd3607d X70fe +23 +2.5 ± 0.5 +7 ± 3 +0.10 +0.37 +6 +2018 Oct 17/09:31:07.1 +2018 Oct 17/11:02:16.5 +uid +A002 Xd36f86 X24dd +25 +1.8 ± 0.8 +12 ± 4 +0.07 +0.28 +7 +2018 Oct 18/23:23:08.3 +2018 Oct 19/00:52:37.4 +uid +A002 Xd37ad3 X7ef1 +25 +1.1 ± 0.3 +6 ± 4 +0.37 +0.91 +7 +2018 Oct 19/00:58:00.0 +2018 Oct 19/01:32:54.5 +uid +A002 Xd37ad3 X82a2 +25 +1.0 ± 0.1 +4 ± 3 +0.51 +0.96 +6 +2018 Oct 21/09:12:54.4 +2018 Oct 21/10:59:18.3 +uid +A002 Xd395f6 Xd41f +29 +0.85 ± 0.10 +3 ± 3 +0.93 +0.97 +7 +Note—See footnotes to Table 1 for an explanation of the columns. The final column indicates the ALMA observing band. +and high wind speeds (≳10–15 m s−1; see Table 2 and +Figure 10 in Appendix C). These factors led to unstable +atmosphere conditions over timescales of a few seconds, +which compared unfavorably with the ∼ 18 second loop +time of the “slow” APS phasing solutions (see Matthews +et al. 2018; Goddi et al. 2019). The phasing efficiency, ηv +(see Appendix C and Eq. C2), was consequently rather +low: typical values reported by TelCal during the ob- +servations ranged from 5%–20%, and for portions of the +session ALMA appeared to be effectively unphased (see +Section 4). +At the onset of the third session (on the night of Oc- +tober 18/19), atmospheric stability was significantly im- +proved compared with the previous two VLBI sessions. +Finally, during the fourth and final VLBI session (cor- +responding to the fifth day of the VLBI observing win- +dow), weather at ALMA was excellent, with precipitable +water vapor (PWV) ∼0.8 mm and wind speeds of only +a few m s−1 (see Table 2 and the bottom panel of Fig- +ure 10 in Appendix C). Throughout the latter session, +the estimated phasing efficiency reported by TelCal was +consistently >90%, and frequently above 95% (see Sec- +tion 4). +3. APS PERFORMANCE METRICS + +Phasing ALMA at 345 GHz +5 +Table 3. VLBI Sources Observed During the 2018 October Band 7 Test Campaign +Source +UTC Starta +UTC Enda +Band +Calibration Intentb +(YYYY MMM DD/hh:mm:ss) +(YYYY MMM DD/hh:mm:ss.s) +CTA102 +2018 Oct 18/23:43:25 +2018 Oct 18/23:58:05 +7 +. . . +3C454.3 +2018 Oct 19/00:06:25 +2018 Oct 19/00:20:15 +7 +Flux +BL Lac +2018 Oct 19/00:29:25 +2018 Oct 19/00:43:15 +7 +. . . +BL Lac +2018 Oct 19/01:02:25 +2018 Oct 19/01:23:58 +6 +. . . +J0423–0120 +2018 Oct 21/09:21:25 +2018 Oct 21/09:43:15 +7 +Bandpass +J0510+1800 +2018 Oct 21/09:52:25 +2018 Oct 21/10:06:15 +7 +Polarization +J0510+1800 +2018 Oct 21/10:16:25 +2018 Oct 21/10:28:41 +7 +Polarization +J0522–3627 +2018 Oct 21/10:36:25 +2018 Oct 21/10:59:18 +7 +. . . +a Only VLBI targets observed on October 18/19 and 21 are listed. Observations on October 16/17 did not produce good-quality data owing to poor weather conditions and +were not included in the analysis (see Table 2 and Appendix A). +b See Appendix A for additional information on the calibration of these data. +Table 4. ALMA Frequency Settings +Band +Central Freq. (GHz) +Chan. Width +No. Spec. +Integ. time +(λ) +SPW 0 +SPW 1 +SPW 2 +SPW 3 +(MHz) +Chans. +(s) +6 (1.3 mm) +213.1 +215.1 +227.1 +229.1 +7.8125 +240a +4.03 +7 (0.85 mm) +335.5 +337.5 +347.7 +349.7 +15.625 +120 +4.03 +Note—The SPW designations correspond to those in the calibrated CASA measurement set rather than those in the original raw data files. +a For bandpass calibration purposes the Band 6 scans were rebinned in frequency to 120 channels for consistency with Band 7 (see Appendix A). +Table 5. ALMA Source Flux Densities from the 2018 October Band 7 Test Campaign +Source +S0 (Jy) +S1 (Jy) +S2 (Jy) +S3 (Jy) +S (Jy) +Sarch (Jy) +Ratio +∆tS (days) +Flux Calibrator = 3C454.3, S343GHz = 3.53 Jy, α=−0.69 +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +CTA102 +1.42 +1.41 +1.40 +1.40 +1.41 +1.68 +0.84 +0 +BL Lac (Band 7) +1.06 +1.06 +1.05 +1.05 +1.06 +1.11 +0.95 ++71 +BL Lac (Band 6) +1.26 +1.25 +1.22 +1.24 +1.24 +... +... +... +J0423-0120 +2.48 +2.50 +2.44 +2.43 +2.46 +2.24 +1.1 +0 +J0510+1800 +1.23 +1.24 +1.22 +1.21 +1.22 +1.28 +0.95 +0 +J0522−3627 +4.91 +4.94 +4.87 +4.84 +4.89 +4.32 +1.13 +0 +Note—Tabulated flux densities include values measured during the 2018 October Band 7 VLBI test campaign and values retrieved from the ALMA GS calibrator archive. +Explanation of columns: (1) source name; (2)–(5) flux density in Jy, measured in SPW=0,1,2,3, respectively (see Table 4 for their central frequencies) using CASA’s fluxscale +task and corrected for Tsys (see Appendix A.1); (6) flux density in Jy, derived at the mean frequency over the four SPWs (342.6 GHz in Band 7 and 221.1 GHz in Band 6); +(7) expected flux density in Jy at 343 GHz from the ALMA archive (when an entry is available); (8) ratio between the measured and the archive-predicted flux density; (9) +time difference in days between the APS test observation and the archival entry. + +6 +Crew, Goddi, Matthews, et al. +One effective way to visualize the APS performance +is through plots of the phasing efficiency, ηv (see Eq. +3 in Appendix E). For monitoring purposes, this quan- +tity is computed by TelCal for a designated comparison +antenna and can be extracted from the archival science +data model (ASDM) file metadata (see also Goddi et al. +2019). Some details about how and why this is done are +discussed in Appendix E. +An additional figure-of-merit computed by TelCal is a +‘quality’ metric, which is a figure of merit intended to +provide a sense of the goodness-of-fit of the phasing cal- +culations. It is constructed from the RMS of the phase +residuals, σRMS, and assumes values ranging between 0 +(no solution) to 1 (excellent fit). Noting that in the case +of pure noise this value is σRMS,max = π/ +√ +3 (Thomp- +son et al. 2017), a quality metric for each fit may be +constructed as q = (σRMS,max − σRMS)/σRMS,max. +Both are plotted in Figure 2 for the 2018 October +data, arranged by correlator subscan (i.e. the interval +of the phasing solution), with the data for each time +interval averaged over all basebands. This plot shows +that in the 2018 October test, the APS has achieved +90% phasing efficiency on October 21, which was the +goal specified in the original operational requirements +(Matthews et al. 2018). On October 18/19 the phas- +ing efficiency was rather modest, which is ascribable to +poor atmospheric conditions (see Section 4), but still of +good quality. This situation occurs in cases where the +phase-solving algorithm is able to find good-quality solu- +tions, but atmospheric conditions are varying sufficiently +rapidly that the 16-second time delay in the application +of these “slow” phasing corrections to the data renders +them “stale” and no longer optimal. The contrast be- +tween these days and the first two makes clear that the +quality metric is a useful discriminator between differ- +ent causes of low-phasing efficiency. During the first two +observing sessions where wind speeds were high (Octo- +ber 16/17 and October 17) ηv is low and the quality +metric is << 1. On the other hand, for October 18/19, +the quality metric is consistently ∼1 despite periods of +low ηv, suggesting that rapid variations in water vapor +rather than wind effects were the dominant source of +efficiency loss. +An alternative way to display the APS performance +during observations is to plot directly amplitudes and +phases of the interferometric visibilities including the +sum and the reference antennas, on baselines to one or +more comparison antennas. In Figure 3 (left panels) we +show a comparison of the correlated amplitude as a func- +tion of time for the 2018 October data on: (i) baselines +between the phasing reference antenna and each of two +different unphased comparison antennas; and (ii) base- +lines between the phased sum antenna and the same +comparison antennas. For an optimally phased array, +the correlated amplitude of (ii) should ideally improve +by a factor of ∼ +� +Nphased when phasing is active. Plots +of (i) are useful in making this assessment. We see that +for October 16/17 and October 17, the correlated ampli- +tude for a baseline with the phased sum is comparable +to that on a baseline with a single antenna, implying +that the array is effectively unphased as a result of the +poor weather conditions. +On October 19, the phased +sum shows a significant improvement in correlated am- +plitude (green points), though the data are noisy and +the improvement does not match the ideal +� +Nphased +scaling. Finally on October 21 nearly ideal phasing per- +formance is seen. +In Figure 3 (right panels) we show phase versus time +on baselines which are part of the phased sum for the +same four data sets. +On October 21 the phases in +individual scans show a low RMS dispersion and are +tightly clustered near 0, except for a few seconds near +the start-up of each scan, when the phases are still be- +ing adjusted2. +(Note that the scan at 10:06 UT was +passively3 rather than actively phased, hence the higher +noise level). On October 19 some hints of phase coher- +ence are seen, but the data are much noisier. Finally on +October 16/17 and 17 the phases appear nearly random, +consistent with an unphased array. +The performance of the APS in Band 7 relative to +Band 6 is discussed in Section 4.3. Although the weather +conditions were sub-optimal during the test observa- +tions, we nonetheless see comparable RMS phase fluc- +tuations in the two bands, suggesting that there is no +systematic degradation in phasing performance in Band +7 compared to Band 6. +4. VARIABLES IMPACTING PHASED ARRAY +PERFORMANCE IN BAND 7 +Because the data recorded during the Band 7 APS +tests presented in Section 2 were acquired using different +arrays of ALMA antennas and across a range of weather +conditions, this allows us to begin to investigate how dif- +ferent array parameters, weather conditions, and other +2 The APS scans are started two subscans (18-s each) prior to the +start of the VLBI recording to allow the APS to calculate and +apply the phase adjustments. The “phase-up” occurs during the +first 22 s of each scan (where typical scan lengths are several +minutes); these intervals are routinely flagged to prevent using +poorly phased data. +See Matthews et al. (2018); Goddi et al. +(2019) for details. +3 The APS supports a “passive” phasing mode where a bright cal- +ibrator located within a few degrees of the fainter target is used +to phase up the array (Matthews et al. 2018). + +Phasing ALMA at 345 GHz +7 +0.00 +0.20 +0.40 +0.60 +0.80 +1.00 + 0 + 100 + 200 + 300 + 400 + 500 +Phasing Efficiency +BLC Subscans with Phase Corrections +0.00 +0.20 +0.60 +1.00 +Quality +16/17 +17 +18/19 +21 +Figure 2. +Phasing efficiency (ηv, as defined in Eq. E3; lower panel) and quality (top panel) during the phased-array test +in Band 7 as part of the 2018 October VLBI campaign. Calendar dates and their respective time ranges are indicated by the +arrows between the two panels. All scans are plotted and colored by science target for each day. The phasing “quality” is a +“goodness-of-fit” parameter derived from the fitting process, which is scaled so that unity corresponds to perfect phasing. The +phasing efficiency ranges from 0 (totally un-phased) to 1 (perfect phasing). The large departures of efficiency below 0.8 (on +days from 16 to 19) correspond to poor atmospheric conditions (see Section 4). +variables affect APS performance in Band 7. These ef- +fects are discussed in the following subsections. +4.1. Impact of Weather Conditions +The four dates of the 2018 October campaign were +conducted with a phased array with a fixed radius (300- +m) and all included a similar number of phased antennas +(∼25). However, weather conditions varied on the dif- +ferent days. We can therefore use the 2018 data to assess +how weather conditions affect the APS performance. +In Section 2.2.3 we point out that observations on Oc- +tober 16 and 17 were plagued by variable PWV and high +winds (see also Appendix C). Under these conditions, it +was not possible to successfully phase the array, with +the phased sum antenna performing no better than a +single antenna. Under conditions of moderate wind but +still relatively high PWV fluctuations (October 18/19), +the array could be successfully phased, but the variable +conditions reduced the duration of the validity of the so- +lution, resulting in a lower than expected improvement +in the correlated amplitude and a phasing efficiency of +only 20%–80%. Finally, under low-wind and low-PWV +conditions (October 21), the correlated amplitude of the +phased antennas reaches the expected square root of +the number of phased dishes (29 in this case), once the +known efficiency losses are considered (Appendix E), in- +dicating an optimally phased array (overall phasing ef- +ficiency, ηv of ≳90%; Figure 2). +In addition to the standard “slow” phasing corrections +computed by TelCal, the APS has an option to apply +in real time “fast” phasing corrections (with ∼1.6 s ca- +dence)4 computed from the WVR data available at each +antenna (Matthews et al. 2018). These real-time fast +corrections were not used during the 2018 October ob- +servations, but we have investigated the expected im- +pact of such corrections by applying WVR-derived cor- +rections off-line to the individual elements of the phased +array, via the wvrgcal task in CASA. Since the phased +sum is formed in real time, it is not possible to use the +fast corrections in post-processing to improve the SNR, +as they apply to the individual antennas used to form +the sum. Nonetheless we can gauge the expected impact +by computing the improvement in the phase coherence +of the individual baselines in the phased array. In the +case that rapid phase fluctuations (such as those ob- +served on October 16/17) result from atmospheric wa- +ter vapor varying on timescales more rapid than the +computed “slow” phasing solutions, we should expect +to see an improvement in the phase coherence on indi- +vidual baselines after application of the WVR correc- +tions. For example, the analysis of ALMA data with +vwind <∼10 km s−1 by Maud et al. (2017) and Matsushita +et al. (2017) suggest that such corrections are typically +helpful in reducing phase fluctuations and coherence loss +for baselines <500 m when PWV>1. +We find, how- +ever, that for the October 16/17 data the WVR-based +4 The underlying measurements are currently made every 1.152 s, +and it takes an additional ∼0.5 s to apply the correction. + +8 +Crew, Goddi, Matthews, et al. +23:52 +00:02 +00:12 +00:22 +00:32 +00:42 +00:52 +UT Time (HH:MM) +0.0000 +0.0002 +0.0004 +0.0006 +0.0008 +Visibility Amplitude +2018-10-17 +23:52 +00:02 +00:12 +00:22 +00:32 +00:42 +00:52 +UT Time (HH:MM) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2018-10-17 +09:40 +09:50 +10:00 +10:10 +10:20 +10:30 +10:40 +10:50 +UT Time (HH:MM) +0.0000 +0.0002 +0.0004 +0.0006 +0.0008 +Visibility Amplitude +2018-10-17 +09:40 +09:50 +10:00 +10:10 +10:20 +10:30 +10:40 +10:50 +UT Time (HH:MM) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2018-10-17 +23:44 +23:54 +00:04 +00:14 +00:24 +00:34 +00:44 +UT Time (HH:MM) +0.00000 +0.00025 +0.00050 +0.00075 +0.00100 +0.00125 +0.00150 +0.00175 +Visibility Amplitude +2018-10-19 +23:44 +23:54 +00:04 +00:14 +00:24 +00:34 +00:44 +UT Time (HH:MM) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2018-10-19 +09:19 09:29 09:39 09:49 09:59 10:09 10:19 10:29 10:39 10:49 +UT Time (HH:MM) +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +Visibility Amplitude +2018-10-21 +09:19 09:29 09:39 09:49 09:59 10:09 10:19 10:29 10:39 10:49 +UT Time (HH:MM) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2018-10-21 +Figure 3. Illustration of the performance of the APS during the 345 GHz (Band 7) VLBI experiment on 2018 October 16/17, 17, 18/19, +21 (from top to bottom). Left panels: correlated amplitude is plotted as a function of time on two sets of baselines: (1) the phasing +reference antenna with two unphased comparison antennas (red points); (2) the phased sum antenna with the same comparison antennas +(green points). Right panels: phase versus time is plotted on baselines between the ALMA reference antenna and the other phased ALMA +antennas (blue points). The uncorrelated phases during the first few integrations of each observing block (lower-right panel), are due to +the fact that phases are still being adjusted (i.e. the array is unphased; see §3). In both columns, data from a single correlator quadrant +(baseband 3, corresponding to SPW = 2 in Table 4) and a single polarization (XX) are shown. Data in the other SPWs and polarization +YY show similar behaviors. + +Phasing ALMA at 345 GHz +9 +corrections do not improve the overall phase coherence; +instead they appear to add additional noise. This sug- +gests that the rapid phase fluctuations, which lead to +a systematic degradation of phasing efficiency towards +the beginning of the VLBI campaign (as displayed in +Figures 2 and 3), are not induced by variations in tropo- +spheric water vapor alone, but most likely arise instead +from a combination of water vapor and wind-induced +atmospheric turbulence (e.g., Nikolic et al. 2013; Maud +et al. 2017). +Figure 4. +Phasing efficiency (top) and phasing quality +(bottom) as a function of wind speed for the data sets pre- +sented in the current paper (Tables 1 & 2). Data sets with +PWV≥2.0 mm are indicated in red; data with PWV<2.0 mm +are plotted in black. The horizontal dashed line in the upper +panel indicates the nominal APS efficiency goal of ≥90%. +All of the data sets plotted here were taken without the use +of WVR-based fast phasing corrections. +To obtain a preliminary assessment of how the com- +bination of wind speed and PWV affects phasing per- +formance at 345 GHz, we plot in Figure 4 the phas- +ing efficiency ηv and the phasing quality as a func- +tion of wind speed vwind for each of the data sets pre- +sented in the current paper (Tables 1 & 2). Data points +with PWV≥2.0 mm are shown in red and data with +PWV<2.0 mm are shown in black. +Figure 4 shows that irrespective of wind speed, when +PWV>2.0 mm, phasing efficiency in Band 7 generally +falls below ∼50%. Thus operation of the APS in Band 7 +in conditions with PWV>2.0 mm is not recommended +in general, although it may be possible to relax this re- +striction with future use of the fast phasing mode (see +above). +We also see in Figure 4 that when wind speeds exceed +∼10 m s−1, phasing efficiency is consistently quite low +( <∼20%), even in one case with PWV<2 mm. Further- +more, phasing solution quality is seen to decline system- +atically for such high wind speeds. This suggests that +for the high wind-speed regime, the use of fast phasing +corrections is unlikely to improve the overall phasing +performance. It is thus recommended that phased array +observations in Band 7 are strictly avoided in conditions +with vwind >10 m s−1. +For intermediate wind speeds (3 ≤ vwind < 10 m +s−1) the situation is more complex. We find that (in +absence of fast phasing corrections) one generally does +not meet the nominal APS efficiency goal of ηv ≥0.9. +However, for VLBI, the sensitivity and strategic impor- +tance of phased ALMA mean that even data with lower +phasing efficiency may be scientifically useful. For ex- +ample, when PWV is low (<2.0 mm), in many cases +ηv >∼0.5; assuming 37 phased 12 m antennas, this still +provides the sensitivity comparable to a 25 m diameter +parabolic dish. Furthermore, the generally good phasing +quality seen for data sets with 3 ≤ vwind < 10 m s−1 and +PWV≤2.0 mm suggests that the fraction of experiments +achieving ηv >0.5 under this combination of conditions +is expected to grow significantly with the use of the fast +phasing mode. We are currently in the process of acquir- +ing additional regression test data to explore how much +improvement the fast mode provides under a range of +observing conditions, including moderate wind speeds +(<10 m s−1) and moderate PWV values (∼2–3 mm). +4.2. Impact of Array Size and Maximum Baseline +Length +Figure 5 compares phase as a function of uv distance +for two tests carried out in 2015 (on March 30 and +August 2), and one carried out in 2021 (on Septem- +ber 3). +The tests were taken under similar weather +conditions (PWV∼0.5 mm) but with different baselines +ranges (<180 m, <1500 m, and <6900 m, respectively). +Table 6 provides a summary of these tests, labelled +B180, B1500, and B6900, respectively. + +10 +Crew, Goddi, Matthews, et al. +Table 6. Comparison of phase RMS as a function of baseline length. +Data set +Max. baseline +Date +Target +Flux +Phase RMS +Phase RMS +density +all baselines +baselines<200 m +(m) +(YYYY MMM DD) +(Jy) +(deg) +(deg) +B180 +180 +2015 Mar 30 +3C273 +4.0 +16 +16 +B1500 +1500 +2015 Aug 02 +J0522–3627 +4.5 +69 +35 +B6900 +6900 +2021 Sep 03 +J1924–2914 +3.0 +55 +39 +Considering the single correlation quadrant (SPW=0) +and polarization (XX) that is plotted for each data set, +we find that the RMS dispersion in the phases for all +baselines in the phased array is significantly higher in +the B1500 data (69 deg) and the B6900 data (55 deg) +compared with the B180 data (16 deg). This is true even +if we limit our comparison to baselines <200 m for all +three data sets; in this case the RMS phase dispersions +are 35 deg (B1500), 39 deg (B6900) and 16 deg (B180). +We thus see evidence that even under relatively good +weather conditions it is advantageous to limit the phased +array to short baselines (less than a few hundred meters) +when observing at wavelengths λ <∼1 mm. Because the +correlated amplitude scales as e−σ2 +p/2 where σp is the +RMS dispersion (in radians) of the phase, the sensitivity +gained by the inclusion of antennas on long baselines +will be significantly diminished by an overall decrease in +phasing efficiency of the entire phased array. This can be +understood as a result of the fact that the APS phase +solver uses a least-squares method to convert baseline +phases to station phases, and too many noisy baselines +(typically those longer than a few hundred meters) will +impact the overall quality of the phasing solutions. +4.3. Comparison between Band 6 and Band 7 +To begin to assess how the performance of the APS +in Band 7 compares with Band 6, we have performed +preliminary analysis of test observations where Band 6 +and Band 7 measurements were obtained within a sin- +gle session. In particular, during the 2017 February 1 +ALMA-only test and the 2018 October 18 VLBI test, +data in both Bands 6 and 7 were acquired using compa- +rable arrays, with baseline lengths ranging from ∼15 m +to ∼300 m, while the PWV content varied in the range +∼1.5–2.0 mm. +To explore the relative performance in the two bands, +in Figure 6 we compare the results from scans of a +few minutes duration on the source J0522–3627 in each +band, acquired on 2017 February 1. The RMS phase +fluctuations for all phased baselines were 43 deg in Band +6 and 36 deg in Band 7. These relatively high phase +dispersions reflect the sub-optimal weather conditions +for observing in these bands (PWV∼ 1.6 mm; wind +speed ∼9 m s−1), but these results nonetheless indi- +cate that the phasing system is capable of comparable +performance in Band 7 compared with Band 6. In this +example, the fluctuations are actually slightly lower in +Band 7 relative to Band 6. However, as the observa- +tions we compare here were not co-temporal, those dif- +ferences can be ascribed to changes in PWV, coupled +with changes in the source elevation. We reach similar +conclusions from a preliminary analysis on the 2018 test +dataset. +Our initial results suggest that for compact array sizes +(baselines ≲300 m) and moderately good or better ob- +serving conditions (PWV ≲2 mm), high-quality and +high-efficiency phased array performance will be pos- +sible in both Bands 6 and 7 and that Band 7 does not +show any appreciable loss in phasing performance com- +pared with Band 6. Under conditions where the phase +fluctuations are dominated by tropospheric water va- +por, some degradation in Band 7 performance compared +with Band 6 is naturally expected to occur as a conse- +quence of the linear wavelength dependence in the tem- +poral phase fluctuations (e.g., Rioja et al. 2012) and the +slightly lower aperture efficiency of the ALMA anten- +nas in Band 7 compared with Band 6 (Remijan et al. +2019). However, in practice, we did not see any clear +evidence of systematic degradation in phasing perfor- +mance in Band 7, in part because only limited compar- +ison data are available to date, and also because such +comparisons are complicated by the modest variations +in weather conditions that typically occur over tens of +minutes during available test periods at ALMA. +5. ADDITIONAL ASSESSMENTS OF PHASED +ARRAY DATA QUALITY +The primary goal of phasing ALMA in Band 7 is to +harness the enormous sensitivity and collecting area of +ALMA for use a sub-mm VLBI station (e.g., Fish et al. +2013). +This will be discussed further in Paper II. A +key part of the process of turning the phased ALMA +array into a functional sub-mm VLBI station will be +to first calibrate the interferometric visibilities (Goddi +et al. 2019). Because the data taken during the 2018 +October VLBI test were intended for testing and en- + +Phasing ALMA at 345 GHz +11 +25 +50 +75 +100 +125 +150 +175 +UV Distance (m) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2015-03-30 +0 +200 +400 +600 +800 +1000 +1200 +1400 +UV Distance (m) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2015-08-02 +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +UV Distance (m) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2021-09-03 +Figure 5. Phase (in degrees) as a function of projected base- +line length (in meters) for Band 7 phasing tests on 2015 March 30 +(top), 2015 August 2 (middle), and 2021 September 3 (bottom). +In each case the target flux density is a few Jy (see text for de- +tails). The tests were taken under similar weather conditions (see +Table 2). The mean RMS dispersions in the phases in the 2015 +August and 2021 September data (with longest baselines <1.5 km +and <6.9 km, respectively) are larger than in the 2015 March data +where the longest baselines <0.2 km, even on the shortest base- +lines. Data from a single correlator quadrant (SPW=0, averaged +over all channels) and a single polarization (XX) are shown in +all panels. +The phased array for the 2015 August observations +included a number of 7 m CM antennas which are typically not +included in the phased array (see Section 2.2.2). +gineering purposes only, a full suite of calibrators was +not observed. However, as described in Appendix A, we +have been able to perform a modified calibration scheme +to the data to allow these data to be meaningfully com- +bined with other VLBI stations (Paper II). This cali- +bration scheme additionally allows us to perform some +further quality assurance checks, as described below. +5.1. Accuracy of the Absolute Flux-density Scale +To assess the accuracy of the flux density calibration +in VLBI mode, Goddi et al. (2019) compared the mea- +sured flux densities of VLBI targets with values derived +from the independent flux monitoring done with the +ACA, taking advantage of the fact that some of the +Grid Sources are also observed in VLBI observations. +The analysis in Goddi et al. (2019) showed that the flux +density values estimated from ALMA during VLBI ob- +servations are generally within 5% in Band 3 and 10% in +Band 6 when compared with the Grid Sources monitor- +ing values (consistent with the expected absolute flux +calibration uncertainty at ALMA; see Remijan et al. +2019). +We have performed a similar analysis for the Grid +Sources observed in Band 7. Table 5 reports the mea- +sured flux values (per SPW) for all sources observed in +Band 7 during the 2018 October campaign along with +the archival flux values for Grid Sources. +The flux +values of the VLBI sources are estimated in the uv- +plane using the CASA task fluxscale, which adopts a +point-source model (this assumption is valid since Grid +Sources are unresolved on ALMA baselines). The ex- +pected flux density of Grid Sources at a given time +and frequency are retrieved from the ALMA archive via +the getALMAflux() function implemented in the CASA +analysis utils. Table 5 also reports the time differ- +ence between the VLBI observations and the archival +entry, ∆tS, which is <1 day for all sources (i.e. they +were observed with the ACA within less than a day of +the VLBI observations) except BL Lac. The nominal +calibration uncertainty at ALMA in Band 7 is ∼10% +(see ALMA Technical Handbook – Remijan et al. 2019), +and most of our new flux density estimates are consis- +tent with the archival values to within this range, with +the exception of CTA102 (with a 16% lower flux) and +J0522−3627 (with a 13% higher flux). +5.2. Interferometric Test Images +As an additional means of assessing the science readi- +ness of the APS in Band 7, we have produced images +of the VLBI targets from fully-calibrated ALMA inter- +ferometric visibilities (see Appendix A), following the +same procedures outlined in Goddi et al. (2021). We + +12 +Crew, Goddi, Matthews, et al. +25 +50 +75 +100 +125 +150 +175 +UV Distance (m) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2017-02-01 (Band 6) +25 +50 +75 +100 +125 +150 +175 +UV Distance (m) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2017-02-01 (Band 7) +02:33 +02:34 +02:35 +02:36 +02:37 +02:38 +UT Time (HH:MM) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2017-02-01 (Band 6) +03:20 +03:21 +03:22 +03:23 +03:24 +03:25 +03:26 +03:27 +UT Time (HH:MM) +150 +100 +50 +0 +50 +100 +150 +Visibility Phase [Phased Antennas] +2017-02-01 (Band 7) +Figure 6. +Phase (in degrees) as a function of projected baseline length in meters (upper panels) and observing time (lower panels) +during scans of a few minutes duration on the source J0522–3627 using the APS on 2017 February 1 in Band 6 (left) and Band 7 (right), +respectively, under conditions with PWV∼1.6 mm and wind speeds of ∼9 m s−1. The RMS phase fluctuations are ∼43 deg in Band 6 and +∼36 deg in Band 7, respectively, indicating comparable phasing performance in the two bands. +show representative images in Figure 7. The images dis- +played cover an area slightly smaller than the primary +beam of the ALMA antennas (18′′ at 350 GHz) and have +a synthesized beamsize of roughly 0.′′45. The correction +for the attenuation of the primary beam is not applied +to these maps. +We have conducted a series of quality-assurance self- +consistency tests on these images. We first assessed that +the images are consistent with unresolved point sources +(as expected for the selected VLBI targets), indicating +that they are not smeared by residual phase errors. We +then established that the size determined from a Gaus- +sian fit matches the size of the synthesized beam (within +< 1%) and the peak flux and integrated flux have the +same value (within < 1%), as expected for point-like +sources. +Finally we confirmed that the peak flux oc- +curs exactly at the phase center. +Besides these self- +consistency checks, we also estimated source flux den- +sities from the images (following the methods outlined +in Goddi et al. 2021) and assessed that they are consis- +tent with the values estimated in Table 5 (within ≲10% +for sources observed on the 18th/19th and within ≲5% +for sources observed on the 21st, respectively). +6. AMPLITUDE CALIBRATION OF +PHASED-ALMA AS A SINGLE VLBI STATION +Traditionally VLBI stations store time-dependent am- +plitude corrections, A(t), as a combination of Tsys (one +value per intermediate frequency and integration time) +and an instrumental gain given in degrees per flux unit +(K/Jy) or DPFU (assumed to be stable over time and +frequency): +A(t) = +� +Tsys/DPFU +In VLBI, one also often defines a system-equivalent flux +density (SEFD) as the total system noise represented in +units of equivalent incident flux density, which can be +written as +SEFD = ⟨Tsys⟩ /DPFU. +(1) +In the following, we estimate DPFU, Tsys, and SEFD +for phased-ALMA in Band 7 using the data collected on +2018 October 18/19 and 21. Representative values are +reported in Table 7. +6.1. DPFU +While in a single-dish telescope the DPFU is fixed, in +a phased array it scales with the number of phased an- +tennas. +Because the number of phased antennas can +change during the observations, the DPFU may also +change. In order to keep the DPFU of phased ALMA +constant over a given observation (for calibration pur- +poses), we set the DPFU of phased ALMA to the + +Phasing ALMA at 345 GHz +13 +22h32m36.30s +36.40s +36.50s +36.60s +RA (J2000) ++11°43'48.0" +49.0" +50.0" +51.0" +52.0" +53.0" +Dec (J2000) +CTA102 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +0.225 +Flux Density (Jy/beam) +22h53m57.60s +57.70s +57.80s +57.90s +RA (J2000) ++16°08'51.0" +52.0" +53.0" +54.0" +55.0" +56.0" +Dec (J2000) +3C454.3 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Flux Density (Jy/beam) +22h02m43.10s +43.20s +43.30s +43.40s +43.50s +RA (J2000) ++42°16'37.0" +38.0" +39.0" +40.0" +41.0" +42.0" +Dec (J2000) +BLLAC +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Flux Density (Jy/beam) +4h23m15.70s +15.80s +15.90s +16.00s +RA (J2000) +36.0" +35.0" +34.0" +33.0" +32.0" +-1°20'31.0" +Dec (J2000) +J0423-0120 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Flux Density (Jy/beam) +5h10m02.20s +02.30s +02.40s +02.50s +RA (J2000) ++18°00'39.0" +40.0" +41.0" +42.0" +43.0" +44.0" +Dec (J2000) +J0510+1800 +0.05 +0.10 +0.15 +0.20 +0.25 +Flux Density (Jy/beam) +5h22m57.80s +57.90s +58.00s +58.10s +58.20s +RA (J2000) +33.0" +32.0" +31.0" +30.0" +29.0" +-36°27'28.0" +Dec (J2000) +J0522-3627 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Flux Density (Jy/beam) +Figure 7. +Representative total intensity images of targets observed during the 2018 October VLBI campaign. The grey-scale +image shows emission at 347.6 GHz (SPW=2) while the blue contours show emission centered at 336.6 GHz (SPW=0); the red +(dashed) contours indicate negative values. The contour levels are ±3σ ×2n where σ =[0.9, 2.5, 0.7, 0.25, 0.27, 0.7] mJy beam−1 +for CTA 102, 3C454.3, BL Lac, J0423–0120, J0510+1800, J0522–3627, respectively, and n = 0 , 1 , 2 , 3 . . . up to the peak flux- +density. The intensity brightness is plotted using a logarithmic weighting function (starting from the 3σ-level). The major axis +of the synthesized beam for BL Lac is ∼0.′′62 and for the remaining sources is 0.′′45-0.′′5. + +14 +Crew, Goddi, Matthews, et al. +Table 7. ALMA Band 7 Antenna Parameters for Observa- +tions in 2018 October +Date +Nphased +Tsys +a +DPFUb +Tsys[sum]c +SEFDd +(2018 Oct.) +Ant. +(K) +(K/Jy) +(K) +(Jy) +Band 7 +21 +29 +155 +0.011 +2.6 +238 +18/19 +25 +200 +0.011 +6.4 +578 +Band 6 +19 +25 +80 +0.006 +0.9 +150 +Note—DPFU, Tsys, and SEFD estimates for phased ALMA in Band 7, as derived +from observations in 2018 October. +a Antenna-wise median of valid Tsys measurements. +b Antenna-wise average of DPFUs, estimated with Eq. 2. +c Median phased-array Tsys, estimated with Eq. 3. +d Phased-array SEFD, estimated with Eq. 1. +antenna-wise average of DPFUs (instead of the antenna- +wise sum). The DPFU of a single antenna i is calcu- +lated using the measured Tsys and amplitude gains ga,i +computed from self-calibration during QA2 (these are +stored in the